Modelling Returns: the CER and the CAPMdidattica.unibocconi.it/mypage/dwload.php?nomefile=slides_capmcer... · 0.20 0.15 0.10 0.05 0.00 0.05 0.10 Series: RET_M_C Sample 1948M02 2014M06

Modelling Returns: the CER and the CAPM

Carlo Favero

Favero () Modelling Returns: the CER and the CAPM 1 / 34

The Process of Econometric Modelling

exploratory analysis of the data (graphics and descriptivestatistics)model specificationmodel estimationmodel simulationmodel validation and applications


Returns

Consider an asset that does not pay any intermediate cash income (azero-coupon bons, such as a Treasury Bill, or a share in a company thatpays no dividends). Let Pt be the price of the security at time t.The linear or simple return between times t and t− 1 is defined as:

Rt = Pt/Pt−1 − 1

The log return is defined as:

rt = ln(Pt/Pt−1)

Note that, while Pt means “price at time t”, rt is a a shorthand for“return between time t− 1 and t”


Returns

The two definitions of return yield different numbers when the ratiobetween consecutive prices is far from 1.Consider the Taylor formula for ln(x) for x in the neighbourhood of 1:

ln(x) = ln(1) + (x− 1)/1− (x− 1)2/2+ ...

if we truncate the series at the first order term we have:

ln(x) ∼= 0+ x− 1

so that if x is the ratio between consecutive prices, then for x close toone the two definitions give similar values. Note however thatln(x) ≤ x− 1. In fact x− 1 is equal to and tangent to ln(x) in x = 1 andabove it anywhere else.


Multi-period returns

Define the simple multi-period return between time t and t+n as:

Rt,t+n = Pt+n/Pt − 1 (1)

=Pt+n

Pt+n−1

Pt+n−1

Pt+n−2...

Pt+1

Pt− 1

=nΠi=1(1+ Rt+i,t+i−1)− 1

in the case of log returns we have instead:

rt,t+n = ln (Pt+n/Pt) (2)

= ln(

Pt+n

Pt+n−1

Pt+n−1

Pt+n−2...

Pt+1

Pt

)=

n

∑i=1

rt+i,t+i−1


Annualized returns

annualized returns the constant annual rate of return equivalent to themultiperiod returns to an of an investment in asset i over the period t,t+n. In the case of simple returns we have

(1+ RA

t,t+n

)n= 1+ Rt,t+n =

nΠi=1(1+ Rt+i,t+i−1)

RAt,t+n =

(nΠi=1(1+ Rt+i,t+i−1)

) 1n

− 1

Consider now log returns:

nrAt,t+n = rt,t+n =

n

∑i=1

rt+i,t+i−1

rAt,t+n =

1n

n

∑i=1

rt+i,t+i−1


Working with Returns

Consider the value of a buy and hold portfolio of invested in shares ofk different companies, that pay no dividend, at time t be:

Vt =k

∑i=1

niPit

The simple one-period return of the portfolio shall be a linear functionof the returns of each stock.

Rt =Vt

Vt−1− 1 = ∑

i=1..k

niPit

∑j=1..k njPjt−1− 1

= ∑i=1..k

niPit−1

∑j=1..k njPjt−1

Pit

Pit−1− 1 =

= ∑i=1..k

wit(Rit + 1)− 1 = ( ∑i=1..k

witRit + ∑i=1..k

wit1)− 1 =k

∑i=1

witRit


Working with Returns

log returns are not additive in the cross-section but they are additivewhen we consider the time-series of returns

rt = ln(Vt

Vt−1)

= ln(

k∑

i=1niPit−1

k∑

i=1niPit−1

Pit

Pit−1) = ln(

k

∑i=1

wit exp(rit))

rt,t+n =n

∑i=1

rt+i,t+i−1

Note that additivity in the time-series does not apply to simple returns.


Stock Returns and the dynamic dividend growthmodel

consider the one-period total holding returns in the stock market, thatare defined as follows:

Hst+1 ≡

Pt+1 +Dt+1

Pt− 1 =

Pt+1 − Pt +Dt+1

Pt=

∆Pt+1

Pt+

Dt+1

Pt, (3)

Dividing both sides by(1+Hs

t+1

)and multiplying both sides by

Pt/Dt we have:

Pt

Dt=

1(1+Hs

t+1

)Dt+1

Dt

(1+

Pt+1

Dt+1

).

Taking logs we have:

pt − dt = −rst+1 + ∆dt+1 + ln

(1+ ept+1−dt+1

)Favero () Modelling Returns: the CER and the CAPM 9 / 34


Taking a first-order Taylor expansion of the last term about the pointP̄/D̄ = ep̄−d̄ :

ln(

1+ ept+1−dt+1)' ln(1+ ep̄−d̄) +

ep̄−d̄

1+ ep̄−d̄[(pt+1 − dt+1)− (p̄− d̄)]

= − ln(1− ρ)− ρ ln(

11− ρ

− 1)+ ρ(pt+1 − dt+1)

= κ + ρ(pt+1 − dt+1)

where

ρ ≡ ep̄−d̄

1+ ep̄−d̄=

P̄/D̄1+ (P̄/D̄)

< 1 κ ≡ − ln(1− ρ)− ρ ln(

11− ρ

− 1)

.

