Lecture 8, Value-At-Risk and Expected Shortfall

Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

Quantitative Finance Lecture 8, Value-at-Risk and Expected Shortfall

Adam Farago

Department of Economics, University of Gothenburg

2019

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Value-at-Risk Introduction

Ways to measure market risk:

Scenario-based risk measures, stress testing

Factor sensitivity analysis

Risk measures based on the loss distribution approach Value-at-Risk (VaR) Expected shortfall (ES)

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Value-at-Risk Introduction

Ways to measure market risk:

Scenario-based risk measures, stress testing

Factor sensitivity analysis

Risk measures based on the loss distribution approach Value-at-Risk (VaR) Expected shortfall (ES)

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Value-at-Risk Introduction

So far we were interested in small changes of the underlying. But how bad things can get? What loss level is such that we are X% conﬁdent it will not be exceeded in N business days? VaR is certainly the risk measurement standard in the industry today. Regulators base the capital they require banks to keep on VaR. Mutual funds and hedge funds use it as a risk measure to report to clients.

It captures an important aspect of risk in a single number. It is easy to understand. It asks the simple question: How bad can things get?

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Value-at-Risk Formalization Consider a ﬁxed time period [t, t + ∆t] , where ∆t can be e.g 1 day, 10 days, one month, one year, etc. We can deﬁne the following objects: The change in the value of the portfolio during the period [t, t + ∆t] is

∆V ≡ Vt+∆t − Vt The return on the portfolio during the period [t, t + ∆t] is ∆V RV ≡ Vt The loss of the portfolio during the period [t, t + ∆t] is

L ≡ −∆V = − (Vt+∆t − Vt ) The relative loss of the portfolio during the period [t, t + ∆t] is L RL ≡ = −RV Vt

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Value-at-Risk Formalization

The distribution FL (x) of the random variable L is called the loss distribution (over the time period [t, t + ∆t])

It is deﬁned as FL (x) = P (L ≤ x)

FL (x) is the cumulative distribution function (cdf) of L.

Be aware of the sign! L takes a positive value if the value of the portfolio goes down.

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Value-at-Risk Deﬁntion “What loss level is such that we are X% conﬁdent it will not be exceeded in N business days?” Value-at-Risk (VaR)

Given the loss L and a conﬁdence level α ∈ (0, 1), VaRα (L) is given by the smallest number x such that the probability that the loss exceeds x is not larger than 1 − α. That is,

VaRα (L) ≡ inf {x ∈ R : P (L ≥ x) ≤ 1 − α}

Note that

VaRα (L) = inf {x ∈ R : 1 − P (L ≤ x) ≤ 1 − α} = inf {x ∈ R : P (L ≤ x) ≥ α} = inf {x ∈ R : FL (x) ≥ α}

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Value-at-Risk for discrete random variables

0.5 0.45 If α = 0.99, then 0.4 {x ∈ R : FL(x) ≥ 0.99} = [6000, ∞[ 0.3 so VaR (L)0.99 = 6000 0.2 0.2 0.2 If α = 0.97, then probability

{x ∈ R : FL(x) ≥ 0.97} = [4000, ∞[ 0.1 0.06 0.06 so VaR (L) = 4000 0.03 0.97 0 1 If α = 0.95, then 0.8 {x ∈ R : FL(x) ≥ 0.95} = [4000, ∞[ so VaR (L)0.95 = 4000 0.6 cdf If α = 0.9, then 0.4

{x ∈ R : FL(x) ≥ 0.9} = [2000, ∞[ 0.2 so VaR (L)0.9 = 2000 0 -6000 -4000 -2000 0 2000 4000 6000 L (loss)

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Value-at-Risk for continuous random variables

10-4 For continuous random variables, 2 the deﬁnition simpliﬁes. 1.5 Reason: The cdf of a continuous 1

random variable is continuous pdf and strictly increasing. 0.5

Therefore: 1- 0 1 VaR (L)α = {x ∈ R : FL (x) = α} 0.8 −1 = FL (α) 0.6 cdf −1 0.4 where FL is the inverse cdf. 0.2 That, is the VaR is a quantile of the distribution. 0 -6000 -4000 -2000 0 2000 4000 6000 L (loss)

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Comments

VaRα(L) will typically be a positive number, since it is connected to the loss-distribution.

