Embed Size (px)
Citation preview
New Frontiers in Practical Risk Management
Italian edition Issue n. 5 - Winter 2015
Iason ltd. and Energisk.org are the editors of Argo newsletter. Iason is the publisher. No one is al-lowed to reproduce or transmit any part of this document in any form or by any means, electronicor mechanical, including photocopying and recording, for any purpose without the express writtenpermission of Iason ltd. Neither editor is responsible for any consequence directly or indirectly stem-ming from the use of any kind of adoption of the methods, models, and ideas appearing in the con-tributions contained in Argo newsletter, nor they assume any responsibility related to the appropri-ateness and/or truth of numbers, figures, and statements expressed by authors of those contributions.
New Frontiers in Practical Risk ManagementYear 2 - Issue Number 5 - Winter 2015
Published in March 2015First published in October 2013
Last published issues are available online:www.iasonltd.comwww.energisk.org
Winter 2015
NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT
Editors:Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Andrea RONCORONI (ESSEC Business School, Paris)
Executive Editor:Luca OLIVO (Iason ltd)
Scientific Editorial Board:Fred Espen BENTH (University of Oslo)Alvaro CARTEA (University College London)Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Mark CUMMINS (Dublin City University Business School)Gianluca FUSAI (Cass Business School, London)Sebastian JAIMUNGAL (University of Toronto)Fabio MERCURIO (Bloomberg LP)Andrea RONCORONI (ESSEC Business School, Paris)Rafal WERON (Wroclaw University of Technology)
Iason ltdRegistered Address:6 O’Curry StreetLimerick 4Ireland
Italian Address:Piazza 4 Novembre, 620124 MilanoItaly
Contact Information:[email protected]
Energisk.orgContact Information:[email protected]
Iason ltd and Energisk.org are registered trademark.
Articles submission guidelinesArgo welcomes the submission of articles on topical subjects related to the risk management. Thetwo core sections are Banking and Finance and Energy and Commodity Finance. Within these twomacro areas, articles can be indicatively, but not exhaustively, related to models and methodologiesfor market, credit, liquidity risk management, valuation of derivatives, asset management, tradingstrategies, statistical analysis of market data and technology in the financial industry. All articlesshould contain references to previous literature. The primary criteria for publishing a paper are itsquality and importance to the field of finance, without undue regard to its technical difficulty. Argois a single blind refereed magazine: articles are sent with author details to the Scientific Committeefor peer review. The first editorial decision is rendered at the latest within 60 days after receipt of thesubmission. The author(s) may be requested to revise the article. The editors decide to reject or acceptthe submitted article. Submissions should be sent to the technical team ([email protected]). LaTex orWord are the preferred format, but PDFs are accepted if submitted with LaTeX code or a Word file ofthe text. There is no maximum limit, but recommended length is about 4,000 words. If needed, forediting considerations, the technical team may ask the author(s) to cut the article.
ENERGY AND COMMODITY FINANCE
RESEARCH CENTER
EXECUTIVE WEBINAR SERIES
Structural Positions in Oil Futures Contracts:
What are the Useful Indicators?
Hilary Till
Premia Capital Management LLC, Chicago
Date
March 27, 2015
Time
12:30 – 13:30
Webinar
http://energy-commodity-finance.essec.edu/
Venue
ESSEC Executive Education – Cnit La Défense - room 220
Registration
www.ecomfin-till-seminar.eventbrite.fr
Contact
Pauline Delécaut - [email protected]
RESEARCH CENTER LED BY ANDREA RONCORONI AND FRANCIS DECLERCK
3
NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT
Table of Contents
Editorial pag. 05
Antonio Castagna, Andrea Roncoroni and Luca Olivo introduce the new topics of this n. 5 Argo edition.
energy & commodity finance
Order-Flow and Liquidity Provision pag. 07
Álvaro Cartea and Sebastian Jaimungal
Monetary Measurement of Risk:A Critical Overview - Part IV: Coherent PortfolioRisk Measures pag. 13
Lionel Lecesne
banking & finance
Market Instrumentsfor Collateral Management pag. 27
Antonio Castagna
A Critical Review of Central Banks Satellite Modelsfor Probabilities of Default pag. 47
Marco Rauti
Front Cover: Giacomo Balla Futuro, 1923.
EDITORIAL
Dear Readers,the Winter 2015 issue of Argo magazine will surprise you not only
for the new interesting topics discussed inside, but also for someenhancements in the graphic. As you will see throughout, a newlook has been shaping for the newsletter: it means Argo is still evolv-ing and improving to always guarantee the best quality of reading.
Focusing on this issue contents, this time we start from the Energy& Commodities finance section. Álvaro Cartea and Sebastian Jaimungalfocus on trading dynamics, illustrating a way to optimally takepositions in the limit order book by placing limit orders at-the-touchwhen the midprice of the asset is affected by the trading activity of themarket. It follows the fourth and conclusive part of the crash course:Lionel Lecesne will discuss about the coherent portfolio risk measures.
The Banking & Finance section carries on with the up-to-datetopic of collateral management. Antonio Castagna presentsthe ideal continuation of his previous paper by introducingthe most common market instruments to manage and opti-mise collateral allocation. The other important contributionis provided by Marco Rauti and debates on a critical re-view of central banks satellite models for default probabilities.
We conclude as usual by encouraging the submission of con-tributions for the next issues of Argo in order to improve eachtime this newsletter. Detailed information about the process isindicated at the beginning. New enhancements will follow inthe next issues too. Thanks again for downloading Argo: it hasbeen a great 2014 and surely the 2015 will not disappoint you.
Enjoy your reading!
Antonio CastagnaAndrea Roncoroni
Luca Olivo
Winter 20155
NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT
AAAA
Energy & CommodityFinance
Trading Dynamics
Crash Course Part IV:Monetary Measurement of Risk
6
Order-Flow andLiquidity Provision
The authors show how to optimally takepositions in the limit order book byplacing limit orders at-the-touch whenthe midprice of the asset is affected bythe trading activity of the market. Themidprice dynamics have a short-term-alpha component which reflects howinstantaneous net order-flow, the dif-ference between the number of mar-ket buy and market sell orders, affectsthe asset’s drift. If net-order flow ispositive (negative), so short-term-alphais positive (negative), the strategy mayeven withdraw from the sell (buy) sideof the limit order book to take ad-vantage of inventory appreciation (de-preciation) and to protect the tradingstrategy from adverse selection costs.
Álvaro CARTEASebastian JAIMUNGAL 1
In modern electronic markets traders provide
liquidity by posting limit orders (LOs) whichshow an intention to buy or sell the asset andmust indicate the amount of shares and price
at which the investor is willing to trade. During thetrading day, all orders are accumulated in the limitorder book (LOB) until they find a counterpartyfor execution or are canceled by the investor whoposted them. The counterparty is a market order(MO) which is an order to buy or sell an amount
of shares, regardless of the price, which is imme-diately executed against limit orders resting in theLOB at the best execution prices.
Investors who specialise on providing liquid-ity to the market earn revenues from completinground-trip trades, buys followed by sells or sellsfollowed by buys, which earn them the spread be-tween their quotes. These investment strategies areexposed to prices moving in a direction that ad-versely affect the profitability of the strategy andexpose it to risks which include inventory risk andadverse selection risk. For example, if the investorhas accumulated a long position (more limit sellthan limit buy orders have been filled) and pricesdrop, then her net worth is reduced. Similarly, ifthe investor sells one share of the asset to a betterinformed trader then it is more likely to observeand increase in the midprice of the asset, hence theexpected revenue of the round-trip trade will reflecta financial loss.
Investors do not know the direction of priceinnovations, but order-flow, which is the numberof market buy and market sell orders, generallyconveys information of whether the midprice willtrend up or down. As new information arrives inthe market this is impounded in the midprice ofthe assets by traders who execute market orders(MOs) and/or traders who reposition their LOs. Ingeneral, when there is positive net order-flow (morebuy than sell MOs) midprices tend to drift up, andwhen there is negative net order-flow (more sellthan buy MOs) the midprice tends to drift down,see [1].
In this paper we show how an investor adjustsher trading strategy when she takes into account
1 SJ would like to thank NSERC and GRI for partially funding this work. ÁC acknowledges the research support of the Oxford-ManInstitute for Quantitative Finance and the hospitality of the Finance Group at Saïd Business School.
Winter 20157
TRADING DYNAMICS
how the market’s order-flow affects midprices. Inaddition, the investor also controls her exposure toinventory risk by restricting how much inventoryshe is willing to hold, long or short, and by adjust-ing the speed at which she expects her inventoriesto revert to the optimal inventory level to hold.
The rest of this article is structured as follows.In the firtst part we present the model and derivethe investor’s optimal strategy. In the mid-sectionwe use simulations to illustrate the performance ofthe strategy, and the last part concludes.
Investment strategy and order-flow
We model how an investor maximises terminalwealth by trading in and out of positions usingLOs. The investor provides liquidity to the LOBby posting buy and sell LOs only at-the-touch, i.e.only posts LOs at the best quotes and we assumethat the bid-ask spread ∆ is constant throughoutthe trading horizon. To formalise the problem, welist the relevant variables that we use throughoutthis section:
• M± =(
M±t)
0≤t≤T denote the counting pro-cesses corresponding to the arrival of otherparticipants’ buy (+) and sell (−) MOs whicharrive at Poisson times with intensities λ±.
• `±t ∈ {0, 1} denote whether she is posted onthe sell side (+) or buy side (−) of the LOB –`±t = 1 when posted on that side of the LOB,and `±t = 0 when not posted.
• N±,` =(
N±,`t
)0≤t≤T
denote the controlled
counting processes for the investor’s filledsell (+) and buy (−) LOs.
• X` =(
X`t
)0≤t≤T
denotes the investor’s con-
trolled cash process and satisfies the SDE
dX`t =
(St +
∆2
)dN+,`
t −(
St +∆2
)dN−,`
t ,(1)
which accounts for the cash increase when asell LO is lifted by a buy MO and the cashoutflow when a buy LO is hit by an incomingsell MO.
• Q` =(
Q`t
)0≤t≤T
denotes the investor’s con-
trolled inventory process and is given by
Q`t = N−,`
t − N+,`t . (2)
We further assume that if the investor is posted inthe LOB, when a matching MO arrives her LO isfilled with probability one. In this case, N±,`
t arecontrolled doubly stochastic Poisson processes withintensity `±t λ± – the analysis does not change sub-stantially if we assume that conditioned on an MOarriving, the LO is filled with probability p ∈ (0, 1).
The midprice of the asset S = (S)0≤t≤T , is as-sumed to satisfy the SDE
dSt = (υ + αt) dt + σ dWt , (3)
where W = (W)0≤t≤T is Brownian motion2 whichaccounts for the reshuffle of LOs and its potentialimpact on the midprice, and the midprice drift isgiven by a long-term component υ and by a short-term component αt which is a predictable zero-mean reverting process independent of W. Herethe long- and short-term components are key ingre-dients in the profitability of the investor’s strategy.For example, if the investor trades at time scaleswhere she does not ‘see’ the short-term componentαt, then her strategy will not only be sub-optimal,but will lose money to better informed traders –traders who are better informed will pick-off theLOs posted by the less informed liquidity provider.On the other hand, if the investor has the abilityto observe αt then she will ensure that on averageher strategy does not lose money to other tradersand will also use this knowledge to execute morespeculative trades when αt is different from zero aswe shall show below.
We assume that the investor has the ability toobserve order-flow and knows how this affects thedrift of the midprice, see also [3] and [2]. The short-term-alpha component is as a zero-mean-revertingprocess which jumps by a random amount at thearrival times of MOs. The short-term drift jumpsup when buy MOs arrive and jumps down whensell MOs arrive. As such, αt satisfies
dαt =− ζ αt dt + η dWαt
+ ε+1+M+
t−dM+
t − ε−1+M−
t−dM−t
(4)
where {ε±1 , ε±2 , . . . } are independent random vari-ables, with all + having identical distributionand all − having identical distribution, and withE[ε±1 ] < +∞. They represent the size of thesell/buy MO impact on the drift of the midprice.Moreover, Wα = (Wα
t )0≤t≤T denotes a Brownianmotion independent of all other processes, ζ, η arepositive constants, and recall that the MOs arrive ata Poisson rate of λ±.
2The Brownian motion can safely be replaced by any martingale independent of short-term alpha component – e.g., a pure jumpprocess so that prices remain on a discrete grid. Under this more general modelling choice, the resulting optimal strategy remainsunaltered.
8energisk.org
The investor’s performance criteria is
H`(t, x, S, α, q)
= Et,x,S,α,q
[X`
T+Q`T
(ST−
(∆2 +ϕ Q`
T
))−φ∫ T
t (Q`u)
2du ]
(5)
and the investor imposes the restriction that in-ventory remains in Q`
t ∈ [q, q] and the strategy’strading horizon is T.
On the right-hand side of the performance cri-teria are three terms. The first is the investor’sterminal cash from taking positions in the mar-ket. The second is the cost that the investor incurswhen employing an MO to liquidate any remain-ing inventory Q`
T at the end of the strategy. Theterminal inventory is liquidated at the midprice STand the costs associated to crossing the spread, liq-uidity taking fees, and market impact, all of whichare captured by the liquidation penalty parameterϕ ≥ 0. Finally, the third term is a running inventory
penalty φ∫ T
t
(Q`
u
)2du where φ ≥ 0 is the target
penalty parameter, see [4] for a discussion of this in-ventory penalty where the authors show this termstems from the investor’s uncertainty in the modelfor the midprice.
The investor’s value function is given by
H(t, x, S, α, q) = sup`∈A
H`(t, x, S, α, q) ,
where the set A of admissible strategies are F -predictable such that the investor is not posted onthe buy (sell) side if her inventory is equal to theupper (lower) inventory constraints q (q).
The Resulting DPE
Applying the dynamic programming principle wefind the investor’s value function H should satisfythe dynamic programming equation (DPE)
0=(
∂t+α ∂S+12 σ2∂SS−ζ α∂α+
12 η2∂αα
)H−φ q2
+ λ+ max`+∈{0,1}
{1q>q E [H(t,x+(S+∆
2 `+),S,α+ε+ ,q−`+)−H ]
+(
1− `+ 1q>q
)E [H(t,x,S,α+ε+ ,q)−H ]
}
+ λ−max`−∈{0,1}
{1q<q E [H(t,x−(S−∆
2 ),S,α−ε− ,q+`−)−H ]
+(1− `− 1q<q
)E [H(t,x,S,α−ε− ,q+`−)−H ]
}subject to the terminal condition
H(T, x, S, α, q) = x + q(
S−(
∆2 + ϕ q
)).
Here, the expectations are over the random jumpsizes ε±. The various terms in the DPE carry thefollowing interpretation. The first line in the DPErepresents the drift and diffusive components of themidprice and the short-term-alpha, as well as thealpha’s mean-reverting feature. The maximisationterms represent the investor’s control whether topost an LO at-the-touch. The second line repre-sents the change in value function, if the investoris posted, due to the arrival of an MO which liftsthe investor’s offer and simultaneously induces ajump in the short-term-alpha. The third line rep-resents the change in the value function when anMO arrives, but the investor is not posted – in thiscase only the short-term-alpha jumps. Finally, thefourth and fifth lines are for the other side of thebook.
The terminal condition and the form of the DPEsuggests the ansatz
H(t, x, S, α, q) = x + q S + h(t, α, q) ,
which splits out the accumulated cash of the trad-ing strategy up until t, the book value of the shareswhich are marked-to-market at the midprice, andthe added value from trading in and out of posi-tions optimally until the end of the trading horizon.
Note that h depends on time, inventory, and theshort-term-alpha. Substituting this ansatz into theabove DPE we find that h satisfies
0=(
∂t−ζ α∂α+12 η2∂αα
)h+α q−φ q2
+λ+ max`+∈{0,1}
{1q>q E [`+
∆2 +(h(t,α+ε+ ,q−`+)−h(t,α+ε+ ,q))
]}
+λ−max`−∈{0,1}
{1q<q E [`− ∆
2 +(h(t,α−ε− ,q+`−)−h(t,α−ε− ,q))]}
+λ+ E[h(t,α+ε+ ,q)−h(t,α,q)]+λ− E[h(t,α−ε− ,q)−h(t,α,q)] ,
subject to the terminal condition
h(T, α, q) = −q(
∆2 + ϕ q
).
The term α q which appears in the first line of theequation above is responsible for making the solu-tion to this problem be explicitly dependent on α.If it were absent, then the optimal postings and hfunction would be independent of α since the termi-nal conditions do not depend on α and there wouldbe no source terms in α. However, it is preciselythis dependence on α which renders the strategyinteresting and allows it to adapt to the adverseselection costs induced by the arrival of order flow.
The form of the optimising terms allow us tocharacterise the optimal postings in a compact form.When ` = 0 both terms that are being maximised
Winter 20159
TRADING DYNAMICS
(a) Buy Side Posts (b) Sell Side Posts (c) Asymptotic Strategy Posts
FIGURE 1: The optimal postings when accounting for short-term-alpha.
are zero, hence, the optimal postings of the investorcan be characterised succinctly as
`+,∗(t, q) = 1({
∆2 +
+ E [h(t,α+ε+ ,q−1)−h(t,α+ε+ ,q) ] > 0}∩{
q > q}
,
(6)
`−,∗(t, q) = 1({
∆2 +
+ E [h(t,α−ε− ,q+1)−h(t,α−ε− ,q) ] > 0}∩{
q < q}
.
(7)
Here, the notation 1(·) is the indicator function, andequals 1 if its argument holds, and zero otherwise.
Performance of the Strategy
For the purpose of focusing solely on the effect ofshort-term-alpha, we set the running penalty φ = 0and the remaining model parameters are
T = 60 sec, q = −q = 20,
λ± = 0.8333, ∆ = 0.01,
ϕ = 0.01, η = 0.001,
ζ = 0.5, E[ε] = 0.005 .
The choice of λ± ensures that the investor has amaximum inventory equal to 20% of the expectednumber of trades. With these parameters, Figure 1shows how the optimal strategy behaves as a func-tion of time and short-term-alpha. The investorsposts limit buys whenever her inventory is belowthe surface in the left panel, and she posts limitsells whenever her inventory is above the surfacein the right panel. There are a number of notablefeatures here:
1. Due to the symmetry of rates of arrival ofMOs, the surfaces are mirror reflections ofone another.
2. As maturity approaches her strategy becomesessentially independent of short-term-alpha,and instead induces her to sell when her in-ventory is positive and buy when inventoryis negative. Therefore, the optimal strategyattempts to close the trading period with zeroinventory.