Total stock market returns can then be written as:

rst+1 = κ + ρ (pt+1 − dt+1) + ∆dt+1 − (pt − dt) ,



By forward recursive substitution one obtains:

(pt − dt) = κm

∑j=1

ρj−1 +m

∑j=1

ρj−1(∆dt+j − rst+j)

+ ρm (pt+m − dt+m) .

Under the assumption that there can be no rational bubbles, i.e., that

limm−→∞

ρm (pt+m − dt+m) = 0,

(pt − dt) =κ

1− ρ+

m

∑j=1

ρj−1(

∆dt+j − rst+j

).


Zero-Coupon Bonds

Define the relationship between price and yield to maturity of azero-coupon bond as follows:

Pt,T =1

(1+ Yt,T)T−t ,

Taking logs of the left and the right-hand sides of the expression forPt,T, and defining the continuously compounded yield, yt,T, aslog(1+ Yt,T), we have the following relationship:

pt,T = − (T− t) yt,T,

The one-period uncertain holding-period return on a bond maturing attime T, rT

t,t+1, is then defined as follows:

rTt,t+1 ≡ pt+1,T − pt,T = − (T− t− 1) yt+1,T + (T− t) yt,T

= yt,T − (T− t− 1) (yt+1,T − yt,T) ,= (T− t) yt,T − (T− t− 1) (yt+1,T)


Coupon Bonds

The relationship between price and yield to maturity of a constantcoupon (C) bond is given by:

Pct,T =

C(1+ Yc

t,T

) + C(1+ Yc

t,T

)2 + ...+1+ C

(1+ Yt,T)T−t .

To measure the length of time that a bondholder has invested moneyfor we need to introduce the concept of duration:

Dct,T =

C(1+Yc

t,T)+ 2 C

(1+Yct,T)

2 + ...+ (T− t) 1+C(1+Yt,T)

T−t

Pct,T

=

CT−t∑

i=1

i

(1+Yct,T)

i +(T−t)

(1+Yt,T)T−t

Pct,T

.


Coupon Bonds

The one-period return on a coupon bond can be approximated asfollows:

rct+1 = Dc

t,Tyct,T −

(Dc

t,T − 1)

yct+1,T,

where in turn the duration can be computed as

Dct,T =

1−(

1+ Yct,T

)−(T−t)

1−(

1+ Yct,T

)−1 ,


Graphics

time series graphics of returns (change of prices)time series graphics of portfolio perfomance (level of prices)scatter plotsmultiple graphsdensity estimates (histograms)


Time-series graphs (returns)

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50 55 60 65 70 75 80 85 90 95 00 05 10

log exact loglin

monthly returns


Time-series graphs (portfolios)

0

200

400

600

800

1,000

1,200

50 55 60 65 70 75 80 85 90 95 00 05 10

SM without reinvestmentSM with reinvestment10_Y Bonds 1month rolling

cumulative performance


Scatter-plots

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RET_M_C

RET_

M_LC


Multiple-graphs


Histograms

0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

0 . 2 0 0 . 1 5 0 . 1 0 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0

Series: RET_M_CSample 1948M02 2014M06Observations 788

M e a n 0 .0 0 9 5 0 6M e d i a n 0 .0 1 2 3 0 3M a x i m u m 0 .1 2 3 1 5 5M i n i m u m 0 .2 0 1 9 4 6S td . D e v . 0 .0 3 5 0 0 3S ke w n e ss 0 .7 3 7 7 0 9K u rto si s 5 .8 1 0 2 9 5

J a rq u e B e ra 3 3 0 .7 8 3 3P ro b a b i l i ty 0 .0 0 0 0 0 0


Matrix Representation of the data

A matrix is a double array of i rows and j columns, whose genericelement can be written as aij, it is a convenient way of collectingsimultaneusly information on the time-series and the cross-section ofreturns:

A =

a11 . . a1j

ai1 aij

, 0 =

0 . . 0

0 0

I =

1 . . 0

0 1


Matrix Operations

Transposition a′ij = aji

Addition: For A and B nxm (a+ b)ij = aij + bij

Multiplication: For A nxm and B mxp (ab)ij =m

∑k=1

aikbkj

Inversion for non-singular A nxn, A−1 satisfies A−1A = AA−1 = I


Modelling returns

The (naive) log random walk (LRW) hypothesis on the evolution ofprices states that, prices evolve approximately according to thestochastic difference equation:

ln Pt = µ∆+ ln Pt−∆ + εt

where the ’innovations’ εt are assumed to be uncorrelated across time(cov(εt; εt′) = 0 ∀t 6= t′), with constant expected value 0 and constantvariance σ2∆.Consider what happens over a time span of, say, 2∆.

ln Pt = 2µ∆+ ln Pt−2∆ + εt + εt−∆ = ln Pt−2∆ + ut

having set ut = εt + εt−∆.