The VaR can also be reported in terms of returns. Because of the linearity of VaR, VaRα(L) VaRα (RL) = Vt

where Vt denotes the current total value of the portfolio.

−1 If we can ﬁnd an analytical expression for the inverse function FL (y), then we can also ﬁnd an analytical expression for the VaR.

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Comments VaR at diﬀerent horizons

The Value-at-Risk is deﬁned for a certain time horizon (as L is deﬁned over a a certain time period [t, t + ∆t]) If ∆t is one day, then we talk about a one-day VaR. If ∆t is h days, then we talk about a h-day VaR.

In market risk, the typical horizon is 1 day, 5 days, or 10 days.

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Linearity of the Value-at-Risk

Lineraity Let a ∈ R and b > 0 constants. Then,

VaRα (a + b · L) = a + b · VaRα (L)

Proof:

VaRα (a + bL) = inf {x ∈ R : P (a + bL ≤ x) ≥ α} x − a = inf x ∈ : P L ≤ ≥ α R b = inf {a + bx˜ ∈ R : P (L ≤ x˜) ≥ α} = a + b inf {x˜ ∈ R : P (L ≤ x˜) ≥ α} = a + bVaRα (L)

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Analytical VaR when L is normally distributed

Let L be normally distributed with mean µ and variance σ2, that is, L ∼ N µ, σ2. Then x − µ F (x) = Φ L σ where Φ (z) is the cdf for a standard normal random variable, i.e.,

Z z 2 1 − y Φ(z) = √ e 2 dy −∞ 2π

−1 So FL (y) is obtained by solving for x in the equation y = FL (x) x − µ x − µ Φ = y ⇔ = Φ−1 (y) ⇔ x = µ + σΦ−1 (y) σ σ

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Analytical VaR when L is normally distributed

So, the Value-at-Risk at conﬁdence level α is

−1 VaRα (L) = µ + σΦ (α)

where Φ−1 (y) can be obtained from tables or in any software such as Matlab, Excel, Stata, etc. (in Matlab, write icdf(’norm’,y,0,1)).

Some often used values are: α Φ−1 (α) 0.999 3.090 0.995 2.576 0.99 2.326 0.95 1.645 0.9 1.282

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Example: portfolio of stocks

Consider a portfolio of US stocks, which invests 10,000$ in each of the following ﬁve stocks (the total value of the portfolio is 50,000$): Apple, Coca-Cola, Exxon Mobil, General Electric, and Procter & Gamble.

Let us calculate the 5-day VaR and ES with α = 0.95 using the assumption that the return on the ﬁve assets follow a multivariate normal distribution.

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Example: portfolio of stocks Multivariate normal returns Means, standard deviations, and correlations of the weekly (five-day) returns on these stocks using the sample Jan 2, 1986 to Jun 29, 2018: Mean Std. Dev. Correlations 1. 2. 3. 4. 5. 1. Coca-Cola 0.30 3.30 1.00 0.40 0.42 0.18 0.54 2. Exxon 0.26 3.00 1.00 0.47 0.18 0.38 3. GE 0.24 4.14 1.00 0.28 0.40 4. Apple 0.60 6.39 1.00 0.12 5. P&G 0.28 3.19 1.00 Means and standard deviations are expressed in % per five trading days. Let us assume that the five return series follow a multivariate normal distribution R ∼ N(µ, Σ) where µ is the mean vector from the above table and Σ is the variance-covariance matrix based on the above descriptives. Quantitative Finance, Lecture 8 15 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

Example: portfolio of stocks Multivariate normal returns What is the ﬁve-day VaR with α = 0.95 The distribution of the portfolio returns is

2 RV ∼ N µp, σp with 0 µp = w µ = 0.0033 √ 0 σp = w Σw = 0.027 where w 0 = [0.2 0.2 0.2 0.2 0.2]. The loss is L = −RV Vt (with Vt = 50000), so 2 2 L ∼ N −µpVt , σpVt Therefore,

−1 VaRα(L) VaRα(L) = −µpVt + σpVt Φ (0.95) = 2057.1 and = 4.11% Vt

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Approximate-analytical solutions

Unfortunately, if we go outside the multivariate-normal framework, we quickly lose the possibility of analytical solutions. Numerical techniques (see the next lecture) come handy in these cases.