3. The optimal strategies become time indepen-dent far from maturity.
4. Far from maturity, the investor tends to postsymmetrically when short-term-alpha and in-ventory is close to zero. As α increases, she iswilling to take on inventory, but keeps post-ing on both sides, until α becomes quite large,then she posts only buy LOs. The oppositeholds when α decreases.
Next, Figure 2 shows a sample path of the in-vestors posts together with the short-term-alpha.In the left panel, the green lines highlight when(and at what price) she is posted, and the red filledcircles indicate arrival of MOs that fill the investor’sposts, while red unfilled circles indicate MOs thatarrive when the investor was not posted. In thissample path, her postings change a total of fivetimes and her inventory begins at zero (Q = 0).In regime A she is posted only on the buy sidesince α is large enough to suggest that purchasingthe asset is worthwhile. As time evolves and sheenters regime B, α decays and she begins to postsymmetrically. A buy MO arrives and lifts her offer(so that Qt = −1), short-term-alpha immediatelyjumps upwards and she removes her sell LO inregime C. A sell MO eventually arrives and hitsher bid (so that Qt = 0) and immediately inducesa downward jump in α. Since α in regime D is rel-atively small, and her inventory is zero, she postssymmetrically once again. Eventually a buy MOonce again lifts her offer (so that Qt = −1) andinduces an upward jump in α. In regime E she now
10energisk.org
Time28 28.5 29 29.5 30
Price
29.95
29.96
29.97
29.98
29.99
D
EA C
B
Time28 28.5 29 29.5 30
Shor
t-te
rm,
0
0.005
0.01
0.015
D
EA C
B
FIGURE 2: Sample path of the optimal strategy. Green lines show when and at what price the investor is posted. Solid redcircles indicate MOs that arrive and hit/lift the posted bid/offers. Open red circles indicate MOs that arrive but do not hit/lift theinvestor’s posts because the investor has not posted an LO. Shaded region is the bid-ask spread.
only posts on the buy side. A sequence of buy MOsarrive in this interval inducing more upward jumpsin short-term-alpha, however, since she has no sellLO posted, her inventory remains one short andshe remains posted only on the buy side.
Conclusions
We show how an investor who incorporates theeffect of order-flow on midprices adjusts her in-vestment strategy to take advantage of short-termtrends in the midprice and protects the strategyfrom adverse selection risks. Net-order-flow af-fects the short-term deviations of the midprice’sexpected growth and we show how an investorwho provides liquidity to the market devises andoptimal strategy to benefit from this short-termdeviations which induce a short-term-alpha com-ponent in the drift of the asset.
We use simulations to show the performance
of the strategy and show that, as expected, whenshort-term-alpha is high (low), the investor with-draws from the sell (buy) side of the LOB andis only posted on the buy (sell) side to take ad-vantage of expected appreciation (depreciation) ofthe asset. Moreover, when short-term-alpha is inthe neighbourhood of zero, the strategy posts two-sided quotes.
ABOUT THE AUTHORS
Álvaro Cartea: University College London
Email address: [email protected]
Sebastian Jaimungal: University of Toronto
Email address: [email protected]
ABOUT THE ARTICLE
Submitted: December 2014.Accepted: January 2015.
References
[1] Cartea, Á. and S. Jaimungal. A Closed-Form Execu-tion Strategy to Target VWAP. SSRN eLibrary. 2014.http://ssrn.com/abstract=2542314
[2] Cartea, Á. and S. Jaimungal. Risk Metrics and Fine Tun-ing of High Frequency Trading Strategies. Working PaperSeries. Forthcoming in Mathematical Finance. SSRN eLi-brary. 2012. http://ssrn.com/abstract=2010417
[3] Cartea, Á., S. Jaimungal and J. Ricci. Buy Low, Sell High:A High Frequency Trading Perspective. SIAM Journal onFinancial Mathematics, v.5, n.1, pp. 415-444. 2014.
[4] Cartea, Á., R. Donnelly and S. Jaimungal. AlgorithmicTrading with Model Uncertainty. SSRN eLibrary. 2013.http://ssrn.com/abstract=2310645
Winter 201511
ADVERTISING FEATURE
http://energy-commodity-finance.essec.edu
12
ADVERTISING FEATURE
FloRiskTM Metrics
Continuous Monitoring of Corporate Risk Exposure
13
ADVERTISING FEATURE
FloRisk™ and FloCurve™ are cloud solutions for the price assessment and risk management of corporate positions. FloCurve™ Solution runs a cutting-hedge model for building forward price curves and simulate precise market scenarios. FloRisk™ Solution delivers a fine assessment of time evolution of position metrics and their risk exposure. FloRisk™ metrics provide operators with the great picture of aggregate exposure of standing and potential positions: a tool to mitigate risk effectively and negotiate price protection instruments offered by financial institutions.
FloRisk™ Solution
Your questions:
• How do my cash flows expect to grow?
• When do I need to update my cash reserves?
• Is my banker offering an appropriate hedge?
• What is the actual risk exposure of my position?
• How do I optimize the timing for hedge decision?
Our answers:
• Capturing the time evolution of expected cash flows
• Estimating the timing of extreme downside shocks
• Measuring the impact of risk mitigation policies
• Disentangling portfolio risk into market components
• Setting automatic signals for market warning
FloCurve™ Solution
Key features:
• Quickly generating on-‐demand forward curves
• Encompassing gas, power, oil, and FX markets
• Allowing for hourly to monthly time granularity
• Capturing periodical patterns at varying frequencies
• Accessing quotes in several formats for ease of integration
• No installation required, fully web accessible
Snapshot Contact
"A framework that gives the time evolution of risk is a creative idea that could be a useful tool for small-to-medium sized energy companies." Professor Vincent Kaminski, Rice University, US.
Energisk SAS, Technologies segment 104, rue du Château, 92100 Boulogne, France Email: [email protected] Web: www.energisk.org
Copyright © 2015 Energisk All rights reserved.
14
Monetary Measurement ofRisk: a Critical OverviewPart IV : Coherent Portfolio RiskMeasures
In this fourth and last course de-voted to monetary measurement ofrisk, Lionel Lecesne presents the the-ory of coherent portfolio risk mea-sures, which was introduced by Acerbiand Scandolo (2008). Coherent port-folio risk measures provide an origi-nal solution to the problem of encom-passing illiquidity in risk assessment..
Lionel LECESNE
In the previous lectures of this crash-course onmonetary measurement of risk, we introducedtwo fundamental classes of financial risk met-
rics, namely coherent and convex measures of risk.Recall that coherent measures of risk (Artzner etal., 1999; Delbaen, 2002) have been criticized fornot considering illiquidity in risk assessment. In-deed, coherent risk measures assume that risk in-creases linearly with size of the position, what isnot supported by empirical observations. As a so-lution, Föllmer and Schied (2002) and Frittelli andRosazza-Gianin (2002) suggested to relax the ax-ioms of coherence and introduced the class of con-vex measures of risk, which enables the existenceof liquidity risk.
We introduce in this article an alternative wayof encompassing illiquidity in risk measurementinitiated by Acerbi and Scandolo (2008). Their ar-gument is that liquidity risk lies in the relation
between value and quantities held rather than di-rectly between cash-flow of a position and risk. Inother terms, according to them, an increase of thesize of a position raises position’s value in a lessthan proportional way. They suggest a rigorousmodel that we recall to value a portfolio in an illiq-uid market, which leads to the notion of liquidity-adjusted value of a portfolio. Then, as illiquidityis already considered in the relation between valueand quantity, they rehabilitate coherent measuresfor assessing risk. The composition of a coherentmeasure of risk with the liquidity-adjusted valueyields the main concept of Acerbi and Scandolo(2008)’s paper, namely a coherent portfolio riskmeasure. Coherent portfolio risk measures havethe particularity of being defined on the space ofportfolios (i.e. the quantities held) and not on thespace of random profit and losses as it is generallythe case for monetary measures of risk.
Adjusting the Value of a Portfolio to Liq-uidity Risk
We introduce in this section the model initiatedby Acerbi and Scandolo (2008) to provide the fairvalue of a position given a certain state of marketliquidity and some investment constraints that maybe imposed to the portfolio. For pedagogic reasons,some intermediary concepts may be simplified withrespect to the original article. Also, we voluntarychange the names and notations of certain notionsfor more clarity in the presentation. We prevent thereader when such modifications are operated.
Winter 201515
CRASH COURSE
Marked-to-Market and Liquidation Portfolio Val-ues
Assume that the financial market is composed ofN risky assets and of one risk-free asset, namelycash. Then, a portfolio is described by a (bolded)vector q = (q0, q1, ..., qN) ∈ RN+1 where q0 denotesan amount of cash and where for any i ∈ [1, N], qidenotes the quantity of assets of type i held by aninvestor.
In Acerbi and Scandolo (2008), any risky assetis described by a list of prices which representsthe order book at a given point of time. The orderbook is modellized by a function called the marginalsupply-demand curve which to any order transmittedto the market associates the price of the last tradedunit. In this framework, the best ask (resp. bid)defines the price of a given asset when a purchase(resp. sell) order of infinitesimal size is transmittedto the order book by the investor. The differencebetween the best ask and the best bid is called thebid-ask spread. For the ease of presentation, weassume that for any risky asset the best bid equalsthe best ask and call this price the marked-to-marketprice. It can be regarded as the price of the asset ifthe market was perfectly liquid. In the sequel, wedenote by Pi the marked-to-market price of asseti ∈ [1, N]. As cash corresponds to an amount, theprice of this particular asset is set to P0 = 1.
The previous definitions enable to introduce theconcept of marked-to-market portfolio value, whichcan be seen as the value of a portfolio when liq-uidity risk is neglected. The marked-to-marketportfolio value is defined as the sum over all as-sets of the quantity of each asset held multipliedby its marked-to-market price. For this reason, themarked-to-market value is also referred to as theuppermost value in Acerbi and Scandolo (2008)3.We give a formal definition of the marked-to-marketvalue below.
Definition 1. For any portfolio q ∈ RN+1, the marked-to-market value is given by a map VM : RN+1 → R
defined by
VM(q) := q0 +N
∑i=1
qiPi. (1)
The map VM is linear and increasing on RN+1 andsatisfies VM(0) = 0.
It makes in general sense to speak about themarked-to-market value of a portflio when this lat-ter is not currently being traded by its portfoliomanager; indeed, in this case the portfolio is notexposed to liquidity risk. The marked-to-market
value of a portfolio can be opposed to its liquida-tion value, which describes the proceeds (resp. ex-penses) from the liquidation (resp. purchase) of aportfolio. The liquidation value corresponds to thelowest value of a portfolio as it assumes that theentire position is liquidated. The difference withthe marked-to-market value arises from liquidityfrictions.
In Acerbi and Scandolo (2008), the liquidationvalue is defined from the marginal supply-demandcurve. For the sake of simplicity we choose todirectly define the liquidation value from its prop-erties and then introduce an expression satisfyingthis definition that we keep until the end of thispaper.
Definition 2. The liquidation value is a nonlinear mapVL : RN+1 → R satisfying the following properties forall q, r ∈ RN+1:
• Concavity: For any θ ∈ [0, 1], VL(θq + (1−θ)r) ≥ θVL(q) + (1− θ)VL(r);
• (0): VL(0) = 0;
• Positive Subhomogeneity: For any λ ≥ 1,VL(λq) ≤ λVL(q).
Remark 1. In Acerbi and Scandolo (2008), the liquida-tion value is denoted by a map L.
Positive subhomogeneity is a consequence ofhaving concavity and (0) together. Concavity ofthe liquidation value means that the value of ablended portfolio is greater than the blend of theindividual values. The underlying idea is that toget a given amount of cash, the cost of liquidating anon-diversified portfolio is greater than the cost ofliquidating a diversified one, due to illiquidity. Therole played by liquidity may be recovered in the pos-itive subhomogeneity property which means thatif a portfolio is multiplied by two, its liquidationvalue is less than doubled.
In the following we will consider a liquidationvalue defined by the function
VL(q) := q0 +N
∑i=1
Piki(1− e−kiqi ), (2)
where ki > 0 is a parameter describing the impactof illiquidity on the liquidation value.
Liquidity-adjusted Portfolio Value
In the previous subsection we presented two ex-treme ways of valuing a portfolio: the marked-to-market value can be seen as the value when the po-sition stays unchanged while the liquidation valuecorresponds to the proceeds of the whole portfolio.
3Which is denoted by a map U.
16energisk.org
In this subsection, we are interested in derivingthe "fair" value of a portfolio given a certain stateof the liquidity of the assets and some contraintsimposed to the portfolio. Before introducing thecentral concept of liquidity-adjusted value, we recallsome intermediary notions introduced by Acerbiand Scandolo (2008).
Important is to understand that the fair valueof a portfolio strongly depends on what its holderintends to do with it. Indeed, assume that twofictitious investors A and B operating in the sameilliquid market own exactly the same portfolio buthave different investment strategies. Suppose thatat a future date T, investor A keeps his portfoliounchanged whereas investor B needs to liquidatehis entire portfolio with immediacy. Then, on thefirst hand, as A does not liquidate any quantity atall, his portfolio may reasonably be valued at themarket; On the other hand, B undergoes strongliquidation costs as he needs to sell his portfolio.Hence, A’s portfolio value is greater than B’s port-folio value and we may conclude that the valueof a portfolio sensibly differs with the investmentbehavior.
Important is to understand that the fair valueof a portfolio strongly depends on what its holderintends to do with it. Indeed, assume that twofictitious investors A and B operating in the sameilliquid market own exactly the same portfolio buthave different investment strategies. Suppose thatat a future date T, investor A keeps his portfoliounchanged whereas investor B needs to liquidatehis entire portfolio with immediacy. Then, on thefirst hand, as A does not liquidate any quantity atall, his portfolio may reasonably be valued at themarket; On the other hand, B undergoes strongliquidation costs as he needs to sell his portfolio.Hence, A’s portfolio value is greater than B’s port-folio value and we may conclude that the valueof a portfolio sensibly differs with the investmentbehavior.
To modellize the behavior of the portfolio man-ager or some investment constraint s that are im-posed to him, Acerbi and Scandolo (2008) introducethe general concept of liquidity policy. A liquiditypolicy, denoted L, is a subset of the space of port-folios RN+1. A portfolio q ∈ RN+1 is said to satisfythe liquidity policy L iff q ∈ L. Two propertiesenter the definition of a liquidity policy: if a port-folio q satisfies the liquidity policy L, then it willstill satisfy L if i) cash is added to it; ii) the entirerisky position is withdrawn from q. Additionaly,any portfolio obtained by blending two portfoliosthat satisfy L also satisfies L. This discussion leads
to the following definition of a liquidity policy:
Definition 3. A liquidity policy L is any closed convexsubset L ⊆ RN+1 satisfying:
1. q ∈ L ⇒ (q0 + c, q1, ..., qN) ∈ L, for any c > 0;
2. q = (q0, q1, ..., qN) ∈ L ⇒ (q0, 0) ∈ L.
Several liquidity policies are provided in Acerbiand Scandolo (2008) and Acerbi (2008). Amongstthese, one example is given by the cash liquiditypolicy which is the set of all portfolios having acash component greater than a fixed lower bound.Formally, the cash liquidity policy is defined as
L(c) :={
q ∈ RN+1 : q0 ≥ c}
, c ∈ R. (3)
An other interesting example is given by theclass of monetary risk liquidity policies, each riskliquidity policy being defined as the set of all portfo-lios whose marked-to-market value’s risk is belowa given risk threshold. Let ρ denote a monetary riskmeasure and R ∈ R be a fixed risk threshold, thenthe risk liquidity policy is defined by
Lρ := {q ∈ RN+1 : ρ[VM(q)] ≤ R}. (4)
Notice that Lρ satisfies the properties of a liq-uidity policy only if ρ is convex. For instance, theuse of Value-at-Risk is not recommended since itsconvexity depends on the distribution of prices. Asshown by Embrechts et al. (2005), Value-at-Risksatisfies subadditivity (and by the way convexity)only if the risk factors are elliptical.
We may now introduce the main concept ofthis section called liquidity-adjusted value 4. For anyportfolio, the liquidity-adjusted value provides itsmaximal value given a liquidity policy constraintand a certain state of market liquidity. If the liq-uidity policy is satisfied by the portfolio, then theliquidity-adjus ted value is equal to the marked-to-market value as no change in the position is neces-sary. Else, if the liquidity policy is not satisfied bythe portfolio, then the liquidity-adjusted value isthe result of an optimization problem which aimsat finding among all positions satisfying the liquid-ity policy the portfolio which maximizes the value.Acerbi and Scandolo (2008) introduce the follow-ing definition of the liquidity- adjusted value of aportfolio:
Definition 4. For any portfolio q ∈ RN+1, theliquidity-adjusted value is given by a mapp VL :RN+1 → R which is defined by
4Referred to as Mark-to-Market or simply value in Acerbi and Scandolo (2008)
Winter 201517
CRASH COURSE
VL(q) = sup{
VM(q− s) + VL(s) : s ∈ CL(q)}
,(5)
with CL(q) := {s∈RN+1 :(q0−s0+VL(s),q1−s1,...,qN−sN)∈L}.It is proved in Acerbi and Scandolo (2008) that the
liquidity-adjusted value map satisfies the properties be-low:
Proposition 1 (Acerbi and Scandolo, 2008). . Let Lbe any liquidity policy. Then, the map VL : RN+1 → R
satisfies the following properties for all q ∈ RN+1:
• Concavity: For any pair of portfolios q1, q2 ∈RN+1 and any θ ∈ [0, 1], VL(θq1 + (1 −θ)q2) ≥ θVL(q1) + (1− θ)VL(q2);
• Translation Supervariance: For any c ≥ 0,VL (q0 + c, q1, ..., qN) ≥ VL(q) + c.
We already commented the concavity propertyin the case of the liquidation value. Translation su-pervariance signifies that adding cash to a portfolioincreases value by more than the amount added.This reveals that in a situation in which liquidatingassets is difficult due to low liquidity, cash may bevalued over its face value as it is costly to convertrisky assets into the risk-free asset.
