P1.T4. Valuation & Risk Models Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach Bionic Turtle FRM Study Notes

By David Harper, CFA FRM CIPM and Deepa Sounder www.bionicturtle.com

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Linda Allen, Chapter 2: Quantifying Volatility in VaR Models KEY IDEAS AND SOME TERMINOLOGY .................................................................................... 4 NORMAL LINEAR VALUE AT RISK (VAR): A FUNDAMENTAL FRM CALCULATION .......................... 5

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Linda Allen, Chapter 2: Quantifying Volatility in VaR Models Explain how asset return distributions tend to deviate from the normal distribution. Explain reasons for fat tails in a return distribution and describe their implications. Distinguish between conditional and unconditional distributions. Describe the implications regime switching has on quantifying volatility. Explain the various approaches for estimating VaR. Compare and contrast parametric and non-parametric approaches for estimating conditional volatility. Calculate conditional volatility using parametric and non-parametric approaches. Explain the process of return aggregation in the context of volatility forecasting methods. Describe implied volatility as a predictor of future volatility and its shortcomings. Explain long horizon volatility/VaR and the process of mean reversion according to an AR(1) model. Calculate conditional volatility with and without mean reversion. Describe the impact of mean reversion on long horizon conditional volatility estimation

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Key ideas and some terminology

The three basic value at risk (VaR) approaches are: parametric (aka, analytical), historical simulation and Monte Carlo simulation. With respect to parametric approaches, in the FRM our initial concern is the classic linear normal approach.

Two assumptions especially are among the convenient but egregious (unrealistic) employed in the linear normal approach: first, our tendency to “impose normality” on actually non-normal returns; and, second, our belief that returns are independent over time (the square root rule of scaling VaR/volatility assumes independence).

On the fallacy of our tendency to “impose normality:” Because we can compute a standard deviation—or because we only have the first two distributional moments—does not by itself imply the dataset is normally distributed (of course!)

Risk varies over time. Models often assume a normal (Gaussian) distribution with constant volatility from period to period; e.g., Black-Scholes-Merton assumes constant volatility. But actual returns are non-normal and volatility is actually time-varying. Therefore, it is hard to use parametric approaches on random returns. It is hard to find robust distributional assumptions for stochastic asset returns.

Aside from implied volatility, we are chiefly concerned with the three basic methods that estimate volatility with historical return data: (simple) historical simulation; exponentially weighted moving average (EWMA); and GARCH(1,1).

Historical simulation has many variations (see Dowd) but our initial concern is the “hybrid approach” which assigns weights according to the EWMA model

Conditional parameter (e.g., conditional volatility): this is a parameter such as variance that depends on (is conditional on) circumstances or prior information. A conditional parameter, by definition, changes over time. This is generally realistic!

Persistence: In EWMA, persistence is the lambda parameter (λ). In GARCH(1,1), it is the sum of the alpha (α) and beta (β) parameters. In GARCH(1,1), high persistence implies slow decay toward to the long-run average variance; EWMA does not embed any long-run average variance.

EWMA can be viewed as a special case of GARCH(1,1) but where the weight assigned to the long-run variance (i.e., gamma) is zero. This is not to be confused with a zero long-run variance; rather, EWMA has no long-run variance. Conversely, GARCH(1,1) can be viewed as a generalized EWMA.

In GARCH(1,1), the implied L.R. Variance = ω/(1-[α+β]) = ω/(1-persistence).

Two GARCH terms: Autoregressive: Recursive. A parameter (today’s variance) is a function of itself (yesterday’s variance). Heteroskedastic: Variance changes over time (homoscedastic = constant variance).

Leptokurtosis: a heavy-tailed distribution where relatively more observations are near the middle and in the “fat tails (kurtosis > 3). The GARCH(1,1) model exhibits heavy-tails in the unconditional distribution of returns; yet the model assumes conditional returns are normal.

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Normal linear value at risk (VaR): a fundamental FRM calculation

You should be able to calculate the normal linear value at risk (VaR) for both a single asset and its extension to a two-asset portfolio.

Single asset value at Risk (VaR)

In the following example, we assume 250 trading days per year and the following assumptions about the asset:

Asset value equals $200.00

Asset volatility of 10.0% per annum

Further, any value at risk (VaR) estimate requires two user-based “design decisions:”

What is the confidence level? In this case, the selected confidence is 95.0%

What is the desired horizon? In this case, the horizon is one day. Therefore, we will need to scale volatility (and VaR) from one year to one day.

Single-asset delta normal VaR (per annum inputs)

Trading days per year 250

VaR horizon (days) 1

VaR confidence level, c 95.0%

Normal (one-tailed) deviate 1.645

Asset Value ($) $200.0

Volatility, per annum 10.0%

Value at Risk (VaR), per annum, % 16.4%

Value at Risk (VaR), per annum, $ $32.90

Value at Risk (VaR), horizon, % 1.04%

Value at Risk (VaR), horizon, $ $2.08 Given these assumptions about the asset and our VaR choices (i.e., confidence and horizon), we can calculate:

Per annum VaR = $200.0 * 10.0% * 1.645 = $32.90. But our horizon is one day.

One-day VaR = $200 * 10.0% * sqrt(1/250) * 1.645 = $2.08; the volatility is scaled per the so-called “square root rule”

Relative VaR (versus absolute VaR): we have here assumed the expected return (aka, drift) for a single day is zero; or, alternatively we have ignored the expected return. This is common when estimating the one-day VaR. Technically, if we exclude the expected return, we are estimating a relative VaR (as opposed to an absolute VaR). This is called a relative VaR because it is the worst expected loss is relative to the expected (future) value at the end of the horizon.