One important case: when we have derivatives in the portfolio, it is not easy to derive the distribution of L.

Even if the return on the underlying asset is assumed to be normally distributed, the return on the derivatives is highly non-normal.

In these cases we can try to come up with approximate-analytical solutions.

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Example Continuing from the previous lecture Consider a portfolio with three assets: S&P 500 index (you can think of a well-diversiﬁed US stock portfolio) Call option on the S&P 500 (strike price: 2900, maturity: 6 months) Put option on the S&P 500 (strike price: 2750, maturity: 6 months)

Asset Pi ($) ni ni Pi ($)

S&P 500 2800 100 280,000 Call option 126.8 50 6,339 Put option 120.0 150 18,003

Portfolio value (Vt ) 304,342

Note: The volatility of the S&P 500 is assumed to be 20%, while the risk-free rate is 2% (both yearly). Quantitative Finance, Lecture 8 18 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

Delta-Normal VaR

Let us take the ﬁrst order (delta-) approximation of the loss

∂V L = −∆V = − ∆S = −δ · ∆S ∂S V Assume that stock returns are normally distributed

∆S 2 2 2 ∼ N µS , σS ⇔ ∆S ∼ N St µS , St σS St

Then, VaRα(∆S) can be calculated using the formula from the previous lecture, and VaRα(L) = −δV VaRα(∆S)

BUT be careful with the sign of δV !!

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Delta-Normal VaR

δV < 0 δV > 0

200 200

100 100

L 0 L 0

-100 -100

-200 -200 S S pdf of pdf of

-200 -100 0 100 200 -200 -100 0 100 200 S S We need Φ−1 (α) We need Φ−1 (1 − α)

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Delta-Normal VaR We need ( −1 St µS + St σS Φ (1 − α) if δV ≥ 0 −1 St µS + St σS Φ (α) if δV < 0 The Delta-Normal VaR is ( −δ S µ − δ S σ Φ−1 (1 − α) if δ ≥ 0 VaR (L) = V t S V t S V α −1 −δV St µS − δV St σS Φ (α) if δV < 0

But also note that since the normal distribution is symmetric, Φ−1 (1 − α) = −Φ−1 (α) , so

Delta-Normal VaR ( −δ S µ + δ S σ Φ−1 (α) if δ ≥ 0 VaR (L) = V t S V t S V α −1 −δV St µS − δV St σS Φ (α) if δV < 0

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Delta-Normal VaR Special cases If ∆S ∼ N 0, σ2 , then St S ( δ S σ Φ−1 (α) if δ ≥ 0 VaR (L) = V t S V α −1 −δV St σS Φ (α) if δV < 0

If ∆S ∼ N 0, σ2 and you have only one stock in your portfolio St S (implying δV = 1), then

−1 VaRα(L) −1 VaRα(L) = St σS Φ (α) or = σS Φ (α) St If ∆S ∼ N 0, σ2 and you have n stocks in your portfolio (implying St S δV = n), then

−1 −1 VaRα(L) −1 VaRα(L) = nSt σS Φ (α) = Vt σS Φ (α) or = σS Φ (α) Vt

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Horizon

We start from the assumption that

∆S 2 ∼ N µS , σS St We can easily take into account the horizon by choosing the parameters of the distribution appropriately, so that they correspond to returns over the horizon [t, t + ∆t]. This will also be true when we use the t-distribution instead (later in this lecture).

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Delta-Normal VaR Example

Consider our example portfolio with three assets: S&P 500 index (you can think of a well-diversiﬁed US stock portfolio) Call option on the S&P 500 (strike price: 2900, maturity: 6 months) Put option on the S&P 500 (strike price: 2750, maturity: 6 months)

The yearly volatility of the S&P 500 is assumed to be 20%.