Obviously, the liquidity-adjusted value of thenull portfolio is equal to zero, what is formallygiven by
(0) : VL(0) = 0.
One may easily verify that property (0), associ-ated with concavity, leads to positive subhomogene-ity of VL that we define below. Positive subhomo-geneity means that upscaling a portfolio increasesvalue in a less than proportional way, which sup-ports the idea that illiquidity impacts value nega-tively. Formally, this is given by
• Positive Subhomogeneity: For any q ∈RN+1 and any λ ≥ 1, VL(λq) ≤ λVL(q).
In the following we derive an example of liquid-ity policy which aims at showing how the liquidity-adjusted value behaves with position’s size.
Example: Variance Liquidity Policy
Let us now provide an example of liquidity pol-icy and derive the associated liquidity-adjustedvalue map. We consider portfolios composed of therisk free asset and of two risky assets. We assumethat the risky assets are independent and that theirvolatilities are given by σi > 0, i ∈ {1, 2}. We studya liquidity policy that we call variance liquidity policyand which describes the set of all portfolios whoseMarked-to-Market value has a variance lower thana fixed threshold.
Definition 5. Let τ > 0 be a fixed threshold. Then, thevariance liquidity policy, denoted Lτ , is defined by
Lτ :={
q ∈ R3 : Var[VM(q)] ≤ τ}
, (6)
where Var[.] denotes variance.
Given a liquidity policy Lτ and a certain state ofliquidity of the market, the liquidity-adjusted valueVL
τis given by the following proposition:
Proposition 2. Let Lτ be a variance liquidity pol-icy with fixed threshold τ. Then, for any portfolioq ∈ R3, the liquidity-adjusted value is given by a mapVL
τ: R3 → R defined by
VLτ(q) =
sups∈R2
q0+∑2i=1(qi−si)Pi+∑2
i=1Piki(1−e−ki si )
sub.to ∑2i=1 σ2
i (qi−si)2≤τ.
(7)Proof. We apply definition 5 to the liquidity policy Lτ .For q ∈ R3, the objective function is given by VM(q−s) + VL(s)= q0 + ∑2
i=1(qi − si)Pi + ∑2i=1 k−1
i Pi(1−exp{−kisi}). As to the constraint, it is easily seenthat Lτ = {q ∈ R3 : ∑2
i=1 σ2i q2
i ≤ τ}. Then,CLτ (q) = {s ∈ R3 : ∑2
i=1 σ2i (qi − si)
2 ≤ τ}.
As this problem admits no analytical solution,we compute it numerically and provide a repre-sentation of VL
τin Figure 1. More specifically,
the left and the right plots depict respectively theliquidity-adjusted value VL
τand the ratio of the
liquidity-adjusted value on the marked-to-marketvalue as functions of q.
Two important comments can be made from theobservation of these plots:
• First, both graphs confirm that the liquidity-adjusted value is a concave map since valueincreases in a less than proportional waywith size of the position due to illiquid-ity. If there was no liquidity risk, thevalue of (q0, q1, q2) = (0, 500, 500) would be10, 000aswehaveassumedP1 = P2 = $10. How-ever, in the present example, the left graphshows that VL
τ(0, 500, 500) is approximately
equal to 7, 000.
• As long as the liquidity policy is satisfiedby the portfolio, i.e. as long as the varianceof the marked-to-market value is lower thanτ = 5000, no change of the position is re-quired and the liquidity-adjusted value is thusequal to the marked-to-market value. In theleft graph, this situation is depicted in blue,where the map is linear; In the right plot, the
18energisk.org
0100
200300
400500
0100
200300
400500
0
1000
2000
3000
4000
5000
6000
7000
q2q1
Liq
uid
ity−
ad
juste
d V
alu
e (
$)
0
100
200
300
400
500
0100
200300
400500
0.6
0.7
0.8
0.9
1
q2
q1
Liq
uid
ity−
ad
juste
d /
Ma
rke
d−
to−
Ma
rke
t R
atio
FIGURE 1: Liquidity-adjusted value (left) and liquidity-adjusted value on marked-to-market value (right) as functions of q withτ = 5000, q0 = 0, P1 = P2 = 10, σ1 = σ2 = 0.2 and k1 = k2 = 0.01.
liquidity policy is satisfied when the ratio ofthe liquidity-adjusted value on the marked-to-market value is equal to 1, which is displayedin dark red. Once the portfolio attains a crit-ical size, its variance becomes greater thanthe threshold and liquidation is required toconvert risky assets into cash, what decreasesvolatility. In both graphs of Figure 1, thisis described by the nonlinear portions of themaps.
Coherent Portfolio Risk Measures
The purpose of this section is twofold: first, werecall the definition of coherent portfolio risk mea-sures (CPRM’s in the sequel) introduced by Acerbiand Scandolo (2008) as well as their properties; sec-ond, we provide some comparisons of CPRM’s withmonetary risk measures.
Definition and Properties
One major feature of CPRM’s is that unlike mon-etary measures of risk, they are not defined onthe space of random variables but on the space ofportfolios. Indeed, any CPRM is defined as the com-position of a coherent risk measure with a liquidity-adjusted value function. The reasoning leadingto this definition, already mentioned in the intro-duction , is the following: as illiquidity is alreadyconsidered in the valuation process of portfolios,there is no need for the risk measure to be convex.In fact, coherent risk measures are adequate forassessing the risk of the liquidity-adjusted value ofa portfolio. Let us now recall the definition of aCPRM below:
Definition 6. Let ρch : X → R be a given coherentmeasure of risk and VL : RN+1 → R any liquidity-
adjusted value function. Then, the map ρL : RN+1 →R defined by
ρL(q) := ρch(VL(q)), q ∈ RN+1,
is called coherent portfolio risk measure.
Acerbi and Scandolo (2008) have shown thatCPRM’s are convex and translationally subvariantmaps on the space of portfolios. In Lecesne (2014),we provided an explanation of the convexity axiomin the framework of monetary measures of risk. ForCPRM’s, convexity means that the risk of the blendof two portfolios is lower than the blend of port-folio’s risk outputs. More interesting is the prop-erty of translational subvariance which reveals howilliquidity is captured by CPRM’s. Recall that a fun-damental requirement of monetary risk measuresis translational anti-variance (refer to Lecesne andRoncoroni, 2013 for more details on monetary mea-sures of risk) which signifies that adding a certainamount of cash to a position reduces risk by exactlythis amount. In the framework of CPRM’s, thisproperty is replaced by translational subvariancewhich means that when cash is added to a portfo-lio, risk decreases by more than the amount added.From an economic point of view, this feature can beexplained by the fact that due to illiquidity, it canbe costly to convert a risky position into cash, whatleads to a valuation of cash greater than its facevalue. We recall the previous properties of CPRM’sbelow:
Theorem 1 (Acerbi and Scandolo, 2008). . Let ρL beany CPRM. Then, ρL admits the following propertiesfor all q, r ∈ RN+1:
• Convexity: For any θ ∈ [0, 1], ρL(θq + (1 −θ)r) ≤ θρL(q) + (1− θ)ρL(r);
Winter 201519
CRASH COURSE
0200
400600
0200
400600
0
500
1000
1500
2000
2500
3000
3500
q2
Expected Shortfall of the Liquidity−adjusted Value
q1
Ris
k (
$)
0
200
400
600
0
200
400
6000
50
100
150
200
250
300
q2
Expected Shortfall of the Marked−to−Market Value
q1
Ris
k (
$)
FIGURE 2: ESα[XLτ(q)] (left) and ESα[XM(q)] (right) as functions of (q1, q2) with α = 0.95, q0 = 0 and τ = 5000. For any i ∈ {1, 2},
Pi ∼ N (µi = 10, σ2i = 0.04) with Cov(P1, P2) = 0 and ki = 0.01.
• Translational Subvariance: For any c ≥ 0,ρL(q + c) ≤ ρL(q)− c.
An Illustrative Example based on Expected Short-fall
In this subsection, we study to what extent CPRM’simprove risk assessment. To this end, we com-pare for a certain set of portfolios the outputs of agiven coherent risk measure applied to a marked-to-market value with those of the same risk mea-sure applied to a liquidity-adjusted value. As amatter of fact, we no longer deal with portfoliovalues but with portfolio profit and losses (PL’s inthe sequel) which are more suitable for risk assess-ment purposes. We define the marked-to-marketPL as the difference between the marked-to-market value, which is random, and its expectation. Asto the liquidity-adjusted PL, it is defined as theliquidity-adjusted value minus the expectation ofthe marked-to-market value. Formally, for any port-folio q ∈ RN+1, the marked-to-market PL is givenby XM(q) := VM(q)− E[VM(q)] and the liquidity-adjusted PL by XL(q) := VL(q)− E[VM(q)]. Ex-pected Shortfall, whose definition is recalled below,is the coherent risk measure we utilize for this study.A more detailed presentation of Expected Shortfallis provided in Lecesne and Roncoroni (2014, PartII).
Definition 7. Let X be a given space of random vari-ables and α ∈ (0, 1) a fixed confidence level. Then,Expected Shortfall at level α is a map ESα : X → R
defined by
ESα[X] := −E[X|X ≤ −VaRα(X)], X ∈ X , (8)
where VaRα : X → R is the Value-at-Risk at level αdefined by
VaRα(X) := −inf {x ∈ R : P[X ≤ x] ≥ 1− α} .
As already discussed in Lecesne (2014), coher-ent measures of risk and hence Expected Short-fall satisfy the positive homogeneity requirement,what does not allow to take liquidity risk into ac-count. Hence, the first way we consider to assessportfolio risk totally ignores liquidity as it consiststo compose Expected Shortfall with the randommarked-to-market PL. This can be described by thesubsequent scheme
q −→No liquidity adjustment
XM(q) −→No Liquidity Risk
ESα[XM(q)]
In comparison, we apply Expected Shortfall to agiven liquidity-adjusted PL, what defines a CPRMand is given by the scheme
q −→Liquidity adjustment
XL(q) −→No liquidity risk
ESα[XL(q)]
We consider the same data as in the exampleprovided in the previous section: the liquidity pol-icy is still the variance liquidity policy defined by (6)with τ = 5000. Recall that the portfolio is composedof three assets where the quantity of the risk-freeasset is set to q0 = 0. We assume that the prices Pi,i ∈ {1, 2} of the risky assets are independent andnormally distributed with mean µi = 10 and volatil-ity σi = 0.2. The parameters describing liquidityare equal to ki = 0.01. As to the confidence level ofExpected Shortfall, it is set to α = 0.95. We simulate10, 000 pairs of prices (P1, P2) which enable to plotthe graphs in Figure 2.
Figure 2 depicts Expected Shortfall of respec-tively the liquidity-adjusted PL (left) and themarked-to-market PL (right) as functions of q. Tworemarks can be made from this figure:
• For both maps, risk increases in a nonlinear(convex) way with size of the portfolio, and
20energisk.org
0200
400600
0200
400600
0
500
1000
1500
2000
2500
3000
3500
q2
Expected Shortfall of the Liquidity−adjusted Value
q1
Ris
k (
$)
0200
400600
0200
400600
0
500
1000
1500
2000
2500
3000
3500
4000
4500
q2
Expected Shortfall of the Liquidity−adjusted Value with k1 < k2
q1
Ris
k (
$)
FIGURE 3: ESα[XLτ(q)] as a function of (q1, q2) with α = 0.95, q0 = 0 and τ = 5000. For any i ∈ {1, 2}, Pi ∼ N (µi = 10, σ2
i = 0.04)with Cov(P1, P2) = 0 and k1 = 0.01. Left plot: k2 = 0.01. Right plot: k2 = 0.05.
this feature is far more pronounced in theliquidity-adjusted case.
• When the liquidity policy is satisfied bythe portfolio, the risk of XL
τequals that of
XM. However, as soon as the variance ofthe marked-to-market value exceeds τ, liqui-dation is required, what increases risk dueto liquidation costs. In the present exam-ple, striking is to find that for a portfolio(q0, q1, q2) = (0, 500, 500), the economic capi-tal computed by Expected Shortfall is approxi-mately equal to $300 in the marked-to-marketcase while it attains roughly $3, 500 in theliquidity-adjusted case.
Figure 3 reveals how the map ESα[XLτ(.)] reacts
to a rise of the liquidity parameter k2 from 0.01 to0.05. The left plot depicts the same map as in Fig-ure 2(k2 = 0.01) while the right plot describes thesituation in which the liquidity of the second assethas deteriorated. For all portfolios requiring someliquidation in order to satisfy the variance liquiditypolicy (i.e. portfolios outside the red surface), riskis greater when k2 = 0.05. Indeed, as an exampleobserve that the risk of q = (0, 500, 500) attains re-spectively $3, 500 when k2 = 0.01 and $4, 000 whenk2 = 0.05. A second observation is that the highestthe proportion of the less liquid asset in the totalportfolio, the greatest the risk: for instance, for theportfolio (q0, q1, q2) = (0, 0, 500), the risk approx-imately equals $1, 000 when k2 = 0.01 (left plot)while it attains $1, 500 when k2 = 0.05 (right plot).
Remark 2. The present example merely gives the intu-ition of the relation between liquidity and portfolio risk.However, an empirical estimation of (k1, k2) would beuseful to assess the real impact of liquidation.
Liquidity Risk: Convex versus CoherentPortfolio Risk Measures
In the previous subsection we showed throughan example that CPRM’s improve risk assessmentby encompassing illiquidity. The purpose of thepresent subsection is to compare CPRM’s with another way of including liquidity in risk measure-ment. Recall that in part III of this series of papersdedicated to monetary risk measurement (Lecesne,2014), we discussed the fact that positive homogene-ity of coherent risk measures is a shortcoming asit postulates that risk increases linearly with size,what is not verified empirically. We presented theclass of convex risk measures (Föllmer and Schied,2002; Frittelli and Rosazza-Gianin, 2002) as an alter-native to coherent risk measures, as they do not re-quire positive homogeneity and thus enable risk toincrease non-linearly with size. Hence we assumethat if risk is assessed using a convex measure ofrisk, it suffices to apply it to the marked-to-marketPL since illiquidity is already considered in the riskmeasure. The convex risk measure we consider isthe entropic risk measure that we define below:
Definition 8. Let β > 0 be a fixed parameter. Then, theentropic risk measure is a map ENTβ : X → R definedby
ENTβ[X] :=1β
log E[e−βX ], X ∈ X . (9)
We hence compare the shape of the CPRM ofthe previous subsection that can be recalled by thefollowing scheme
q −→Liquidity adjustment
XLτ(q) −→
No Liquidity RiskESα[XM(q)]
Winter 201521
CRASH COURSE
0200
400600
0200
400600
0
500
1000
1500
2000
2500
3000
3500
q2
Expected Shortfall of the Liquidity−adjusted Value
q1
Ris
k (
$)
0
200
400
600
0
200
400
6000
100
200
300
400
500
600
700
q2
Entropic Risk of the Marked−to−Market Value
q1
Ris
k (
$)
FIGURE 4: ESα[XLτ(q)] (left) and ENTβ[XM(q)] (right) as functions of (q1, q2) with α = 0.95, β = 1, q0 = 0 and τ = 5000. For any
i ∈ {1, 2}, Pi ∼ N (µi = 10, σ2i = 0.04) with Cov(P1, P2) = 0 and ki = 0.01
with that of the entropic risk measure applied tothe marked-to-market PL summarized by
q −→No liquidity adjustment
XM(q) −→Liquidity Risk
ENTβ[XM(q)]
Figure 4 compares the CPRM ESα[XLτ(q)] ob-
tained previously with the entropic risk of themarked-to-market PL given by ENTβ[XM(q)].
As the risk measures are different, simply com-paring the outputs of the two approaches wouldlack significance here. More interesting is to findout whether the two maps share common featuresor not. It is apparent from the figure that the mapof the entropic risk measure applied to the marked-to-market value is convex as well, what confirmsthat risk increases in a more than proportionnalway with size.
Taking a look at the right plot of Figure 2 alsoreveals that the entropic risk measure applied tothe marked-to-market PL is more U-shaped thanis Expected-Shortfall. However, Expected Shortfallapplied to the liquidity-adjusted PL enables a muchmore accurate assessment of portfolio risk as thislatter soars when the liquidity policy is not satisfied.
ABOUT THE AUTHOR
Lionel Lecesne is Ph.D. candidate at University of Cergy,France. He was formerly student at Ecole NormaleSupérieure in Cachan, France. Research interests includerisk measurement, energy finance and financialeconometrics.
Email address: [email protected]
ABOUT THE ARTICLE
Submitted: January 2015.Accepted: February 2015.
22energisk.org
References
[1] Acerbi, C. and G. Scandolo. Liquidity Risk Theory andCoherent Measures of Risk.Quantitative Finance, vol. 8,No. 7, pp. 681-692. 2008.
[2] Acerbi, C. Portfolio Theory in Illiquid Markets. WorkingPaper, 2008.
[3] Artzner, P., F. Delbaen, J.M. Eber and D. Heath. CoherentMeasures of Risk. Mathematical Finance, 9, pp. 203-228.1999.
[4] Delbaen, F. Coherent Risk Measures on General Proba-bility Spaces. Advances in Finance and Stochastics, pp.1-37. 2002.
[5] Embrechts, P., R. Frey and A. J. McNeil. Quantita-tive Risk Management: Concepts, Techniques, and Tools.Princeton Series in Finance, Princeton University Press.2005.
[6] Föllmer, H. and A. Schied. Convex Measures of Risk andTrading Constraints.. Finance and Stochastics, 6, pp.429-447. 2002.
[7] Frittelli, M. and E. Rosazza Gianin. Putting Order inRisk Measures.. Journal of Banking and Finance, 26,pp.1473-1486. 2002.
[8] Lecesne, L. Monetary Measurement of Risk - A CriticalOverview - Part III. Argo, 4 (Fall 2014), pp 63-67. 2014
[9] Lecesne, L. and A. Roncoroni. Monetary Measurement ofRisk - Part I. Argo, 1 (Winter 2014), pp. 67-72. 2013.
[10] Lecesne, L. and A. Roncoroni. Monetary Measurement ofRisk - Part II. Argo, 3 (Summer 2014), pp. 14-19. 2014.