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Two asset value at Risk (aka, 2-asset diversified portfolio VaR)

We build on the previous example by considering a portfolio of two assets rather than a single asset. Notice this requires an additional assumption: what is the correlation between the two assets? Correlation is almost always denoted by Greek rho, ρ. In this example, we assume the correlation is zero. Increasing the correlation will increase the portfolio VaR; at perfect correlation, the portfolio VaR will equal the sum of the individual VaRs (which illustrates diversification: at any correlation less than perfect 100%, the portfolio VaR will be less than the sum of the individual VaRs).

Two-asset delta normal ¶ VaR (per annum inputs)

Trading days per year 250

VaR horizon (days) 1

VaR confidence level, c 95.0%

Normal (one-tailed) deviate 1.645

Portfolio Value ($) $200.0

Asset A Asset B

Volatility (per year) 10% 20%

Portfolio Weight (w) 50% 50%

Individual VaR, per annum, $ $16.45 $32.90

Individual VaR, horizon, $ $1.04 $2.08

Correlation, ρ(A,B) 0

Portfolio volatility, per annum, % 11.18%

Portfolio VaR, per annum, % 18.39%

Portfolio VaR, annual, $ $36.78

Portfolio VaR, horizon, $ $2.33

Portfolio VaR, horizon, $ (alt) $2.33 Given these assumptions about the asset and our VaR choices (i.e., confidence and horizon), we can calculate:

Per annum portfolio variance = 0.52*0.12 + 0.52*0.22 + 2*0.5*0.5*0.1*0.2*0 = 0.01250

Per annum portfolio volatility = sqrt(0.01250) = 11.18%

Per annum (diversified) portfolio VaR = 11.18% * 1.645 = 18.39%

Scaled one-day portfolio VaR = $200.0 * 18.39% * sqrt(1/25) = $2.33

There is another approach that utilizes the individual VaRs of $1.04 for Asset A and $2.08 for Asset B. Where iVaR is Individual VaR, this uses:

portfolio VaR = �iVaR�� + iVaR�

� + 2 ∗ ����� ∗ iVaR� ∗ �. In this case,

portfolio VaR = �1.04� + 2.08� + 2 ∗ 1.04 ∗ 2.08 ∗ 0 = $2.33

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Portfolio VaR increases with correlation

In this two asset portfolio, a key relationship is between the correlation parameter and diversified portfolio VaR: higher correlation implies higher portfolio VaR. Using the same individual VaR parameters given previously (i.e., an equally-weighted portfolio with respective volatilities of 10% and 20%), the following exhibit displays the portfolio VaR:

And the corresponding plot (below) shows the portfolio VaR is increasing and non-linear. Notice at perfect correlation (ρ = 1.0), the portfolio VaR is equal to the sum of the individual VaRs: at ρ=1.0, portfolio VaR equals $3.12 which is the sum of $1.04 and $2.08. This is unsurprising because under this normal linear VaR estimate, VaR is simply a multiplier on the standard deviation.

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The key time scaling assumption is i.i.d. normal, specifically independence

When scaling the VaR from one year to one day (or vice versa) we typically assume independence (recall that i.i.d. designates both independent and identical distributions). Independence is a critical but unrealistic assumption. If returns are independent across time, then their autocorrelation (aka, serial correlation) is zero. By the contrapositive, the following is also true: if the autocorrelation is non-zero, then the returns are not independent across time. Therefore, non-zero autocorrelation is a violation of the i.i.d. assumption and renders inaccurate the square root rule of scaling VaR. For the FRM, you do not need to be able to quantify the implications of this i.i.d. violation. Nonetheless, it is illustrated below so that you can see the directional impact which is discussed in this reading. Please note we assume:

The daily volatility is conveniently 1.0%

The first column illustrates the typical approach used to scale the daily volatility to a 10-day VaR: 10-day VaR = 1.0% * 2.33 deviate * sqrt(10/1) = 7.36%

Parametric VaR

Standard Deviation (Volatility), Daily 1.0% 1.0% 1.0%

Confidence Level, c 99.0% 99.0% 99.0%

Significance Level, 1-c 1.0% 1.0% 1.0%

Target Horizon (days) 10 10 10

Autocorrelation 0.0 0.2 -0.2

1-day Value at Risk (VaR) 2.33% 2.33% 2.33%

Extended over Target Horizon (i.i.d.)

Standard deviation (i.i.d.) 3.16% 3.16% 3.16%

n-day VaR (i.i.d.) 7.36% 7.36% 7.36%

Extended over Target Horizon (autocorrelated)

Scaling factor 10.00 14.38 6.94

Standard deviation (with autocorr.) 3.16% 3.79% 2.64%

VaR over horizon (with autocorr.) 7.36% 8.82% 6.13% At the bottom, compare the 10-day VaR of 7.36% to the second and third columns:

When autocorrelation is +0.20, the scaled 10-day VaR is 8.82%

When autocorrelation is -0.20, the scaled 10-day VaR is 6.13%. Negative autocorrelation reflects mean reversion and we can see that, when there exists mean reversion, the i.i.d. scaled VaR of 7.36% will overstate the “adjusted” VaR.