The VaR parameters: We look at a 5-day horizon. We use the conﬁdence level of 95% (α = 0.95). The 5-day return on the S&P 500 is normally distributed: ∆S ∼ N 0, σ2 St S q 5 σS = 0.2 · 250

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Delta-Normal VaR Example In our example, the 5-day VaR with α = 0.95 can be calculated as (note that δV > 0) −1 VaR0.95 (∆S) = St · σS · Φ (0.05) = 2800 · 0.0283 · (−1.645) = −130.3 and

VaR0.95 (L) = −δV VaR0.95 (∆S) = (−63.8) · (−130.3) = 8307.2 Equivalently −1 VaR0.95 (L) = δV St σS Φ (0.95) = 63.8 · 2800 · 0.0283 · 1.645 = 8307.2

This also means VaR0.95(L) =2 .73% Vt

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Delta-Normal VaR Example 10000 8307.2

5000 V 0

-5000 S pdf of

-200 -100 0 100 200 Quantitative Finance, Lecture 8 S 26 / 56 Linear assets (e.g., the portfolio contains only the S&P 500). Then the delta-approximation is exact.

Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

Delta-Normal VaR When is it a good approximation?

As you can see, in our example, the delta-approximation is not too accurate. In what cases does this approximation work well?

Short period VaR (e.g., 1-day). Because for a shorter the period, smaller movements in ∆S are expected.

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Delta-Normal VaR When is it a good approximation? 10000

5000 3715.1 V 0

-5000 S pdf of

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Delta-Normal VaR When is it a good approximation?

As you can see, in our example, the delta-approximation is not too accurate. In what cases does this approximation work well?

Short period VaR (e.g., 1-day). Because for a shorter the period, smaller movements in ∆S are expected.

Linear assets (e.g., the portfolio contains only the S&P 500). Then the delta-approximation is exact.

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Delta-Gamma-Normal VaR

We can improve the accuracy if we use a Delta-Gamma approximation instead ∂V 1 ∂2V L = −∆V = − ∆S + (∆S)2 ∂S 2 ∂S2 1 = − δ · ∆S + γ (∆S)2 V 2 V

But, (∆S)2 is a squared normally distributed (chi-squared) variable ⇒ NOT normally distributed. Thus, we cannot use the analytical formula we used previously. But we can calculate the VaR in two steps.

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Delta-Gamma-Normal VaR

First, calculate the VaR for the stock price changes, but again you have to be careful, which tail of the distribution you look at:

−1 VaR0.95 (∆S) = St · σS · Φ (1 − α) = −130.3

Then use this value to calculate the portfolio VaR:

1 VaR (L) = − δ · VaR (∆S) + γ (VaR (∆S))2 0.95 V 0.95 2 V 0.95 = 6645.5

VaR (L) 0.95 =2 .18% Vt

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Delta-Gamma-Normal VaR

10000

6645.5 5000 V 0

-5000 S pdf of

-200 -100 0 100 200 S

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Full-Valuation-Normal VaR

We can take the previous idea one step further:

First, calculate VaR0.95 (∆S) as in the case of the Delta-Gamma approximation. Second, calculate the price of all assets in the portfolio assuming that the stock price drops to this value. Third, calculate the value of the portfolio in this scenario, and the associated loss. This is called the Full-Valuation approach. In our example, the VaR with this approach is

VaR0.95 (L) = 6601.8 VaR (L) 0.95 =2 .17% Vt

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Full-Valuation-Normal VaR

10000

6601.8 5000 V 0

-5000 S pdf of

-200 -100 0 100 200 S

Quantitative Finance, Lecture 8 34 / 56 Example: S&P 500 from 2010 to 2017

0.6% 30.2% 30.2% 0.6% 1.8% 20.5% 23.7% 1.2% 38.4% 52.8%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall t-distribution How reasonable is the assumption of normally distributed returns? Many ﬁnancial data series have excess kurtosis or fat tails: The empirical distribution is more peaked than the normal distribution. The empirical distribution has heavier tails than the normal distribution. This means that small changes and large changes are more likely than the normal distribution would suggest.