Winter 201523
ADVERTISING FEATURE
32nd International Conference of the French Finance Association
June 1 – 3, 2015, Cergy (France)
Keynote Speakers:
Darrell Duffie Dean Witter Distinguished Professor of Finance, Stanford University
Laura Starks Associate Dean for Research and Professor of Finance, University of Texas
The 32nd
Spring International Conference of the French Finance Association will be held at the ESSEC Business School (France) on June 1-3, 2015.
On June 2nd, two Special Sessions on Energy Finance and Commodity Finance are chaired by:
Jérôme Detemple Everett W. Lord Distinguished Faculty Scholar, Boston University
The Energy Finance session will host a speech by Derek Bunn (Professor of Decision Sciences, Management Science and Operations at the London Business School, and editor of the Journal of Energy Markets & Journal of Forecasting). The Commodity Finance session will host a speech by Jaime Casassus (Associate Professor of Financial Economics at the Universidad Catolica Chile and managing editor of Quantitative Finance).
FURTHER INFORMATION: The main conference is held on June 1st
and 2nd
. The last day of the conference, June 3rd
, is dedicated to the Ph.D. Workshop. A prize will be awarded for the best Ph.D. Workshop paper.
REGISTRATION: http://affi-2015.essec.edu Andrea Roncoroni Professor of Finance, ESSEC Energy Finance and Commodity Finance Workshops Chair
24
ADVERTISING FEATURE
25
NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT
AAAA
Banking & Finance
Collateral Management
Satellite Models
26
Market Instruments forCollateral Management
The article is the ideal continuation ofthe Castagna’s previous one on Collat-eral Management (published in Argo4 - Fall 2014). Now the authorwill analyse the most common mar-ket instruments to manage and opti-mise collateral allocation: the repo,the sell/buy back and security lending.
Antonio CASTAGNA
In a previous article (Castagna [3]) we intro-duced a conceptual framework to analyse thesupply and the demand of collateral internally
originated by the banking activity; we focussedon the tools to manage both and the targets thatshould be aimed at by setting up effective processesto minimise costs and to monitor risk exposures.
In the present work, we ideally continue ourstudy of the modern collateral management byanalysing three types of contracts that are the basicmarket instruments to manage and optimise col-lateral allocation: the repurchase agreement, thesell/buy back and the security lending. For eachof them, most updated legal frameworks and bestpractices will be thoroughly investigated. More-over, attention will be given to their evaluation interms of fair value and additional adjustments dueto credit, liquidity and funding factors.
The three instruments are the building blocksfor more complex contracts and sophisticated trad-ing strategies aiming at an effective, enterprise-wide, collateral management.
The Repurchase Agreement
(Repo)
Definition and Introduction
Definition 1. A repurchase agreement, commonlycalled repo, is a transaction in which one party agreesto sell security to another against the transfer of cash;at the same moment, the party also agrees to repurchasethe same (or equivalent) security at a specific price ata future date, when the contract will expiry. The partyreceiving security is also referred to as the buyer; theparty lending security is referred to as seller. When thetransaction is seen as a secured cash loan, the repo buyeris also referred to as the lender, which receives securityas collateral against a possible cash borrower’s default.
Counterparties typically involved in a repotransaction are financial institutions (banks): al-though repos can be traded also between othertypes of economic agents (e.g.: corporate compa-nies), in this paper we will assume that they alwaysare banks. Figure 1 shows how the obligations of arepo agreement are fulfilled by the two parties.
In a repo transaction, there is an interest ratecomponent which is implicit in the terminal priceat which the repurchase occurs. At inception, thesecurity are bought at the current market price plusthe accrued interest to date; at the expiry, the secu-rity are re-sold at a predefined price equal to theoriginal sale price (market price + accrued interest),plus an interest rate (the repo rate) agreed upon bythe parties.
Repo can be security-driven or cash-driventransactions, depending on which is the primaryinterest of the two parties. If one of them is dealingto borrow the security (a security-driven transac-tion), the repo rate is typically set a level lower
Winter 201527
COLLATERAL MANAGEMENT
FIGURE 1: Obligations of a repo agreement.
than current money market rates to compensate thelender of security. It is true that the repurchaseprice is higher than the initial selling price, becauseit will include the repo interests; nonetheless, thecash received by the seller can be reinvested for theduration of the contract in the money market, andthe interest earned in this investment will be higherthan the implied rate paid in the repo transaction,the difference being the compensation for the seller.
When one the two parties deals to receive cashagainst delivering security (a cash-driven transac-tion), the repurchase price is set so that the lenderof cash (borrower of security) earns the equivalentof money market secured funding rate. The col-lateral represented by security protects the cashlender against the cash borrowers’ default, so therepo rate should trade at an implicit rate lower thanan unsecured funding rate, such as a money marketdeposit rate. We will return on this point later on.
In cash-driven repo transactions, a margin isoften provided to the lender of cash by pricingthe collateral security at the market price minusa “haircut”, or an “initial margin”(we will discussthe difference between the two below). Therefore,the initial price is lower than the market price ofthe security. On the other hand, in security-drivendeals, the lender of security will typically receive amargin by pricing security higher than their marketvalue. The mechanics is similar in the two casesanyway.5
Repo agreements are quoted in terms of therepo rate. The repo rate is generally quoted on
the basis of the day count/annual basis conventionprevailing in the wholesale money market in thecurrency of the Purchase Price (i.e.:, the deposit andforward rate market).6 In the Global Master Re-purchase Agreement (GMRA, the standard generallegal framework used by most banks), the repo rateis called the Pricing Rate.
A buyer in a repo may also allow the seller toreplace some or all collateral securities during theduration of the repo. In practice, the seller, at anytime between the purchase date and repurchasedate, has the right to call for the buyer to returnequivalent collateral securities in exchange for sub-stitute collateral. For this right, the seller typicallyagrees to pay a higher repo rate.
As seen above, since the repo can be security-driven, the repo rate can be negative: this mayhappen when a particular collateral asset is in largeborrowing demand and/or in scarce supply, so thatit is said to go on special.
During the life of the contract, the collateral islegally property of the buyer, which means that allpayments produced by the security are paid by theissuer directly to the buyer. However, under an eco-nomic point of view, the buyer can be consideredjust as the possessor, not the owner, of the security,because the repurchase price is fixed and there isthe seller’s obligation to repurchase the security.Hence both the risk and return on the collateralasset are kept by the seller.7
The guiding principle is that the seller shouldreceive all income payments due on collateral asset
5Security driven repo transactions are very similar to security lending transactions when collateral is provided by the securityborrower: the legal and economic differences will be clear after analysing the security lending later on.
6The two conventions prevailing in practice are actual/365 days and actual/360 days.7Castagna and Fede [4] consider just the economic perspective when they analyse how the market instruments we study in this
paper affect the amount of securities available for collateral (or, more generally, liquidity) purposes.
28iasonltd.com
as if the asset had not been repoed out. This meansthat the terms of a repo contract provides for theobligation for the buyer to make an equivalent andimmediate8 income payment to the seller (called“manufactured payment”) every time the collateralasset produce a payment. Following the same cri-terion, if the issuer for some reason fails to makean income payment due on the collateral security,the buyer does not have to make the equivalentpayment to the seller.
The transfer of legal title to collateral in theinitial sale means that the buyer has the right tore-use, or re-hypothecate, the collateral during theduration of the repo. The right exits independentlyfrom the default of the seller. So the buyer can repoagain the bought asset or sell it to a third party atany time: it only has to be able to return to theseller an equivalent amount of securities on the re-purchase date. The right to re-use the bought assetis the reason why repo transactions are market in-struments to manage and transform collateral andemploy them in collateral strategies, as discussedin Castagna [3].
Quotation and Trading
When repo contracts are cash-driven, then they aretypically quoted in terms of (repo) rate, given astandard haircut to be applied on the collateral as-set: the collateral asset can be any from a pool ofeligible assets that the market considers of suitablecredit quality and liquidity. The haircut for thesecollateral assets is the same, since they have sim-ilar credit and liquidity characteristics. For thesereasons, cash-driven repos are defined General Col-lateral (GC) and quotes are easily available for themin the market for a range of standard maturitiesrunning from O/N to 1 year.
If the repo is security-driven, then the quotationhas to be defined between the counterparties, bothin terms of repo rate and haircut to be applied onthe asset: the stronger interest in borrowing thesecurity, rather than the cash, may shift the reporate to levels lower than the GC repo rate. More-over, the haircut depends on the credit and liquiditycharacteristics of the collateral asset and it can bealso quite different from the GC haircut (see alsobelow for how the fair haircut is calculated). Asmentioned before, the collateral asset in this repoare said to go on special, and repos are Specific Col-lateral (SC). Quotations are in terms of special reporates and haircuts to be agreed by the parties.
Either the repo is GC or SC, once the repo rateand the haircut are agreed upon, the trading of the
transaction involves the calculation of the PurchasePrice and Repurchase Price: both can be easily cal-culated as follows. Consider a repo starting at timet and expiring in T and let MV(t) be the marketvalue of the collateral asset: in case it is a bond, it isthe dirty price (the quoted market clean price plusaccrued interests from the last coupon paymentdate to date t); if it is (less commonly) an equity, itis just the market price.
The Purchase Price PP is the MV(t) deductedthe haircut H(t), or the initial margin IM(t) (seebelow for the more details):
PP = MV(t)(1−H(t)) (1)
or
PP =MV(t)IM(t)
(2)
Let the repo rate of the transaction be rE, thenthe Repurchase Price RP is computed as:
RP = PP× [1 + rE × τE(t, T)] (3)
where τE(t, T) is the year fraction of the period[t, T] calculated according to the day count andbasis convention of the contract.
EXAMPLE 1 We present a practical exampleof how the Purchase and Reurchase Prices arecomputed for a repo traded on the 6th Au-gust, 2014. The collateral asset is a Ger-man government bond, the Bund with annualfixed coupon with details shown in Table 1.
Bond DetailsFace Value 100Coupon 3
Frequency yearlyLast Payment 04/07/2014
DC Conv. Act/ActExpiry 04/07/2020
Ref. Date 06/08/2014Clean Price 115.05
Accrual Days 33Accrued Interests 0.2712
Dirty Price 115.32
TABLE 1: Details for the bond Bund 3% Jul 20, denominated inEuro.
In Table 2 we show the Purchase Price in thetwo cases when the haircut or the initial marginis used, given the market price and the accrualconventions to compute the market value of theasset MV (the bond’s dirty price). Assume therepo has an expiry in three months, or 92 daysfrom the trading date, and that the agree repo rate
8On the same day as the corresponding income payment by the issuer.
Winter 201529
COLLATERAL MANAGEMENT
Haircut Initial MarginDirty Price 115.3212 Price 115.3212
Haircut (3%) - 3.4596 Initial margin (3%) - 3.3589Purchase Price 111.8616 Purchase Price 111.9624
TABLE 2: Purcahse price when collateral is subject to haircut or initial margin.
Haircut Initial MarginRepo Cash 1,118,615.96 1,119,623.62
Bond Notional 1,000,000.00 1,000,000.00Repo Rate 1.75% 1.75%Maturity 06/11/2014 06/11/2014
Days 92 92Repo Interests 5,002.70 5,007.21Cash Returned 1,123,618.66 1,124,630.83
Repurchase Price 112.3619 112.4630
TABLE 3: Details of the repo contract when collateral is subject to haircut or initial margin.
rE = 1.75%; the notional amount is 1, 000, 000.00Euros. Given the money market convention foraccrued interests of Act/360 (i.e.: τE(0, 92dd) =92/360, the initial cash paid by the buyer (lender)and the cash returned by the seller (borrower) aregiven in Table 3. The Repurchase Price is clearlythe cash returned for 1 eur of notional amount.
Fair Haircuts and Initial Margins
As mentioned above, there are two alternative waysto adjust the value of the collateral asset sold in therepo to protect the buyer against the loss it may suf-fer when the seller defaults. the initial margin. Thehaircut and initial margin are typically applied incash-driven transaction, where the collateral assetis the protection of the lender against the borrowerdefault.
The initial margin is defined as:
IM(t) =MV(t)
PP(4)
where IM(t) is the initial margin at time t, MV(t)is the market value of the collateral and PP is thepurchase price. The initial margin is expressed as apercentage on the market value: 100% means thatno margin has been applied.
The second adjustment is the haircut H(t), de-fined as:
H(t) =MV(t)− PP
PP(5)
where the notation is the same as above. The hair-cut is the percentage difference between the marketvalue of the collateral asset and the purchase price.
For practical purposes, in the following, we maylimit our analysis to the case when a haircut is ap-plied: if an initial margin is applied instead, it is
straightforward convert it to a haircut through therelation:
IM(t) =1
1−H(t)(6)
The problem the seller (or, the cash lender) facesis how to set a fair level of the haircut. Assume theseller is a bank and that it enters a repo transactionat time t, expiring in T, with a counterparty, indi-cated with d, which may go bust before the end ofthe contract.
We will present an approach to determine thefair haircut that is derived from a model outlinedin chapter 8 of Castagna and Fede [4], althoughwe here substantially modify it. In any case, weretain the main feature of the model allowing for awrong-way risk between the default of the counter-party and the default of the issuer of the collateralsecurity, in this case of the bond.
The wrong-way risk is very common in repotransactions because often the collateral bonds areissued by the government of the country of the cashborrower. Typically a strong correlation exists be-tween the default probabilities of the governmentbonds and the banks, for the simple fact that bankshave on their balance sheet huge quantities of suchbonds, if only to build up liquidity buffers. Whenthere are crisis on the sovereign debt, the PD of thebanks of the countries under pressure increases asa result of the higher probability of a default of thegovernment whose bonds are held by the bank.
To model the default of the counterparty (bor-rower) d, we introduce an indicator functionDd(0, T) equal to 1 if the default of the counter-party occurred between times t and T, and it isequal to 0 otherwise. We take the approximation
Dd(0, T) = DId(t, T) + ξdDC
d (t, T)
30iasonltd.com
where DId(t, T) is the indicator of the event that
the idiosyncratic default process for d has jumpedby T9, whereas DC
D(t, T) is the indicator for theevent that the common default process has jumpedby T; finally ξi is the indicator of the event that dgoes bust at the first common credit event.10 In theframework we introduced above Pr(ξd = 1) = pd,where pd is the factor linking the intensity of thecommon default event to the total intensity of thecounterparty d, and in t (given that the counter-party did not default before):
PDId(t, T) = Pr[DI
d(t, T) = 1]
PDCd (t, T) = Pr[DC
d (r, T) = 1]
Assume the collateral asset is a bond expiringin TB: the dirty price is B(t, TB). The cash trans-ferred by the lender (buyer) to the borrower (seller)at inception is the same as the value of the bonddeducted the fair haircut. Let VColl(t) = MV(t) bethe value of the bond used as collateral in t:
VColl(t) = NCollB(t, TB).
where NColl is the notional of the collateral bond.In a repo transaction, the lender seeks to make theexpected loss, given the borrower’s default, equalto zero over a given period from t to T. Assumingthe default is observed at the expiry T of the con-tract (although it may occur at any time betweent and T), the expected loss (EL) is equal to the ex-pected exposure at default (EAD, assumed to befully lost) minus the value of the collateral, in theeven of a default of the counterparty d. The EADis simply the amount lent (assuming no scheduledrepayments between 0 and T) plus the interests, sothat:
EL(t, T)
= E[
max[EAD−VColl(T), 0]∣∣Dd(t, T) = 1
](7)
The idea we follow is that the lender, in settingthe fair haircut, seeks to set at 0 the EL consider-ing the minimum value attainable by the collateralbond at a given confidence level, say 99%. LetvColl(T) be this minimum value, corresponding tothe maximum expected loss EL at the chosen confi-dence level; we rewrite (7) as:
EL(t, T) = E[
EAD− vColl(T)∣∣Dd(t, T) = 1
](8)
The expected exposure is E[EAD|Dd(t, T) =1] = VColl(t)(1−H(t))[1 + rE × τ(t, T)], while theexpected (minimum) value of the collateral is:
E[vColl(T)|Dd(t, T) = 1] =
E[vColl(T))(1− DColl(t, T))
+ (vColl(T)− Lgd))DColl(t, T)∣∣Dd(t, T) = 1]
where DColl(t, T) is the indicator function equalto 1 if the issuer of the collateral bond has gonebankrupt by time T, and Lgd is the loss generatedby the default event and it is a percentage l% ofthe notional NColl. The issuer’s default is modelledsimilarly to the counterparty’s default, so we have:
DColl(t, T) = DIColl(t, T) + ξCollDC
Coll(t, T)
and Pr(ξColl = 1) = pColl, PDIColl(t, T) =
Pr[DIColl(t, T) = 1], and PDC
Coll(t, T) =Pr[DC
Coll(r, T) = 1]. It is immediate to see thatthe parameters pd and pColl play the role of the“correlation”11 between the default of the borrowerand of the bond’s issuer, since the higher they are(everything else being equal), the more likely the de-fault is triggered by the common event DC
Coll(t, T),thus having two simultaneous defaults.
Define:
P1 = (1− PDC(t, T))PDIColl(t, T)
and
P2 = PDC(t, T)((1− pColl)PDIColl(t, T) + pColl)
Equation (8) can be more explicitly written as::
EL(t, T) = EAD− vColl(T)+[[P1PDI
d(t, T) + P2((1− pd)PDId(t, T) + pd)]Lgd
]PDd(t; T)
(9)
On the right hand side, we have the exposure atdefault minus the minimum value of the collateralat the expiry. The amount in the square bracketsis the expected value of the loss on the bond inthe event of the counterparty’s default, given thatalso the collateral bond issuer’s default event is trig-gered: we will denote this amount in what followsas E[Lgd DColl(t, T)|Dd(t, T) = 1].
The higher the volatility of the price of the col-lateral bond, the lower the quantity vColl(T) will
9We consider the default a jump process, accordingly with a reduced-form default modelling we are implicitly adopting.10This approximation has been used by Duffie and Pan [6]. We ignore the double-counting of defaults that occurs from both
common and idiosyncratic credit events. The approximation also under-counts defaults associated with multiple common creditevents before time T. These two effects are partially offsetting each other.
11We are aware that “correlation” is in this case, and later on, used in a loose meaning. Nonetheless, the effect produced by theparameters pd and pColl is similar to that of a rigorously defined correlation between the defaults of the borrower and the bond’sissuer, hence the use of the term.