Modelling returns

Consider now the case in which the time interval is of the length of1-period. If we take prices as inclusive of dividends we can write thefollowing model for log-returns

rt,t+1 = µ+ σεt

εt = i.i.d.(0, 1)

E(rt,t+n) = E(n

∑i=1

rt+i,t+i−1) =n

∑i=1

E(rt+i,t+i−1) = nµ

Var(rt,t+n) = Var(n

∑i=1

rt+i,t+i−1) =n

∑i=1

Var(rt+i,t+i−1) = nσ2


Monte-Carlo simulation

given some estimates of the unknown parameters in the model (µσ in our case).an assumption is made on the distribution of εt.The an artificial sample for εt of the length matching that of theavailable can be computer simulated.The simulated residuals are then mapped into simulated returnsvia µ, σ.This exercise can be replicated N times (and therefore aMonte-Carlo simulation generates a matrix of computersimulated returns whose dimension are defined by the samplesize T and by the number of replications N).The distribution of model predicted returns can be thenconstructed and one can ask the question if the observed data canbe considered as one draw from this distribution.


do exactly like in Monte-Carlo but rather than using a theoreticaldistribution for εt use their empirical distribution and resample from itwith reimmission.


Simulation can be used for several tasks,

provide statistical evidence of the capability of the model toreplicate the dataderive the distribution of returns to implement VaRassess statistical properties of estimators


Stocks for the long-run

The fact that, under the LRW, the expected value grows linearly withthe length of the time period while the standard deviation (square rootof the variance) grows with the square root of the number ofobservations, has created a lot of discussionWe have three flavors of the “stocks for the long run” argument. Thefirst and the second are a priori arguments depending on the lograndom walk hypothesis or something equivalent to it, the third is ana posteriori argument based on historical data.



Assume that single period (log) returns have (positive) expected valueµ and variance σ2. Moreover, assume for simplicity that the investorrequires a Sharpe ratio of say S out of his-her investment. Under theabove hypotheses, plus the log random walk hypothesis, the Sharperatio over n time periods is given by

S =nµ√nσ=√

nµ

σ

so that, if n is large enough, any required value can be reached.



Another way of phrasing the same argument, when we add thehypothesis of normality on returns, is that, there any given probabilityα and any given required return C there is always an horizon forwhich the probability for n period return less than C is less than α.

Pr (Rp < C) = α.

Pr (Rp < C) = α⇐⇒ Pr(

Rp − nµ√nσ

<C− nµ√

nσ

)= α

⇐⇒ Φ

(C− µp

σp

)= α,

C = nµ+Φ−1 (α)√

nσ

But nµ+Φ−1 (α)√

nσ, for√

n > 12

Φ−1(α)µ σ is an increasing function in

n so that for any α and any chosen value C, there exists a n such thatfrom that n onward, the probability for an n period return less than Cis less than α.



Note, however, that the value of n for which this lower bound crossesa given C level is the solution of

nµ+Φ−1 (α)√

nσ ≥ C

In particular, for C = 0 the solution is

√n ≥ −Φ−1 (α) σ

µ

Consider now the case of a stock with σ/µ ratio for one year is of theorder of 6. Even allowing for a large α,say 0.25, so that Φ−1 (α) is nearminus one , the required n shall be in the range of 36 which is onlyslightly shorter than the average working life.As a matter of fact, based on the analysis of historical prices and riskadjusted returns, stocks have been almost always a good long runinvestment.


The CAPM Reduced Form

In the case of the CAPM the general specification of the reduced formis the following one:

(ri

t − rrft

)= µi + βium,t + ui,t(

rmt − rrf

t

)= µm + um,t(

ui,tum,t

)∼ n.i.d.

[(00

),(

σii σimσim σmm

)]


From the Statistical Model to the Conditional Model:the CAPM

Statistical model(ri

t − rrft

)= µi + βium,t + ui,t(

rmt − rrf

t

)= µm + um,t(

ui,tum,t

)∼ n.i.d.

[(00

),(

σii σimσim σmm

)]Estimated Equation

E((

rit − rrf

t

)|(

rmt − rrf

t

), Yt−1, Zt−1, βi

)= αi + βi

(rm

t − rrft

)if σim = 0,then the estimated equation is a valid approximation to thestatistical model for the estimation of βi


VaR ith the CAPM the CAPM

Model Specification(ri

t − rrft

)= αi + βi

(rm

t − rrft

)+ ui,t(

rmt − rrf

t

)= µm + um,t(

ui,tum,t

)∼ n.i.d.

[(00

),(

σii 00 σmm

)]Model EstimationModel Simulation


Documents

Modelling Returns: the CER and the CAPMdidattica.unibocconi.it/mypage/dwload.php?nomefile=slides_capmcer... · 0.20 0.15 0.10 0.05 0.00 0.05 0.10 Series: RET_M_C Sample 1948M02 2014M06