Quantitative Finance, Lecture 8 35 / 56 0.6% 30.2% 30.2% 0.6% 1.8% 20.5% 23.7% 1.2% 38.4% 52.8%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

0.6% 30.2% 30.2% 0.6%

38.4%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

Quantitative Finance, Lecture 8 35 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall t-distribution How reasonable is the assumption of normally distributed returns? Many ﬁnancial data series have excess kurtosis or fat tails: The empirical distribution is more peaked than the normal distribution. The empirical distribution has heavier tails than the normal distribution. This means that small changes and large changes are more likely than the normal distribution would suggest. Example: S&P 500 from 2010 to 2017

0.6% 30.2% 30.2% 0.6% 1.8% 20.5% 23.7% 1.2% 38.4% 52.8%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

Quantitative Finance, Lecture 8 35 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall t-distribution

The t-distribution has lots of applications in statistics.

Characteristics: Symmetric and bell-shaped (like the normal distribution). Has heavier tails: i.e., it is more prone to producing values that fall far from its mean. ν Has a mean of zero and ν−2 variance (the variance is only deﬁned if ν > 2). Described by one parameter: ν, the degrees of freedom: Fat tails increase when degrees of freedom decreases. When degrees of freedom go to inﬁnity, t-distribution converges to normal distribution.

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Let us assume that the loss distribution can be described by a linear transformation of a t-distributed random variable, i.e.,

L = µ + ωT , with T ∼ t (ν)

The expected value and variance of the loss distribution are ν E (L) = µ , and Var (L) = ω2 ν − 2

Let gν (x) denote the density function (pdf) and tν (x) denote the distribution function (cdf) of T .

Note: This distribution is referred to as “t Location-Scale Distribution” in Matlab

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t-distribution Have a look at the S&P 500 example again. Consider a t-distributed variable, L, with q 2 ν−2 ν = 3, µ, and ω = σ ν Note that this implies E (L) = µ, and Var (L) = σ2

0.6% 30.2% 30.2% 0.6% 1.8% 20.5% 23.7% 1.2% 38.4% 52.8%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

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t-distribution Have a look at the S&P 500 example again. Consider a t-distributed variable, L, with q 2 ν−2 ν = 3, µ, and ω = σ ν Note that this implies E (L) = µ, and Var (L) = σ2

0.6% 30.2% 30.2% 0.6% 1.8% 20.5% 23.7% 1.2% 1.2% 21.3% 38.4% 21.3% 1.2% 52.8% 55.0%

µ − 2.5σ µ − .5σ µ + .5σ µ + 2.5σ

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Value-at-Risk Let us assume that the Loss distribution can be described as

L = µ + ωT , with T ∼ t (ν) The Value-at-Risk at conﬁdence level α is

−1 VaRα (L) = µ + ωtν (α)

−1 where tν (x) denotes the inverse of the distribution function of T . Values can be obtained from using, e.g., Matlab (write icdf(’t’,x,ν)). Some values with ν = 3: −1 α tν (α) 0.999 10.215 0.995 5.841 0.99 4.541 0.95 2.353 0.9 1.638

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Delta-t VaR

Let us take the delta-approximation of the loss L = −δV · ∆S Assume that stock return can be described by ∆S = µS + ωS T , with T ∼ t (ν) St Then, ∆S = St µS + St ωS T

We will use the linearity of VaR. Again, be careful with the sign of δV . The Delta-t VaR is ( −δ S µ − δ S ω t−1 (1 − α) if δ ≥ 0 VaR (L) = V t S V t S ν V α −1 −δV St µS − δV St ωS tν (α) if δV < 0

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Delta-t VaR

−1 −1 The t distribution is symmetric, tν (1 − α) = −tν (α) , so

Delta-t VaR ( −δ S µ + δ S ω t−1 (α) if δ ≥ 0 VaR (L) = V t S V t S ν V α −1 −δV St µS − δV St ωS tν (α) if δV < 0

The special cases we considered for the Delta-Normal VaR apply here similarly.