Winter 201531
COLLATERAL MANAGEMENT
be. Actually, it is quite sensible to assume that abond with a longer maturity (and hence duration)would be subject to a higher haircut than shortermaturity bond, due to the greater risk that, in caseof default of the borrower, the collateral has a lowerminimum market price, at the chosen confidencelevel.
The fair haircut to apply to the value of the port-folio of bonds at the inception of the repo is thelevel of H(t) that makes EL = 0. In formula:
(1−H(t))VColl(t)[1 + rE × τ(t, T)] =
= vColl(T)− E[Lgd DColl(t, T)|Dd(t, T) = 1](10)
or
H(t)= VColl(t)[1+rE×τ(t,T)]−vColl(T)+E[Lgd DColl(t,T)|Dd(t,T)=1]VColl(t)[1+rE×τ(t,T)]
(11)
It is interesting to measure how much thewrong-way risk impact on the haircut. To this handwe have to apply the formalae above assuming thatno correlation exists between the counterparty andthe issuer of the bond. In our set-up this means thatthe parameter p is equal to zero for each collateralasset’s issuer and the borrower (pColl = pd = 0),or that the probability of a common credit eventPDC = 0. Nonetheless, the total PD of both theissuer and the debtor are the same as the ones pro-duced by the combined effect of the idiosyncraticand common factors.
Equation () modifies as follows:
EL(t, T) =EAD− vColl(T) + PDColl(t, T)Lgd
In practice, the haircut depends only on the defaultprobability of the issuer of the collateral asset whenthere is no correlation between its default and theborrower’s default. When the wrong way exists,that is: when the correlation between the two de-faults is greater than zero, then the haircut dependsalso on the borrower’s default and the level can bestrongly affected. We will show this in an example.
EXAMPLE 2 We present a practical application ofthe framework to set fair haircuts outlined in themain text. We assume that the maturity of the repocontract is in one year and that the borrower and theissuer of the collateral bond have a default probabil-ity, respectively PDd and PDColl, shown in Table 4.The same Table also shows the value of the probabil-ity of occurrence of the common default event PDC.
PDC PDd PDColl
1.00% 1.50% 1.40%
TABLE 4: Default probabilities of the borrower and of the is-suer of the collateral bond.
Given this initial setting of the default values, wewould like to investigate which is the fair haircutto apply on the bond, considering different corre-lations between the defaults of the borrower andof the bond’s issuer. To this end, we present inTable 5 three sets of parameters PDI
d, pd, PDIColl
and pColl that reproduce a medium, high andzero correlation. All the three sets return the de-fault probabilities of Table 4, so the only differ-ence is due to the probability of a joint default.
PDId pd PDI
Coll pColl
1.00% 0.5 1.00% 0.40.50% 1.0 0.50% 0.91.50% 0.0 1.40% 0.0
TABLE 5: Parameters of the model to reproduce the threecases of medium, high and zero correlation between the defaultevents of the borrower and of the issuer of the collateral bond.
In Table 6 we show the input data referring to thecollateral bond. We consider two cases for thecollateral bond maturity: a 2 year and a 10 yearexpiry. The different maturity will affect the min-imum value vColl(1) according to the volatility ofthe market prices, which we use in the calcula-tions to compute the expected loss in formula ().
Bond Expiry2Y 10Y
Ncoll 1 1B(t, T) 98.00 98.00
vColl(T) 93.53 87.13Price Volatility 2.0% 5.0%
Lgd 60 60
TABLE 6: Input data to set the fair haircut for the collateralbond.
Finally, in Table 7, we show the far haircuts tothe price of the bond in the case its maturity is 2or 10 years; the haircuts are calculated consider-ing that the repo rate rE = 0. The three possibleconfigurations of default correlation are reported.The results make clear that the default correlation(wrong way risk) have a big impact on the value ofthe haircut; also considering the possible adverseprice movements affects considerably the haircut,making it greater for the longer maturity bond.
32iasonltd.com
Bond ExpiryDefault Correlation 2Y 10Y
Medium 11.27% 14.00%High 27.54% 28.46%Zero 5.37% 9.57%
TABLE 7: Fair haircuts when the collateral bonds expire in 2and 10 years, in the three default correlation cases.
The analysis above hinges on the simplifyingassumption that the repo borrower’s and issuer’sdefault occur only at the expiry of the contract: thisassumption can be reasonable for short term con-tracts, but for longer maturities it is too restrictive.To complete the analysis of the haircut modelling,we need to consider the more general case whenthe default of the borrower and of the issuer of thecollateral bond may occur not only at the expiry ofthe repo contract, but at any time before the expiry.In this case it is more convenient, also for evalu-ation purposes, to deal with the present value ofthe losses given default. To keep the analysis stillrelatively simple, we assume that the number oftimes when the default events may happen is notinfinite, but finite; to this end we divide the period[t, T] in N smaller periods, at the end of each either(or both) defaults may occur. Let Tn, for n = 1, ..., Nbe the times (with TN = T): we generalise equationas follows:
EL(t, T) = EAD
−N
∑n=1
[vColl(Tn) + [P1PDI
d(Tn−1, Tn)
+ P2((1− pd)PDId(Tn−1, Tn)
+ pd)]Lgd]
/PDd(Tn−1, Tn)
(12)
where
P1(Tn−1, Tn) = (1− PDC(Tn−1, Tn))PDIColl(Tn−1, Tn)
and
P2(Tn−1, Tn) =
PDC(Tn−1, Tn)((1− pColl)PDIColl(Tn−1, Tn) + pColl)
The haircut is defined similarly to what shownin equation (10).
The resulting haircut is an average of the ex-pected losses suffered in the event of default occur-ring on one of the possible dates Tn.
The EL defined as in (12) is strongly resemblingthe CVA at the inception t of the repo transaction,
evaluated by the buyer (lender):
CVAl(t, T) =
EQ[ ∫ T
tD(t, u)max
[(1−H(t))V(t)
× [1 + rE × τE(t, T)]−VColl(u), 0]dPDd(t, u)
](13)
where D(t, u) = exp(−∫ u
t rvdv) is the risk-free dis-count factor at time t for cash-flows occurring attime Tn. All the probabilities of default are com-puted within the intervals included in the period[t, T].
Assuming we use the same risk-neutral mea-sure to compute the EL and to compute the CVA,12
the main difference between the two measures isthat EL is the value of the expected losses giventhe default of the borrower, whereas the CVA is thepresent value of the expected (unconditional) lossesdue to the counterparty credit risk. Besides it isalso worth noting that EL is computed at a mini-mum level (at the chosen confidence level) of thecollateral bond’s value over the life of the contract,whereas the CVAl considers only the expected val-ues. As such EL ≥ CVAl and EL = 0 (after settingthe fair haircut as shown above) implies a fortiorithat the CVAl = 0 as well.
Margin Maintenance
Both counterparties involved in a repo transaction(i.e.: the buyer and seller), are exposed to the riskthat the market value of the collateral asset mayfall below or rise above the repurchase price, thusoriginating a counterparty credit exposure for, re-spectively, the buyer or the seller. This counter-party credit exposures is eliminated by a transferof margin to the exposed party by the other party:collateral can be posted either by a cash paymentor by a transfer of additional collateral assets.
In practice a margin call is made when one partyhas a net exposure to the other. The net exposure isgiven by the sum of all transaction exposures refer-ring to each repo contract existing between the twoparties (see section 4(c) of GMRA 2000 [11] and 2011[10]), plus any income due from the other party butunpaid (i.e.: manufactured payments and interestpayments) plus net margin still held by the otherparty. The transaction exposure can be defined intwo ways, depending on whether an initial marginor a haircut has been defined in the contract (seesection 2(ww) of GMRA 2000 [11] and section 2(xx)2011 [10]) and it is basically a sort of mark to the
12One may actually use a real world measure to calculate EL and hence set the fair haircuts. In this case additional differences mayarise between the two measures.
Winter 201533
COLLATERAL MANAGEMENT
market of the value of the contract on a given datebefore the expiry. As a best practice, net exposureshould be calculated at least every business day.
Transaction exposure when the initial margin isused
Let us start with the case the repo contract providesfor an initial margin: the transaction exposure (ofthe repo buyer) TE(s) on a date s, with s < T isdefined as:
TE(s) =(
RP(s)IM)−MV(s) (14)
where IM has been defined before, and:
• RP(s) is the repurchase price at time s, and ithas computed as:
RP(s) = PP(
1 + rE × τE(t, s))
with rE the contract repo rate, and τE(t, s) isthe year fraction of the period [t, s] calculatedaccording to the day count and basis conven-tion of the contract;
• MV(s) is the market value of the collateral,which in case of a bond is:
MV(s) = NColl ×(
P(s) + c× τB(t−1, s)100
)with P(s) the bond market (clean) price, c thecoupon rate and τB(t−1, s) the year fraction(computed according to the proper conven-tions) of the period [t−1, s], where t−1 is thelast coupon payment date.
It is worth stressing that the repurchase priceRP(s) is not the contract price, but the theoreticalprice that the seller should pay to buy back thebond, should the contract be terminated at times: the accrued interests run from the start of thecontract t to s.13
Transaction exposure when the haircut is used
If in the repo contract a haircut is applied on thepurchase price of the collateral asset, then the trans-action exposure (of the repo buyer) TE(s) on a dates, with s < T is:
TE(s) = RP(s)−(
MV(s)[1−H(t)])
(15)
where H is the haircut and has been defined before,and the repurchase price at time s, RP(s), and themarket value of the collateral, MV(s), are definedas in the case before when the initial margin is used.
Remark 3. It should be noted that the variations of thecollateral depend only on the changes of its market value.In reality the collateral should cover the expected lossesgiven the default of the borrower. When the borrower’sand issuer’s defaults are uncorrelated, provided that themarket price of the collateral includes also the varia-tions of the issuer’s PD, then the margin maintenanceworks quite well. When the wrong-way risk is relevant(high correlations between the two defaults), then themechanism, which does not consider any variation of theborrower’s PD, is not able to properly adjust the level ofcollateral.
The collateral Cl(s) that the lender (buyer)should receive at time s is equal to TE(s). Consid-ering the collateral already received by the buyer,the variation between two margin calls in times qand s, (q < s) is:
∆Cl(s) = TE(s)− Cl(q) (16)
Negative variations and values means the buyerposts instead of receiving collateral.
If the repo contract is assisted by a maintainancemargin agreement, the CVAl the lender should com-pute is:
CVAl(t, T) =
EQ[ ∫ T
tD(t, u)max
[(1−H(t))V(t)
× [1 + rE × τE(t, T)]
−VColl(u)− Cl(u), 0]dPDd(t, u)
] (17)
EXAMPLE 3 The calculation of the transaction ex-posure for the repo contract in Example 1 is shownin this example, starting with the recapitulationthe details of the contract in Table 8. At the in-ception of the contract t = 0, or Day 0, as in-dicated in Table 9, the exposure is obviously nil,as resulting by applying formulae (14) and (15)by inputting the starting dirty price of the collat-eral bond. At day 1 we suppose that the dirtyprice collapses at 114.00, which should imply anincrease of the transaction exposure to the buyer:actually this is what happens as shown in the Ta-ble. If the haircut and the initial margin havethe same value (3% in this example), the varia-tion of the transaction exposure is smaller wherethe haircut is used in stead of the initial margin.
13In reality, in counting the number of days, one should stop at the day before the margin call is made. So the number of days is upto, but excluding, the margin delivery date.
34iasonltd.com
Haircut Initial MarginAmount 1,000,000.00 1,000,000.00
Dirty Price Price Collateral 115.32 115.32N. Days 92 92
Repo rate 1.75% 1.75%Initial Margin - 3%
Haircut 3% -Market Value of Collateral 1,153,212.33 1,153,212.33
Cash Received 1,118,615.96 1,119,623.62Cash Returned 1,123,618.66 1,124,630.83
TABLE 8: Summary of the repo contract details presented in Example 1.
Haircut Initial MarginExposure Day 0
Dirty Price Collateral 115.32 115.32Market Value of Collateral 1,153,212.33 1,153,212.33
Repo Price Time 0 1,118,615.96 1,119,623.62Transaction Exposure - -
Exposure Day 1Dirty Price Collateral 114.00 114.00
Market Value of Collateral 1,140,000.00 1,140,000.00Repo Price Time 1d 1,118,670.34 1,119,678.05
Transaction Exposure 12,870.34 13,268.39
TABLE 9: Variation of the transaction exposure from the trading date (Day 0) to the following date (Day 1), in case where thehaircut or the initial margin is used.
Types of Collateral That Can Be Posted after aMargin Call
The margin call can fulfilled by posting collateral tothe counterparty either in cash or in other securities.The following rules are followed:
• where the margin is posted in cash, the partyreceiving collateral should remunerate it at asuitable reference rate. As an example, if thetransaction exposure is checked daily and themargin is called with this frequency, then anO/N rate (such as the EONIA for collateraldenominated in Euro) should be applied;
• where the margin is posted in securities, thevalue considered is their dirty price (clean “market” price plus accrued interests); if inter-ests are paid by the securities during the lifeof the contract, they should be immediatelytransferred to the party posting the collateral.Moreover, if an initial margin is set on thecollateral security delivered at the inceptionof the repo, then a margin is applied also onthe securities posted as collateral on margincalls: the haircut can be different from the oneset at inception even if the security posted onmargin call is the same as the repo collateral,
since market conditions might have changedafter the purchase date.
Valuation
We show here how to evaluate a repo contract, fol-lowing the incremental valuation approach outlinedin Castagna [2] assuming that the agent is a market-maker (or a hedger): basically the approach im-plies that when a contract is evaluated consideringits inclusion in the existing market-maker’s bal-ance sheet, if the default of the counterparty is nottriggering also its own default, then the market-maker should evaluate the contract by consider-ing just the counterparty credit risk (CVA) and thefunding costs paid to hedge (replicate) the contract(FVA and LVA)14. The debit valuation adjustmentnever plays a role, whereas the feasibility to attain afunding benefit due to possible positive cash-flowsshould be verified on a case-by-case base: excludingthe funding benefit is the simplest and safest policy.
Valuation for the Lender (Buyer)
Let us start in a simplified setting and assume thatthe market-maker is a repo buyer (lender) which isnot subject to credit risk (i.e.: its funding spread is
14See Castagna [1] for a definition of LVA.
Winter 201535
COLLATERAL MANAGEMENT
equal to zero); to keep the analysis simple we alsoassume that the contract does not provide for anymaintenance margin. In very general terms, thevaluation formula for 1 unit notional can be writtenas:
Repol(t, T) = EQ[− PP +D(t, T)RP
]−CVAl(t, T) + LVAl(t, T)
(18)
The LVAl(t, T) is formally defined as:
LVAl(t, T) = EQ[ ∫ T
tD(t, u)Cl
Cash(u)(ru − cu)du]
(19)where c is the rate at which the collateral postedin cash (Cl
Cash) is remunerated. If we assume thatthe risk-free rate is also the remuneration rate, ad ithappens in practice when c is indexed at the O/Nrate, which is conventionally assumed to be alsothe risk-free rate, than the LVAl(t, T) = 0.
We discussed above that if the haircut set at theinception t is the fair one, than the CVAl(t, T) iszero: we can write equation (20)
Repol(t, T) = EQ[−MV(1−H(t))
+D(t, T)[MV(1−H(t))× (1 + rE × τE(t, T))
]](20)
The fair repo rate is the level of rE that makes zerothe value of Repo(t, T):
rE =
(1
PD(t, T)− 1)
1τE(t, T)
(21)
where PD(t, Tn) = EQ[
exp(−∫ T
t rsds)]
is the price
of a risk-free zero-coupon bond. Equation (21) isequal to the simply-compounded risk-free for theperiod equal to the duration of the repo contract. Sothe repo, when the haircut is fairly set, should notbe evaluated differently from a deposit the lendercloses with the borrower, as if the latter were arisk-free counterparty (even though actually it isnot).
This result is perfectly consistent with a no-arbitrage argument in evaluating the repo, whenthe evaluator does not consider the counterparty’sdefault and does not pay a funding spread in themarket. The argument is the following: assumethe repo rate is above the risk-free rate for a givenmaturity: the buyer (lender) can take the cash in
the money market and use that to pay the purchaseprice of the repo: the repurchase price cashed inat the expiry is enough to pay back the debt plusinterests and leave also an arbitrage profit. To avoidthe arbitrage, the repo rate should not be above therisk-free rate.
In reality the lender is not a risk-free agent, soit is likely that it pays a spread when funding thecash needed when closing the contracts, in case itdoes not have a positive cash amount available. Theinclusion of the funding costs within the evaluationof the repo entails adding the FVA component toequation (20), which is a cost and as such it abatesthe value of the contract to the buyer, so it enterswith the negative sign:
Repol(t, T) = EQ[− PP +D(t, T)RP
]−CVAl(t, T)− FVAl(t, T) + LVAl(t, T)
(22)
The FVA is defined as:15
FVA = PD(t, T)MV(1−H(t))× sl× τE(t, T) (23)
where sl is the (constant) funding spread paid overthe risk-free rate by the lender, over the period cor-responding to the duration of the contract. Thefunding cost to the lender is originated by the fund-ing spread paid on the cash it has to raise to lendmoney to the borrower.
If the haircut is fair (i.e.: CVAl = 0), the reporate including the lender’s funding is:
rE =
[(1
PD(t, T)− 1)+
1PD(t, T)
FVAl(t, T)MV(1−H(t))
]× 1
τE(t, T)= r + sl
(24)
where we has assumed that the funding costs orig-inated by the margin calls is enough small to beconsidered negligible.
In this case the no-arbitrage argument shouldbe refined by setting the upper limit to the repo rateat the funding rate (i.e.: risk-free rate plus fundingspread) of the lender, or more generally to the aver-age funding rate paid in the money market by themarket-makers/lenders that can enter in reverserepo transactions. This is the conclusion also inDuffie [5].
15The definition of the FVA could be made more precise by including the possibility to pay the funding spread for a shorter periodthan τE(t, T) by including in the formula the default of the borrower. We do not consider here such case.
36iasonltd.com
Valuation for the Borrower (Seller)
From the bowrrower (seller) point of view, the valueof the repo is:
Repod(t, T) =
EQ[+ PP−D(t, T)RP
]−CVAd(t, T)
(25)
where the CVA, from the seller (borrower) perspec-tive, is defined as:
CVAd(t, T) = EQ[ ∫ T
tD(t, s)max
[VColl
− (1−H)V(t)× [1 + rE × τE(t, T)]
− Cd(s), 0]dPDl(t, s)
] (26)
and PDl(t, s) is the probability of default of thelender between t and s; Cd(s) is the collateral re-ceived by the borrower from the maintainance mar-gin mechanism described above ( Cd(s) = −Cl(s)).It is quite easy to check that if CVAl = 0 (i.e.: thehaircut is fairly set), CVAd ≥ 0, the equality hold-ing only if the lender’s default probability is zero.It is finally worth stressing that the FVAd is im-plicitly determined by the lender’s funding spread.More specifically, assuming for a moment that thelender is default-risk free (i.e.: CVAd = 0), the repois worth:
Repod(t, T) =
= EQ[+ PP−D(t, T)RP
]= PP
− PD(t, T)(1−H)V(t)× [1 + rE × τE(t, T)] = PP
− (1−H)V(t)− PD(t, T)(1−H)V(t)× sl × τE(t, T)
= −PD(t, T)(1−H)V(t)× sl × τE(t, T)
= −FVAd(t, T)(27)
Equation (27) states that the value of the repoto the cash borrower is negative and equal to thelender’s funding costs, or FVAd(t, T) = FVAl(t, T):if the agent lending cash is on the strong side in thebargaining power, it is able to charge its fundingcosts to the borrower, which will eventually paythe lender’s funding spread sl instead of its own(unsecured) funding spread sd.16 The repo allowsthen the borrower to lower its funding costs buyaligning them to those of the credit-worthier lender.Hence the implied funding cost to borrow cash viaa repo transaction (secured funding) is just the reporate rE = r + sl .
If the lender is not a default-risk free agent, thenthe repo’s value to the borrower should include alsothe counterparty credit risk costs:
Repod(t, T) = −FVAd(t, T)−CVAd(t, T) (28)
so that the value of the repo is even more negativethan the case examined above. In this case, theimplied funding rate paid by the borrower is:
r + sE =
r + sl +1
PD(t, T)CVAl(t, T)
MV(1−H(t))1
τE(t, T)> rE
(29)
Hence the funding spread is higher than thelender’s funding spread, when the borrower hasto accept to pay the CVAl if it wants to receivethe money. In any case, it will also be very likelythat, when the lender has a high credit standing,the implied funding is lower than the borrower’sunsecured funding cost.
The Sell/Buy Back
Definition and Introduction
Definition 2. A sell/buy back transaction is virtuallyidentical to a repo transaction: the legal title of the col-lateral is transferred by the seller to the buyer, against apayment of cash; the seller agrees to buy back the same(or equivalent) collateral at a future date at a specificprice.
The main differences between a repo and asell/buy back are that the latter may also not bedocumented under a master agreement, but it canbe simply dealt as two separated trades. A directconsequence of this possibility is that, for undoc-umented sell/buy-backs, it is not possible to setup a (both maintenance and initial) margin pro-cess, a haircut at the start, or close-out and set-off ifdefault by either party occurs. Finally, no manufac-tured payments occur during the life of a sell/buyback: all payments made by the collateral assetare included into the repurchase price paid on therepurchase date.
Nowadays, sell/buy backs are simply seen asa form of repo, as such they are documented inthe Buy/Sell Back Annex to the new 2011 GMRAagreement (see [10]). Also the quotation is nowsimilar to repo contracts, i.e.: in terms of contractrate17 instead of forward bond price, as it used tobe in the past. The difference between a repurchase
16Under a strict theoretical point of view, since we assumed the repo lender (buyer) is risk-free, its funding spreas sl should be zero.17The contract rate is named repo rate also for sell/buy back transactions.
Winter 201537
COLLATERAL MANAGEMENT
Haircut Initial MarginPurchase Price 111.8616 111.9624
Sell/Buy Back Cash 1,118,615.96 1,119,623.62Bond Notional 1,000,000.00 1,000,000.00
Repo Rate 1.75% 1.75%Maturity 06/11/2014 06/11/2014
Days 92 92Repo Interests 5,002.70 5,007.21Cash Returned 1,123,618.66 1,124,630.83
Accrued (92 days) 1.03 1.03Repurchase Price 114.8096 114.8096
TABLE 10: Details of a sell/buy back when a haircut or an initial margin is applied.
agreements and documented sell/buy-backs is basi-cally only due the method the latter use to mitigatecredit exposures originated by the volatility of thecollateral asset’s market price. In more details, re-purchase agreements use the margin maintenancemechanism we have described above (see SectionMargin Maintenance) to restore the equivalence be-tween the the values of cash lent and collateral re-ceived; documented sell/buy-backs mitigate creditexposures by terminating the transaction and si-multaneously opening a new transaction for theresidual time to maturity, keeping the same termsas the original transaction, but re-aligning the valueof the cash to the new market value of the collat-eral asset. We will briefly describe this process,denominated “re-repricing”, below.
Purchase and Repurchase Price
Similarly to a repo transaction, the seller of the col-lateral asset at inception receives the purchase price;the final cash paid at the expiry by the seller, to buythe collateral asset back, considers also the interimpayments made by the asset compounded at thecontract repo rate. Since the mechanics is quite sim-ilar to the repo case, the purchase and repurchaseprice are determined exactly as in equation (1) ((2)if an initial margin is applied instead of a haircut).
If the sell/buy back is quoted in terms of therepo rate, we have everything it is needed. If, onthe contrary, the contract is quoted in terms of col-lateral asset price, then we need to determine theinitial and forward price at which, respectively, itis sold and bought back. The initial price is simplythe price of the asset prevailing in the market whenthe contract starts in t. When the collateral asset is abond, the clean price is considered, even though thecash exchanged is the market value of the position,which included the accrued interests.
The forward price is set in terms of clean price
too, so that accrued interests must be deducted.Moreover, all the interim payments have to be com-pounded at the contract repo rate and deducted,since they have to be returned at the original seller.In formula, we have that the forward clean priceP(T), where the haircut is applied, is equal to:
P(T) =RP(T)NColl
× 11−H(t)
−AI(t, T)− IP (30)
where AI(t, T) = c × τB(t−1, T) is the bond’s ac-crued interest from the last coupon date to theexpiry T, and IP is the sum of the compoundedinterim payments C(ti):
IP = ∑ C(ti)× [1 + rEτ(ti, T)] (31)
Where the initial margin is applied, the forwardclean price is:
P(T) =RP(T)NColl
× IM−AI(t, T)− IP (32)
EXAMPLE 4 Assume the collateral asset is the bondin Example 1 and that the sell/buy back has thesame expiry and rate as the repo therein analysed.If the sell/buy back is quoted in terms of reporate, then the same calculations for the initial pur-chase price and the final repurchase prices apply.If the sell/buy back is quoted in terms of bond’sprice, then the terms of the contract will refer tothe initial and forward price. The initial price isthe clean price as in the Example 1, i.e.: 115.05; thedirty price is 115.3212, since accrued interests are0.2712. The initial cash exchanged is the same as fora standard repo, depending on whether a haircutor an initial margin is applied: in Table 10 we showthe initial cash in both cases. The forward price forthe haircut is determined by means of formula (30):
P(T) =1, 123, 618.661, 000, 000.00
× 100× 11− 3%
− 1.03
= 114.8096
38iasonltd.com
Haircut Initial MarginAdjusted Collateral Notional 1,011,638.94 1,011,638.94
Cash Returned 1,123,618.90 1,124,631.07Accrued 1.03 1.03
New Clean Price 113.48 113.48
TABLE 11: “Adjustment” of a sell/buy back contract.
If the initial margin is applied, usingformula (32) will yield the same result.
Transaction Exposure and Re-pricing
The transaction exposures on sell/buy-backs arenot cancelled by terminating the original transac-tion and simultaneously starting a new transactionwith the same maturity, in which two possible al-ternatives can be adopted (see 4(j) and 4(k) of theGMRA 2000 [11] and 4(k) and 4(l) of the GMRA2011 [10]):
• the purchase price of the new transaction isset equal to the new Market Value of the col-lateral asset or
• the nominal value of the asset is changed tore-align the new market value to the originalpurchase price.
In the first case the net difference between therepurchase price of the terminated transaction andthe purchase price of the new one is paid; in thesecond case the net difference between the originalamount of collateral asset and the new amount isdelivered.
In the GMRA the first method is called “re-pricing” and the second method is called “adjust-ment”. Under the re-pricing method, accrued repointerest is paid to the buyer and it is not includedin the new purchase price.
Finally, when margining is applied to a portfolioof transactions, the re-pricing, or the adjustment, isapplied to the single transactions. It is customary toreprice or adjust transactions in sequence, startingwith the transaction with the highest transactionexposure.
Assume we are in that the sell/buy back startedin t and expires in T; if the “re-pricing” method isused, at time t < s < T, the new deal replacing thelatest one has a new purchase price PP∗ (replacingthe old PP) such that:
PP∗ =MV(s)
IM(33)
orPP∗ = MV(s)× [1−H(t)] (34)
depending on whether an initial margin or a hair-cut was set in the original contract (the notation isthe same as the one used before for repos). Therepurchase price of the new transaction is s is equalto the market value of the collateral at time s.
If the “adjustment” method is used, then at times the new deal will have an adjusted notional suchthat:
MV∗(s) = RP(s)× IM (35)
or
MV∗(s) =RP(s)
1−H(t)(36)
again, depending on whether an initial margin ora haircut was set in the original contract. In thiscase, the equivalence in (35), or (36), is restoredby changing the notional of the deal from NColl toN∗Coll, so that the market value of the collateral is:
MV∗(s) = N∗Coll ×(
P(s) + c× τB(t−1, s)100
)
EXAMPLE 5 We refer to the bond data in Examples3 and 4 and we show how the credit risk mitigationmechanism of a sell/buy back works in practice. Tothis end, consider the variation in the dirty price ofthe collateral bond after 1 day from 115.32 to 114.00.The sell/buy back is “adjusted” by determining thenew notional amount to collateral to deliver in thenew transaction cancelling the old one. In Table 11we show the result in case a haircut or an initialmargin is applied. The notional amount of collat-eral of the new transaction is clearly the same inboth cases and it is the result of the application offormulae (35) and (36). The forward clean price, incase the sell/buy back is quoted in terms of price, isalso the same in both cases and it is determined as:
P(T) =
=1, 123, 618.901, 000, 000.00
× 100× 11− 3%
− 1.03 = 113.48
if a haircut is applied. When an ini-tial margin is applied, the forward price is
P(T) =
=1, 123, 618.901, 000, 000.00
× 100× (1 + 3%)− 1.03 = 113.48
Winter 201539
COLLATERAL MANAGEMENT
Haircut Initial MarginAAAARepriced PPAAAA 1,105,800.00 1,106,796.12
Cash Returned 1,110,624.62 1,111,625.08Accrued 1.03 1.03
New Clean Price 113.47 113.47
TABLE 12: “Repricing” of a sell/buy back contract.
The transaction exposures, from Table 9, are12,870.34 and 13,258.39 respectively for the casea haircut or an initial margin is applied. It shouldbe noted that the transaction exposure are the samefor a repo and a sell/buy back, if the two con-tracts have the same economic terms. The “ad-justment” mechanism is able to reset at zero theexposure. Consider the case of a haircut: the differ-ence in the notional between the old and new con-tract is: 1, 011, 638.94− 1, 000, 000.00 = 11, 638.94(see Table 11 first column). In practice there is anetting of cash-flows between the seller and thebuyer of the collateral asset, so that an amount of:
11, 638.94× 114.00/100× (1− 3%) = 12, 870.34
is paid by the seller to the buyer, exactly compensat-ing the latter (i.e.: the lender) for the increase of thetransaction exposure. A similar check can be madefor the initial margin case; the cash exchanged is:
11, 638.94× 114.00/100 = 13, 258.39
The alternative mechanism is the “re-pricing”:in this case the notional is kept constantwhile the price is changed as shown inTable 12. The new purchase price is:
1, 400, 000.00× (1− 3%) = 1, 105, 800.00
when a haircut is applied. The cash settled be-tween the buyer and the seller is in this case simplythe difference between the updated purchase priceof the old deal (1,118,615.96 see Table 9) and thepurchase price of the new deal (1,105,800.00), so:
1, 118, 615.96− 1, 105, 800.00 = 12, 870.34
so that the transaction exposure is fullycompensated. When the initial marginis applied, the new purchase price is:
1, 400, 000.001 + 3%
= 1, 106, 796.12
and the compensated transaction exposure is:
(1, 118, 615.96− 1, 105, 800.00)× (1 + 3%)
= 13, 258.39
Fair Haircut and Valuation
The fair haircut of a sell/buy back can be calculatedby the same approach as the one sketched abovefor a repo. Also the valuation of a sell/buy backis virtually identical to the valuation of a repo con-tract analysed before. Once the haircut is fairly setand a credit risk mitigation mechanism is chosen,the CVA can be set equal to zero also in this case.
Securities Lending
Definition and Introduction
Definition 3. At the inception of a securities lendingtransaction, the Lender transfers title to a single secu-rity, or a basket of securities to the borrower, receivingin exchange either the title to another security (or a bas-ket of securities) or alternatively cash, and the paymentof a fee; at the expiry of the contract, or on demand, thelender will transfer title to equivalent collateral or re-pay the cash plus an agreed return, in exchange for titleto the security, or a basket of securities, equivalent tothe one it transferred at the inception.
The security (or basket of security) or cash re-ceived by the lender at the start of the contract isthe collateral the borrower posts to receive title tothe lent security (or basket of securities). The de-nomination of the contract and the wording usedin the definition may be misleading, since the ac-tual title to security is transferred from the lenderto the borrower, as in a repo transaction.18 Figure2 provides a visual summary of the obbligationsperformed by both parties in the contract.
The effects of a securities lending transactionare basically analogous to those of a repo contract,the main differences lying in the exchange betweenthe two parties, which can be a security againstanother security in the securities lending, whereasit is against cash in a repo. Another relevant differ-ence is that the expiry of a repo contract is defined,while a securities lending is generally dealt on anopen basis. Open basis transactions may be termi-nated by the borrower, when it returns securities,
18Although this statement is generally true, it worth noting that the transfer of the title may not occur in the US market.
40iasonltd.com
FIGURE 2: Obligations of a security lending contracts.
or by the lender, when it recalls them. Similarlyto a repo, all payments made by the lent securityare transferred to the lender by means of a “manu-factured” payment by the borrower. Analogously,when non-cash collateral is delivered, any paymentmade by the collateral security is transferred to theborrower by the lender.
Under an economic point of view, a securi-ties lending transaction is typically security-driven,which means that the demand of the borrower fora specific security (or a basket of securities) is themain reason for the deal. This is also reflected bythe terms denoting the parties, which are chosenwith reference to the security rather than the cash:the borrower receives cash against securities in arepo, it receives securities against cash (or othersecurities) in a securities lending. Furthermore, thesecurities lender receives a fee contrarily to whathappens in a repo, where the seller (cash borrower)pays an interest rate: this happens when the lendertake securities as collateral; when it is given cash ascollateral, it pay the borrower an interest but at arate (the rebate rate) that is lower than market rates.So, in any case, the economic rationale produces ayield for the securities lender.
For securities lending, the standard masteragreement adopted by institutional market agentsis the ISLA GMSLA [9].
Quotation: Fee and Collateral Margin
The quotation of a securities lending contract re-quires the definition of a fee to be paid to the (secu-rities) lender on a periodic basis. Typically, the feewill be paid monthly at the end of each month ofthe life of the contract; also the final fee referring tothe month in which the contract either expires or itis recalled (in open ended transactions) is paid atthe last day of the month.
A margin is applied to the value of the collateraldelivered by the borrower against the lent securities:the margin is defined as in the repo contract caseanalysed in the first section of the article (Fair Hair-cuts and Initial Margins). It has the same purposeas in the repo case to protect the (securities) lenderfrom the default of the (securities) borrower.
Fair Margin
The security lender wishes to set at inception a mar-gin IM such that the expected loss suffered in theevent of the borrower’s default is nil. To set the fairlevel of initial margin, we can adopt an approachsimilar to the one sketched above for a repo con-tract (also for the notation, we refer to the one usedabove). Assume at the start of the contract t = 0,a time horizon T: this time may clash with the ex-piry of the contract, or an expected duration if ithas been dealt on an open basis.19 Let S(T) be thevalue of the lent security in T, and let vColl(T) be
19We recall here that both parties have the right to terminate the contract by giving a notice to the other party, when it was dealt onan open basis.
Winter 201541
COLLATERAL MANAGEMENT
the minimum market value of the posted collateralin T, calculated at a given confidence level, say 99%.The fair margin is set so that the expected loss ELon the borrower’s default is:
EL(t, T) =
E[
max[S(T)− vColl(T)IM, 0]∣∣∣∣Dd(t, T) = 1
](37)
The notation is the same as in section Fair Hair-cuts and Initial Margins (Dd(t, T) is the stochasticvariable equal to 1 if the security borrower goesbankrupt in the time interval [t, T]). The main dif-ference between equations (37) and (7) is that in theformer the EAD = S(T), and is a stochastic vari-able, whereas in the latter EAD equals the amountof money lent in the repo, so that it is a given quan-tity. We will tackle the problem with the stochasticEAD similarly to the uncertainty of the value of thecollateral: we will calculate the maximum exposureat a given confidence level, say 99%. Let Sα(T) bethis level, when α is the chosen confidence level.
Assuming independence between the event ofdefault of the issuer of the asset underlying thesecurity lending contract, and the default of the col-lateral asset, (and, more generally, any correlationbetween the values of the two assets), following thereasoning in section Fair Haircuts and Initial Mar-gins, we can write (37) that:
EL(t, T) = E[
Sα(T)− vColl(T)IM∣∣∣∣Dd(t, T) = 1
](38)
The equation (38) states that the expected loss isequal to the expected positive difference betweenthe maximum value of the lent asset and the mini-mum value of the collateral, times the initial margin,in T, given the collateral issuer’s default indicatorfunction (DColl(t, T)) is 1, or the default occurs.