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Example The Delta-Gamma and Full Valuation approaches can be done similarly as in case of the normal distribution. In our example q 5 Normal distribution: µS = 0 and σS = 0.2 250 q ν−2 t distribution: µS = 0, ωS = σS ν , and ν = 3 Normal t α = 0.95 Delta 8307.2 2.73% 6862.1 2.25% Delta-Gamma 6645.5 2.18% 5728.2 1.88% Full Valuation 6601.8 2.17% 5702.2 1.87% α = 0.99 Delta 11749.0 3.86% 13240.0 4.35% Delta-Gamma 8425.1 2.77% 9019.0 2.96% Full Valuation 8319.1 2.73% 8878.5 2.92%

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Coherent Risk Measures Properties of VaR

Monotonicty If there are two portfolios with loss distributions L and L˜ such that h i P L ≤ L˜ = 1, then VaRα (L) ≤ VaRα L˜

If one portfolio always produces a bigger loss than another its risk measure should be greater. In other words, if one portfolio always performs worse than another portfolio, it clearly should be viewed as more risky.

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Coherent Risk Measures Properties of VaR

Lineraity Let a ∈ R and b > 0 constants. Then,

VaRα (a + b · L) = a + b · VaRα (L)

Linearity has two important implications: Translation invariance: If we add an amount of cash K to a portfolio its risk measure should go down by K. The cash provides a buﬀer against losses and should reduce the capital requirement by K. Homogeneity: Changing the size of a portfolio by λ should result in the risk measure being multiplied by λ. If we double the size of the portfolio, we should require twice as much capital.

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Coherent Risk Measures Properties of VaR Subadditivity Consider two portfolios with loss distributions L and L˜. The risk measure RM satisﬁes subadditivity if RM L + L˜ ≤ RM (L) + RM L˜

This captures a diversiﬁcation eﬀect, saying that the riskiness should not increase when merging two portfolios.

VaR as a risk measure does not fulﬁll subadditivity, i.e., we can ﬁnd examples when VaRα L + L˜ > VaRα (L) + VaRα L˜

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VaR satisﬁes the ﬁrst three conditions but not the fourth one, i.e., it is not a coherent risk measure.

Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

Coherent Risk Measures

Coherent risk measures A risk measure is coherent if it satisﬁes the following properties: 1 Monotonicity 2 Translation invariance 3 Homogeneity 4 Subadditivity

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Coherent Risk Measures

Coherent risk measures A risk measure is coherent if it satisﬁes the following properties: 1 Monotonicity X 2 Translation invariance X 3 Homogeneity X 4 Subadditivity ×

VaR satisﬁes the ﬁrst three conditions but not the fourth one, i.e., it is not a coherent risk measure.

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Expected Shortfall Formalization Recall the following L is the loss of a portfolio during the period [t, t + ∆t].

FL (x) = P (L ≤ x) is the loss distribution.

Expected Shortfall (ES) Let L be a loss which is a continuous random variable with distribution function FL (x). Then, for any conﬁdence level α ∈ (0, 1), the expected shortfall is ESα (L) = E [L | L ≥ VaRα (L)]

ES asks: “If things do get bad, what is the expected loss?” ES is also sometimes referred to as Conditional Value-at-Risk (CVaR), or Expected Tail Loss

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ES satisﬁes all four conditions, i.e., it is a coherent risk measure.

Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

ES is a coherent risk measures

Coherent risk measures A risk measure is coherent if it satisﬁes the following properties: 1 Monotonicity 2 Translation invariance 3 Homogeneity 4 Subadditivity

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ES is a coherent risk measures

Coherent risk measures A risk measure is coherent if it satisﬁes the following properties: 1 Monotonicity X 2 Translation invariance X 3 Homogeneity X 4 Subadditivity X

ES satisﬁes all four conditions, i.e., it is a coherent risk measure.