Given the independence assumption, the ex-pected (maximum) exposure is E[Sα(T)|Dd(t, T) =1] = Sα(T), while the expected (minimum) value ofthe collateral is:
E[vColl(T)IM|Dd(t, T) = 1] =
E[vColl(T))IM(1− DColl(t, T))
+ (vColl(T)− Lgd)IM)DColl(t, T)∣∣∣∣Dd(t, T) = 1]
or, more explicitly:
EL(t, T) = Sα(T)− vColl(T)IM
+
[[P1PDI
d(t, T) + P2((1− pd)PDId(t, T)
+ pd)]LgdIM]
/PDd(t; T)
(39)
The initial margin IM is the level that makesEL(t, T) = 0, or:
IM= Sα(T)
vColl(T)−
[[P1PDI
d(t,T)+P2((1−pd)PDId(t,T)+pd)]Lgd
]/PDd(t;T)
(40)
Collateral Maintenance
Collateral has to be marked to the market duringthe life of the contract: on any business day, the ag-gregated market value of the collateral delivered to,or deposited with, the security lender (Posted Col-lateral) must be equivalent to aggregated marketvalue of the securities, considering also the appli-cable margin (Required Collateral Value). In casenetting is excluded, the provision applies on a sin-gle contract basis, instead of an aggregated basis.Limiting the analysis to the latter case, for a givencontract we have in formula that:
VColl(s)IM = S(s)IM (41)
where, at time s, VColl(s)IM is the value of theposted collateral and S is the value of the lent se-curity (IM is the margin applied). If on any busi-ness day VColl(s)IM < S(s)IM, additional collat-eral is posted by the security borrower so as toreestablish the equivalence. On the contrary, ifVColl(s) > S(s)IM, excess collateral is returned bythe lender to the borrower.
The collateral maintenance mechanism allowsto set the IM by considering just a margin periodof risk, that is the period between the default andits actual recognition. Typically this period is setequal to 10 business days. Once the initial marginis computed, the collateral maintenance mechanismwarrants a nil expected loss (EL(t, T) = 0), for anytime horizon.
The collateral Cl(s) that the security lendershould receive at time s is equal to S(s)IM −VColl(s)IM. Considering the collateral already re-ceived by the lender, the variation between twomargin calls in times q and s (we recall the margincall occur on each business day), is:
∆Cl(s) = [S(s)IM−VColl(s)IM]− Cl(q) (42)
Negative variations and values means the buyerposts instead of receiving collateral. We can set theinitial collateral at the inception of the contract int = 0 equal to:
Cl(t) = VColl(t)IM = S(t)IM
Valuation
A security lending contract is valued according tothe principles recalled above and stated in Castagna
42iasonltd.com
[2]: so we need to consider the CVA and the FVAof the evaluating party, assuming that its default isnot triggered by the counterparty’s default.
Valuation for the Security Lender
We consider a contract with a fixed expiry in T (anadditional complexity is given by the possibilitythat the contract is open-ended). The value of thecontract to lender is:
SecLending(t, T) =
= EQ[ N
∑i=0D(t, Ti)S(T0)Rτ(Ti−1, Ti)
]−CVAl(t, T) + LVAl(t, T)
(43)
τ(Ti−1, Ti) is the day count fraction (according tothe convention chosen in the contract), betweentwo payment dates (Ti−1 and Ti, with T0 = t andTN = T), and it is multiplied by the contract feerate R, applied to the value of the loan at the startS(T0). The LVAl is defined as in formula (19): thiscomponent exists only when collateral is paid incash and the lender has the opportunity to rein-vest it at the risk-free rate (on a risk-adjusted basis),paying an interest to the borrower.
The CVA, from the lender point of view, of thesecurity lending at time t, assuming a time horizonT; this is defined similarly to what we have seenfor a repo contract. Formally, we have:
CVAl(t, T) =
= EQ[ ∫ T
tD(t, u)max
[S(u)− Cl(u), 0
]dPDd(t, u)
](44)
Taking into account the initial margin and the dailycollateral maintenance, the CVAl is quite negligible.
Hence, since the CVAl(t, T) can be assumed tobe zero for practical purposes, if the collateral isposted in securities, so that also LVAl = 0, then thecontract has always a positive value at the incep-tion for the lender, which seems an arbitrage at afirst look. What are missing here is the fact thatthe security can be lent because it was purchasedsome time before the start of the contract, and itis unencumbered: it has not been repoed out tofinance its own purchase or pledged anyway. Thepurchase is then funded unsecured, over a timehorizon that has been chosen by the lender giventhe original intent; for example, it can be a securityheld for liquidity buffer purposes, or it can be heldfor investment reasons. So it can be expected tobe funded for a longer or shorter period, but in
any case the lender will pay an unsecured fundingspread.20 Let sl be the funding spread paid by thelender over the risk-free rate, assumed a constantfor simplicity’s sake. For the period correspond-ing to the duration of the security lending contract,the present value in t of the cost paid to fund thesecurity is:
FVAS(t, T) = EQ[ ∫ T
tD(t, u)slS(tBuy)du
](45)
where S(tBuy) is the cash outflow the lender fundedwhen purchased the security in time tBuy ≤ t (wehave not considered additional cash flows possiblyoccurring in the period). The funding costs shouldhave been already taken into account when evaluat-ing the economic results of the original transaction:for example, if the security was bought for invest-ment reasons, the cost of carry (i.e.: the costs tofund its purchase) should be deducted from thereturn generated. Now, if the security is lent out fora given period, then the lender can earn a fee thatcan be used to abate the funding costs. Let ρ by thefraction of funding costs the lender aims at savingwhen dealing a contract; the valuation equation(43) modifies by including also this component asfollows:
SecLending(t, T) =
= EQ[ N
∑i=0D(t, Ti)S(T0)Rτ(Ti−1, Ti)
]−CVAl(t, T)
− FVAl(t, T) + LVAl(t, T)
(46)
where FVAl(t, T) = ρFVAS(t, T) is the fraction ofthe (present value of) funding costs to charge onthe security lending contract. Letting the CVA andthe LVA be both equal to zero, the fair lending feeR is the one making nil the value of the contract atinception:
R =FVAl(t, T)
∑Ni=0 PD(t, Ti)S(T0)τ(Ti−1, Ti)
(47)
The valuation problem is now shifted to thechoice of the parameter ρ. We here offer only at avery short discussion, hinting at very general ideas:when setting up a security lending business, thelender must decide how much of the funding costs(attributable to the purchase of the securities) haveto be recovered via the security lending activity.This percentage can be set based on considerationscompletely unrelated to the market demand to bor-row securities, or it can totally depend on it, sothat the lender accepts any fee determined by the
20It is out of the scope of this paper to model the unsecured funding spread. A possible approach to account for the funding mixavailable to the lender, and the related costs and risks, is outlined in chapter 11 of Castagna and Fede [4].
Winter 201543
COLLATERAL MANAGEMENT
market demand and supply forces. So ρ can be afree and independent choice, or it can be simply de-duced from the market prevailing fees, so that thevalue of the security lending contracts are alwaysfair to the lender. In the end, the valuation of thesecurity lending contracts can be quite flexible, andarbitrary somehow.
Valuation for the Security Borrower
The valuation to the borrower is operated similarlyas before; the value is:
SecLending(t, T) =
= EQ[−
N
∑i=0D(t, Ti)S(T0)Rτ(Ti−1, Ti)
]−CVAb(t, T) + LVAb(t, T)
(48)
which is simply the present value of the stream offees paid, considering also the CVA and the LVAfrom the borrower perspective. Neglecting onceagain the LVA, as if collateral were never posted incash, we need to focus on the CVA: it can be com-puted after noting that the exposure of the borrowerto the lender is given by the difference between thevalues of the collateral posted and of the security.In formula we have:
CVAb(t, T) =
= EQ[ ∫ T
tD(t, u)max
[Cb(u)− S(u), 0
]dPDl(t, u)
](49)
where PDl is the default probability of the secu-rity lender and Cb is the collateral received by thesecurity borrower (Cb = −Cl). It is worth notingthat, if the one hand the CVAl is made negligible,if anything, by the initial margin and the collateralmaintenance mechanism, on the other hand these
two factors make the CVAb(t, T) surely a positivequantity (i.e.: an expected loss on the lender’s de-fault always greater than zero).
We do not take into account in (48) any FVA,even though strictly speaking a small cost is paidby the borrower to fund the negative cash flowsrelated to the fee payments.
Thus, summing up the results above, the valueof the contract to the borrower is always negative,which obviously dos not imply any arbitrage op-portunity. The negative value should be considereda cost the borrower has to pay to borrow the se-curity, similarly to the costs paid borrow money.They are factored in the total P&L of the borrowerwhen computing the economic result of its businessactivity.
Conclusion
This paper is a summary of the current legal frame-works and best practices adopted in the trading ofcontracts that are the main market instruments tomanage and transform the collateral within a finan-cial institution. We have dwelt also on valuationaspects that are crucial to assess the total economicresults originated by trading in these instruments.More complex contracts and sophisticated tradingstrategies, based on the investigated instruments,will be the object of future research.
ABOUT THE AUTHOR
Antonio Castagna is Senior Consultant, co-founder andCEO at Iason ltd.Email address: [email protected]
ABOUT THE ARTICLE
Submitted: December 2014.
Accepted: January 2015.
44iasonltd.com
References
[1] Castagna, A. Pricing of Deriva-tives Contracts under CollateralAgreements: Liquidity and FundingValue Adjustments. Iason researchpaper. 2012. Available online at:http://iasonltd.com/resources.php
[2] Castagna, A. Towards a The-ory of Internal Valuation andTransfer Pricing of Products ina Bank. Argo Magazine, n. 2,Spring 2014. Available online at:http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2143979
[3] Castagna, A. Collateral Man-agement: Processes, Tools andMetrics. Argo Magazine, n. 4,Fall 2014. Available online at:
www.iasonltd.com/research/argo-magazine/
[4] Castagna, A. and F. Fede. Measuringand Management of Liquidity Risk.Wiley. 2013.
[5] Duffie D. Special Repo Rates. Jour-nal of Finance, n. 5, 51, pp. 493-526.1996.
[6] Duffie D. and J. Pan. AnalyticalValue at Risk with Jumps and CreditRisk. Finance and Stochastics, n. 5,pp. 155-180. 2001.
[7] Fouque, J-P., G. Papanicolaou andK.R. Sircar. Derivatives in FinancialMarkets with Stochastic Volatility.Cambridge University Press. 2000.
[8] Hull, J. and A. White. Collateraland Credit Issues in DerivativesPricing. Working Paper. 2013.
[9] International Securities LendingAssociation. Global Master Securi-ties Lending Agreement. 2010.
[10] Securities Industry & FinancialMarkets Association and Interna-tional Capital Market Association.General Master Repurchase Agree-ment. 2011.
[11] The Bond Market Association andInternational Securites Market As-sociation. General Master Repur-chase Agreement. 2000.
Winter 201545
ADVERTISING FEATURE
chasing the future in FinanceIason ltd provides solutions that enable managers, risk-control officers and front office peopleto understand and value their portfolios. All solutions are built on the principles of soundness,accuracy and clearness.
PRICING
Modern Derivatives PricingSound models to value derivative instrumentsaccording with the best and newest marketpractice (CVA, DVA and liquidity).
Complex ProductsSophisticated analytical models, supported byresearch, in order to effectively and efficientlyevaluate all kind of exotic structures,accounting for all the relevant risk factors.
Revaluation of BooksThe service is for those institutions notinterested in building a trading and riskmanagement system, but simply needing a fairrevaluation of their (possibly very complex)derivatives positions.
RISK MEASUREMENT
Credit RiskInnovative approach to Credit VaR (CreVar)modeling and computation.
Market RiskInnovative and sophisticated solutions formarket risk measure and management.
Counterparty RiskPioneers in the development of CCR and CVAmeasurement models and applications withdevoted internal teams in many major banks.
ALM and Liquidity RiskConsulting services to support ALM andtreasury department of financial institutions.Innovative models taking into account theinterrelation between funding, liquidity andCCR.
To learn more: www.iasonltd.com.To contact us: [email protected]. Follow us on:
@Iason_ltd company web page iason iason.network
46
A Critical Review of CentralBanks Satellite Modelsfor Probabilities of Default
The aim of this article is to analyse dif-ferent approaches to the probabilities ofdefault in satellite models building. Af-ter illustrating several models proposedby different Central Banks, the author fo-cuses on the advantages and the draw-backs of these models with the sup-port of real-case application experience.
Marco RAUTI
In recent years the stress test of banks balancesheets has become a primary issue, both forinternal governance and in view of the ‘official’
stress tests conducted by Competent Authorities.The scope of this paper is to summarise dif-
ferent methodologies to stress the credit risk, andmore specifically to analyse different approachesto the PDs (Probabilities of Default) satellite modelsbuilding.
We start with a focus on models proposed bydifferent Central Banks, and then we highlightstrengths and weaknesses of these models, thanksto the experience gained in a real case application.
The satellite models are quantitative methodsthat allow to build a link between macroeconomicvariables (such as GDP growth, unemploymentrate, consumer price inflation, etc.) and banks creditrisk measures, such as PDs and LGDs (Losses GivenDefault); when the link is identified and modeled,it’s possible to translate directly the hypotheticalmacroeconomic shocks on banks’ credit portfolio.
The main PDs models we analyse are sum-marised in Table 1 (adapted from [8]) where theyare classified both by the dependent variables andby the mathematical transformation applied on thisvariables.
The simplest model is the one proposed by ECB(European Central Bank) [5] and Sveriges Riks-bank [3], where EDFs (Expected Default Frequencies)are modeled without any transformation, as proxyfor PDs. While this approach guarantees immedi-ate comprehension of what regressors coefficientsmean (due to intrinsic linearity), on the other handit doesn’t guarantee the dependent variable staysin the range between 0 and 1.
Another possible approach, proposed by BoE(Bank of England) [1], Bank of Italy [7], De Neder-landsche Bank [14] and Bank of Spain [9], consistof using logit or probit transformation of the DRs(Default Rates), that is to say that the dependentvariable is forced to range between 0 and 1 due tothe transformation of the right hand of the equationin the cumulated density function of logistic andnormal distribution respectively. Alternatively, inplace of the DRs, the LLPs (Loan Loss Provisions)Ratio may be used as dependent variable, resultingin a model qualitatively similar.
The last approach analysed, proposed by BOJ(Bank of Japan) [13], models the rating transitionprobabilities transformed by the probit.
In the following, we analyse each of the modelsproposed by the Central Banks, and we emphasize,in addition to different dependent variables used,as above, also the different solution algorithms usedand the different choices in the macroeconomic con-test building and in the macroeconomic variables
Winter 201547
SATELLITE MODELS
Model Bank
Expected Default FrequenciesSveriges Riksbank
European Central Bank
Bank of England
Logit transformation of aggregate default rates Bank of Italy
De Nederlandsche
Probit transformation of aggregate default rates Bank of Spain
Logit transformation of the LLP ratioSwiss National Bank
De Nederlandsche Bank
Probit transformation of the probability of a rating transition Bank of Japan
TABLE 1: Classification of Central Banks Models
selection.
Central Banks models
ECB model
To create a link between EDFs and relevant macroe-conomic variables ECB proposes to use an OLS(Ordinary Least Squares) linear regression. From theECB document we have the following EDF defini-tion:
“the EDF measures the probability thata firm defaults within a given time horizonand, hence, it provides a forward-lookingmeasure of default. Intuitively, the EDFscan be interpreted as estimators that mea-sure how close a firm’s assets approachits liabilities given the macroeconomic sce-nario”
The equation driving this model is here after repre-sented:
EDFt = α + β1∆GDPt + β2∆CPIt
+ β3∆EQt + β4∆EPt + β5∆IRt + εt(1)
where ∆ indicates the logarithmic difference, and
• GDP denotes the gross domestic product,
• CPI denotes the consumer price index,
• EQ denotes the equity prices,
• EP denotes the EURUSD exchange, and
• IR denotes the short-term interest rate.
ECB estimates seven stand alone equations for theseCorporate segments:
• Basic goods and construction,
• Energy and utilities,
• Capital goods,
• Consumer cyclicals,
• Consumer non-cyclicals,
• Financial, and
• Technology, media and telecommunications.
From the above equation, as result, we have neg-ative coefficient sign for GDP, equity, EURUSDexchange and interest rates, for all segments, whilefor CPI we have some segment with positive co-efficient and some other with negative coefficient.While the negative sign for GDP and equity pricewas largely expected, on the other hand it seemsslightly counter-intuitive the negative sign for theexchange and interest rate. Lastly, for a possiblereading key of the twofold behavior of the pricesinflation, we refer to the Sveriges Riksbank expla-nation in next paragraph.
In the official stress test conducted by Com-petent Authorities the macroeconomic scenario isgiven as a fact, but if a bank want to simulate itsown set of scenarios for internal governance stresstest or for ICAAP development, the main macroe-conomic variables have to be linked all together toobtain a realistic scenario, namely a stressed (un-likely) but plausible scenario. In this regard ECBproposes a GVAR approach, (Global Vector AutoRe-gressive model), namely a model based on country
48iasonltd.com
specific vector error correction model, where domesticand foreign variables interact simultaneously. Thisis a very complete and complex set of equations,recommended for international banks with differ-ent subsidiaries in different countries. For a deeperdiscussion on this topic, see for example [12].
Both the GVAR equations and the satellite equa-tions can be corrected by adding some lagged term,because it’s reasonable to expect that there is nosynchronism between macroeconomic shocks andthe microeconomic effects. That should improve themodel, which in the ECB document has an averageR2 equal to about 40%.
Sveriges Riksbank model
Sveriges Riksbank uses a VECM (Vector Error Cor-rection Model) to study the long-term relationshipbetween log− EDFs and the macroeconomic devel-opment. The equation representing the model isthe following:
log (EDFt) = α + β1R3Mt
+ β2 log(INDt) + β3 log(CPIt) + εt(2)
where:
• R3M denotes the nominal domestic three-month rate for treasury bills,
• IND denotes the domestic industrial produc-tion index, and
• CPI as above denotes the consumer price in-dex.
Instead of using the log-differences of the termsin the right hand of the equations (as done insteadby ECB), this model works with the variables levels,so these steps have to be followed:
• test for unit root in macroeconomic variables(i.e. stationarity check): generally macroeco-nomic variables result to be not stationary;
• identification of the cointegrating relation be-tween EDFs and macroeconomic variables(the Johansen’s method [10] is recommended):the output of this step are the long-term rela-tion coefficients;
• choice of the lag structure of the equationthat describes the dynamic adjustment of theendogenous variables to deviations from long-run equilibrium: this step can be performedby choosing the lag structure that minimisethe RMSE (Root Mean Square Error), as sug-gested in the document, or by the minimi-sation of the AIC or the BIC (Akaike andBayesian Criterion Information respectively);
• estimation of the complete model by log-likelihood maximisation;
• check for the economic sensible signs of theregressors coefficients (both the long-term re-lation recomputed coefficients and the short-term equilibrium adjustment coefficients).
The VECM complete resulting equation is thefollowing:
∆xt = δ0 + Γ1∆xt−1 + Γ2∆xt−τ + αβxt−1 + εt (3)
where:
• xt represents the vector with EDFs andmacroeconomic variables, and
• ∆xt−τ represents the differences in xt−τ ,where τ means the lag structure chosen.
The application of the model to real data showsresults broadly in line with expectations: theyfound negative correlation between EDFs and man-ufacturing output and a positive correlation be-tween EDFs and interest rates (the latter result isas expected, but in contrast with the one from ECBwork). The link between EDFs and inflation ismainly twofold, also in this case. The possible ex-planation Sveriges Riksbank gives of this fact isthat on one hand higher factor prices lead to in-creased production costs and tend to impair creditquality, but on the other hand product prices canboost earnings and thereby improve creditworthi-ness. The relative strength of these two effects ofinflation (determined by the structure of the mar-kets for factors of production and the company’soutput) gives the resulting sign of the CPI.
Finally, this model seems to be a little too parsi-monious in the choice of macroeconomic variables,although this was one of the aims of the documents.In the Corporate EDF model could be reasonableinsert also long-term interest rates, or equity prices,or oil price, while in the Retail default frequenciesmodel one could insert unemployment rate, forexample. The VECM instead is a robust way tomaintain the information provided by the macroe-conomic variables levels (this information would beinstead lost making the macroeconomic variablesstationary).
BoE, Bank of Italy and De Nederlandsche BankModels
The common factor in the models proposed bythese central banks is the use of the logit transfor-mation of DRs as dependent variables. Hence, thegeneral equation describing these models is the
Winter 201549
SATELLITE MODELS
following:
log(
DRt
1− DRt
)= α + βXτ + εt (4)
where:
• DRt represents the default rates, and
• Xt represents the set of macroeconomic vari-ables at different possible lags.
The use of the logit transformation allows totake into account possible non-linearities in themacro-micro relations, and prevents, even whenextreme shocks were applied to macroeconomicvariables in a scenario design, that the simulatedDRs go outside the range (0,1). An equivalent wayto write the previous equation is:
DRt =eα+βXτ+εt
1 + eα+βXτ+εt(5)
In this way is simpler to see why DRs cannot ex-ceed their natural ranges, in fact the right hand ofthe equation is nothing but the CDF (CumulativeDensity Function) of the logistic distribution.
The main differences between the approachesused by these three banks concern both the macroe-conomic variables chosen and the algorithm usedin the solution: in the following of this paragraphwe briefly describe these topics.
BoE recommend to use the logarithmic differ-ence in GDP and equity price, and short-term inter-est rates, all at time t− 1. They estimate a differentequation for each Corporate sector and they solvethese equations separately with the OLS algorithm.By their own admission, this model is too poor tocatch all the process volatility (the resulting mod-els they build have R2 coefficients between 5% and30%), moreover neither the cross correlation be-tween sectors neither the persistence in the PDs(both empirically demonstrated) are captured.
The Nederlandsche Bank’s model employs asregressors for the estimation of the (logit) DRs theGDP growth and the spread between long-termand short-term interest rates. They jointly estimatealso the (logit) LLP Ratio, as function of the (logit)DRs, found as explained above, GDP growth andlong-term interest rates. The same critical aspectsshown for the BoE document are present here, al-though in this case a larger group of macroeco-nomic variables was tested (in addition to GDPgrowth and interest rates, real effective exchangerate, unemployment rate, house prices, stock pricesand oil prices were initially included in the model,but then excluded due to a worst fit or a ‘wrong’
coefficient sign). An interesting aspect that emergesfrom this work is the analysis they provide on thecoefficient of spread between long-term and short-term interest rates: the expected negative sign isexplained as follows:
“A decreasing term spread either meansthat short term rates increase, e.g. throughtightening monetary and financial condi-tions, or that long term rates decrease,which may reflect a subdued outlook for in-flation and the business cycle. Both couldraise credit risk. Besides, the yield curveis an indicator for the business cycle. Aflattening curve might point to deceleratingeconomic growth, which again could raisecredit risk.”
This explanation seems to be reasonable, and clar-ifies why it’s possible to find negative correlationbetween default rates and (long-term) interest ratesand positive correlation between default rates and(short-term) interest rates in other cases, when theyare separately included in the equation.
The main contributions from Bank of Italy tothe logit model shown above are in the choice ofmacroeconomic variables and in the solution al-gorithm: they use the PCA (Principal ComponentAnalysis) to decide which macroeconomic variablesretain in the model, and they insert among theregressors also the lagged dependent variable, tocatch the persistence in PDs. For the joint solu-tion of the equations for different Corporate sectors,they propose the SUR (Seemingly Unrelated Regres-sions) algorithm, that allows to capture the inter-relations among different economic sectors. TheSUR Method estimates the equations jointly, andaccount for possible correlations between equationserrors. It allows to have different regressors in eachequations, and when the regressors are common inall equations it collapses to single OLS equations.
In the logit contest the Bank of Italy model is themost robust, the most complete and the more stable.The only aspects that may be improved are in thechoice of macroeconomic variables and lags and ina better use of PCA: they use this analysis to avoidmulticollinearity among macroeconomic variables,and to catch the common factors explained by thesevariables, but this approach, although useful, is notcomprehensive, because it doesn’t tell in any wayhow many factors are needed to explain the depen-dent variables. To solve this problem we proposeto use the PCA approach on the left hand of theequations, i.e. on the DRs (see the ‘PCA revised andmodified stepwise regression algorithm’ paragraph fora better explanation).
50iasonltd.com
Bank of Spain Model
This model estimates the DRs for different Cor-porate sectors, and insert among the explicativevariables, in addition to macroeconomic variablesand lagged dependent variables, also 2 latent fac-tors, common to all sectors and orthogonal to themacroeconomic variables. The complete set of equa-tion is represented by a VAR (Vector AutoRegressive)model of order 1, while the common factors areupdated with Kalman Filter method. The distinctivefeature of this model is the use of probit transfor-mation on DRs. In analogy with the logit case, wehave:
Φ−1(DRt) = α + βXτ + εt (6)
or equivalently:
DRt = Φ(eα+βXτ+εt) (7)
The use of the probit transformation, exactly as thelogit does, allows to take into account possible non-linearities in the macro-micro relations, and pre-vents, even when extreme shocks were applied tomacroeconomic variables in a scenario design, thatthe simulated future DRs go outside the range (0,1).The difference is that, as shown in the formulasabove, instead of using the logistic distribution herethe normal distribution is used. Ultimately, the logitand the probit transformations are two sides of thesame coin, there is no evident reason to prefer oneto the other.
Swiss National Bank Model
The model proposed by SNB (Swiss National Bank)at first glance differs from the previous model onlybecause of the different dependent variable (herethe logit transformation of LLP ratio is used in placeof the logit transformation of the DRs), instead, themain features of this model, that make it differ-ent from the previous ones, are in the algorithmsused and also in the choice of variables. Two alter-native equations have to be estimated, expressed as:
yt = X′tβ + Z
′tγ + εt (8)
with:εt = ρεt−1 + νt (9)
and:
yt = yt−1φ + X′tβ + Z
′tγ + εt (10)
where
• yt represents the logit transformation of LLPratio,
• X′t represents the set of macroeconomic vari-
ables, and
• Z′t represents the set of bank-specific vari-
ables.
The first equation (called ‘static’), is the logit formof the classical equation already shown, but in thiscase, because the lagged dependent variable doesn’tappear among regressors, its persistence (i.e. thepositive and significant correlation between yt andyt−1 is taken into account by the autocorrelatederrors. The estimation is performed by a GLS (Gen-eralized Least Squares) estimator for unbalanced pan-els with AR1 disturbances, developed by Baltagiand Wu [4]. The second equation (called ‘dynamic’)doesn’t need autoregressive errors because it con-tains the lagged dependent variable among regres-sors, but has to be estimated by Arellano-BondGMM (General Method of Moments estimator) [2] toavoid the bias introduced by OLS in presence ofcorrelation between dependent variable and errors.
The macroeconomic variables used in both equa-tions are
• ∆GDP,
• 3-months interest rates,
• spread on Corporate or government bonds,and
• unemployment rate.
The results differ from two models, but the regres-sors signs are the same and in both cases are correct,i.e. negative for GDP and positive for interest rates,spreads and unemployment rate.
Bank of Japan Model
The Bank of Japan, unlike other Central Banks,links the macroeconomic variables to the elementsof the transition matrices; the single elements of atransition matrix are so defined:
Pij,t =Nij,t
∑Jj=1 Nij,t
(11)
where Nij,t is the number of borrowers classifiedat the ith rating at time t − 1 that pass at the jth
rating at time t, while the sum at the denominatorrepresents the total number of firms rated at theith rating at time t− 1, with a total of J classes ofrating. More in detail, they regress on macroeco-nomic variables the parameter k from the matrix Q,which is the matrix that produces fluctuations in
Winter 201551
SATELLITE MODELS
the transition probabilities matrix:
Pt = P×Q (12)
Qt =
1 + kt −kt 00 1 + kt −kt0 0 1 + kt
(13)
kt = c + βGDPt + γDebt + ε (14)
where:
• Pt is the transition matrix,
• P is the average of long-term data on the tran-sition matrix, and
• Q is the ‘fluctuations’ matrix.
Here is nothing new respect other Central Banks’approaches, but it’s interesting the generalisationfrom PDs to a complete transition matrix.
Our Contribution to PDs Modeling
PCA revised and modified stepwise regression al-gorithm
In this paragraph we explain two methods wefound very useful to determine how many vari-ables to include in the model, which variables toinclude, and at which lag (these methods may beapplied to any kind of model). First of all, the PCAanalysis may be used on dependent variables, andnot only on macroeconomic variables (as done bythe Bank of Italy). In the analysis of Corporate DRsfor different sectors, the PCA reduces the complex-ity of the problem, and helps to identify how manycommon factors describe the set of PDs. The num-ber of eigenvalues that divided by the sum of alleigenvalues gives a ratio of about 90%, indicateshow many factors it needs to fully describe (at leastat 90%) the PDs set. The next step is to bring backthese factors (generally they are between 2 and 4)to the macroeconomic variables. To identify whichmacroeconomic variables drive the PDs (or the fac-tors, if the PCA has been applied), we wrote analgorithm that we called modified stepwise regres-sion, which works as follows:
• identification of all possible explicative vari-ables: this set has to contain all the macroeco-nomic variables to be tested;
• identification of all possible lags at which maybe economically meaningful include the pre-vious variables: the set has to contain all pre-vious variables declined at all possible lags;
• building of the complete model: the equationwith all the possible regressors (i.e. all thevariables at different lags) has to be estimated;
• removal of variables: we remove the first vari-able and we estimate the model again. Were-include the first variable and we removethe second one, then we estimate again themodel. We continue in this way until all vari-able have been removed once.
• variables and lags selection by selection crite-rion: among all the models estimated, namelythe reduced models (those without one vari-able) and the complete one, we select the onewith the best (i.e. the lowest one) AIC andwe repeat the process at the next step, thistime starting from the model chosen at theprevious step. If the complete model (or thestarting model for the subsequent steps) is thebest, the process ends and we have found thebest variables combination possible.
The flowchart in Figure 1 visually explains the al-gorithm.
This algorithm is robust, because it reaches thebest possible model regardless of the variable re-moval order, moreover we have verified that itworks also proceeding backwards, is to say thatstarting from no variables, and adding them one ata time, we get the same solution.
Our final model
The model we found to be the most reliable, ro-bust and immediate to implement is the one fromBank of Italy, which we implemented with theseadjustments:
• we introduced the dependent variable amongthe regressors with fixed lag 1: the persistencein the PDs is captured in this way, and thishelps to explain much of the volatility thatwould otherwise not be explained;
• we use the modified stepwise regression toselect both the variables and the lags at whichto insert them: we added a ‘check’ constraintduring the loops to ensure that the chosenvariable coefficient had the correct sign, atleast for the variables whose sign is uniquelyknown a priori (GDP growth with negativesign, unemployment rate with positive sign,etc.);
• we estimated each equation separately andwe didn’t use the SUR framework.
52iasonltd.com
FIGURE 1: Choice of Variables Algorithm
Winter 201553
SATELLITE MODELS
We found that for Corporates the GDP has thelargest impact, together with interest rates and in-flation, while for Consumer the unemployment rateis the main driver, also in this case with interestrates and inflation.
While the GDP and unemployment rate coef-ficient signs are obviously negative and positiverespectively, the sign of (short term) interest ratecoefficient is positive, so we found empirically thesame conclusion as in the Nederlandsche Bank’smodel (namely tightening monetary and financialconditions raise the PDs). For the price inflation,we found a very interesting and coherent result:in the Corporate case we have a negative sign, soraising prices improve the creditworthiness of theproducers, conversely in the Retail case we have apositive sign, so higher inflation worsens the credit-worthiness of consumers.
All the variables used are included at some lag(generally we have a delay effect form 3 months to1 years, that means, in a rational manner, that theshocks on macroeconomic variables need some timeto be propagated in the microeconomic framework,
and this is an important issue to be consideredwhen assessing the stress tests.
Conclusion
In this work we’ve shown that there is no modelbetter than others in all situations, because eachone have its own strengths and weaknesses. Withthe summary we made, and with the hints we pro-vided, gained from a real application, every bankmay build its own model, based on its situationand aims, and aware of possible results and criticalissues arising.
ABOUT THE AUTHOR
Marco Rauti is Senior Quantitative Analyst and he worksin the Model Validation division of IntesaSanPaolo Group.At the time of the writing of this article, he was workingon satellite models’ development in Iason ltd.
ABOUT THE ARTICLE
Submitted: November 2014.
Accepted: January 2015.
References
[1] Alessandri, P., P. Gai, S. Kapadia,N. Mora and C. Puhr. A frameworkfor quantifying systemic stability.Preliminary paper presented at theStress Testing of Credit Risk Portfo-lios workshop hosted by the BCBSand the DNB. 2007.
[2] Arellano, M. and S. Bond. SomeTests of Specification for PanelData: Monte Carlo Evidence and anApplication to Employment Equa-tions. The Review of EconomicStudies, n. 2, 58, pp. 277-297. 1991.
[3] Asberg, P. and H. Shahnazarian.Macroeconomic Impact on ExpectedDefault Frequency. Sverige Riks-bank working papers series, n. 219.2008.
[4] Baltagi, B. and P. Wu. UnequallySpaced Panel Data Regressionswith AR(1) Disturbances. Econo-metric Theory, n. 15, pp. 814-823,1999.
[5] Castren, O., S. Dees and F. Zaher.Global Macro-Financial Shocks andExpected Default Frequencies in the
Euro Area. ECB working papers se-ries, n. 875. 2008.
[6] Commission Bancaire. The FrenchApproach of Stress-Testing CreditRisk: the Methodology. Directionde la Surveillance Generale du Sys-teme Bancaire. 2011.
[7] Fiori, R., A. Foglia and S. Ian-notti. Beyond Macroeconomic Risk:the Role of Contagion in the Ital-ian Corporate Default Correlation.CAREFIN working paper. BocconiUniversity. 2009.
[8] Foglia, A. Stress testing credit risk:a survey of authorities’ approaches.Questioni di Economia e Finanza(occasional papers). Banca d’Italia.2008.
[9] Jimenez, G. and J. Mencia. Mod-elling the Distribution of CreditLosses with Observable and LatentFactors. Documentos de Trabajo.Banco de Espana. 2007.
[10] Johansen, S. Estimation and Hy-pothesis Testing of Cointegration
Vectors in Gaussian Vector Autore-gressive Models. Econometrica, n. 6,59, pp. 1551-1580. 1991.
[11] Lehmann, H. and M. ManzAsberg.The Exposure of Swiss Banks toMacroeconomic Shocks - an Empir-ical Investigation. Swiss NationalBank working papers series, n.2006-4. 2006.
[12] Pesaran, M., T. Schuermann andV. Smith. Forecasting Economic andFinancial Variables with GlobalVARs. Federal Reserve Bank ofNew York Staff reports, n. 317.2008.
[13] Shiratsuka, S., A. Otani, R. Tsu-rui and T. Yamada. Macro Stress-Testing on the Loan Portfolio ofJapanese Banks. Bank of Japanworking paper series. 2009.
[14] Willem van den End, J., M. Hoe-berichts and M. Tabbae. ModellingScenario Analysis and Macro Stress-testing. DNB working papers se-ries, n. 119. 2007.
54iasonltd.com
ADVERTISING FEATURE
Advisory Research and Training in Finance
Concentric is a company specialized in the advisory, research and train-ing in sales and finance. The company adopts a dynamic approach toits organsational evolution, reflecting ongoing changes in the business en-vironment, the requirements of clients and developments in best practices.
QUANTITATIVE ADVISORY
quad is the fusion of Concentric and Iason
international risk advisory, research and training
practitioner teams, rigorous project methodologies
and tried-and-tested tools.
CORPORATE GOVERNANCEADVISORY
cga comprises organisational analysis, design and
development projects for bank, insurance company
and corporate compliancy.
CONCENTRIC RESEARCH
core includes service consulation, needs analysis, project modeling, satisfaction analysis, industry survey and
financial analysis.
FINANCE MASTERCLASS
fmc is one of Europe’s most appreciated and
successful public technical financial training programs
for banking and insurance professionals.
SALES AND FINANCEEDUCATION
the safe team adopts a holistic and value-adding
approach to the design and development of
in-company finance, commercial and management
training projects.
For more information visit www.concentric-italy.com
55
NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT
AAAA
in the previous issue
Fall 2014
banking & finance
Collateral Management
Bond Settlement Migration
energy & commodity finance
Natural Gas Statistical Arbitrage
Monetary Measurement of Risk: A CriticalOverview - Part III: Convex Measure of Risk
last issues are available at www.iasonltd.com and www.energisk.org
56