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ES when L is normally distributed Let L be normally distributed with mean µ and variance σ2, that is, L ∼ N µ, σ2. It can be shown that x−µ φ σ E[L | L > x] = µ + σ x−µ 1 − Φ σ We also know that

ESα (L) = E [L | L ≥ VaRα (L)] −1 So, x = VaRα (L) = µ + σΦ (α), implying φ Φ−1 (α) ES (L) = µ + σ α 1 − Φ (Φ−1 (α)) φ Φ−1 (α) = µ + σ 1 − α where φ (x) is the probability density function of the standard normal distribution. Quantitative Finance, Lecture 8 49 / 56 Value-at-Risk Normal VaR Approximate - Normal VaR t VaR Coherent risk measures Expected shortfall

ES when L is normally distributed

The φ (x) values can be obtained from tables or in any software such as Matlab, Excel, Stata, etc. (in Matlab, write normpdf(x)). Some helpful values:

φ(Φ−1(α)) α 1−α 0.999 3.367 0.995 2.892 0.99 2.665 0.95 2.063 0.9 1.755

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Delta-Normal ES

Let us take the ﬁrst order (delta-) approximation of the loss

∂V L = −∆V = − ∆S = −δ · ∆S ∂S V Assume that stock returns are normally distributed

∆S 2 2 2 ∼ N µS , σS ⇔ ∆S ∼ N St µS , St σS St

Then, ESα(∆S) can be calculated analytically and then

ESα(L) = −δV ESα(∆S)

BUT be careful with the sign of δV !!

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Delta-Normal ES

δV < 0 δV > 0

200 200

100 100

L 0 L 0

-100 -100

-200 -200 S S pdf of pdf of

-200 -100 0 100 200 -200 -100 0 100 200 S S We need E[∆S | ∆S > x] We need E[∆S | ∆S < x]

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Delta-Normal ES Without going into the details of the derivation here

Delta-Normal ES  φ(Φ−1(α)) −δ S µ + δ S σ if δ ≥ 0 ES (L) = V t S V t S 1−α V α φ(Φ−1(α)) −δV St µS − δV St σS 1−α if δV < 0

You can easily derive the special cases we considered for the Delta-Normal VaR. In our example:

α = 0.95 α = 0.99 VaR 8307.2 11749.0 ES 10417.5 13460.4

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Comments

Important: the Delta-Gamma and Full Valuation approaches do not work for the ES.

If you assume ∆S ∼ N 0, σ2 , as we did today, then the delta-normal St S method gives −1 VaRα(L) = δV St σS Φ (α) φ(Φ−1(α)) ESα(L) = δV St σS 1−α The VaR and the ES are tightly linked, so the ES does not provide too much additional information. This is due to the normality assumption. If you have more general models, the ES can provide useful extra information.

Similarly, you can derive an analytical formula for ES, when returns follow a t-distribution (we will not do it now).

Quantitative Finance, Lecture 8 54 / 56 Slides that are important for the exam

Introducing VaR Basic concepts and notation: Slides 4-5 Know and understand the deﬁnition of VaR: Slides 6, 8-10 Linearity of VaR (be able to provide the proof also): Slide 11

Analytical VaR calculations When L is normally distributed: Slides 12-13 Delta-Normal VaR (be able to derive the formula and apply it): Slides 19-25 Delta-Gamma-Normal VaR (be able to calculate it): Slides 30-31 Full-Valuation-Normal VaR (understand the logic and be able to implement): Slide 33 t-distribution (understand the formulation): Slides 36-39 Delta-t VaR (be able to derive the formula and apply it): Slides 40-41 Delta-Gamma-t and Full-Valuation-t VaR (be able to calculate it): not on slides but the procedure is exactly the same as for the normal case

Quantitative Finance, Lecture 8 55 / 56 Slides that are important for the exam

Coherent risk measures (know the properties that deﬁne a coherent RM, be able to give an economic interpretation of them): Slides 43-46, 48

Expected Shortfall Deﬁnition of ES (know and understand): Slide 47 ES when L is normally distributed (know the formula and be able to apply it): Slide 49-50 Delta-Normal ES (be able to derive the formula and apply it): Slides 51-54

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