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Non-Gaussian Financial Mathematics 2 AIMS 2011 William Shaw University College London This talk: Risk Measures, Portfolios & T Distribution Technology

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Overview Last time we looked at some of the properties of the statistics associated with single financial variable, with a focus on equity indices that were readily available in our computing environment. We looked at some simple properties of the distributions, in particular their moments, their tails and the use of Maximum Likelihood Estimation. Now we want to look at related properties of distributions that are related to financial risk measures. Some of these can be boiled down to simple functions of the moments, others not. Sometimes there is a reduction to moments for the Gaussian case but not for others.

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ü More than one dimension We also will introduce some multivariate aspects at the same time. More often than not we will want to assess the risk of a portfolio of similar assets, or indeed the overall position of a bank in many different assets. For simplicity the portfolio model will be considered. This will keep matters simple and also lay some foundations for portfolio optimization when returns are nonGaussian and/or our risk functions are not just simple functions of moments. In the previous note we defined mean, variance, skewness and kurtosis associated with the distribution of a single random variable. First we summarize some multivariate extensions relevant to a portfolio.

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Rationale The univariate Gaussian distribution is characterized by just two parameters, the mean and the variance. Many risk functions start off as explicit functions of just these two, or can be boiled down to functions of the two using some calculus. When we go non-Gaussian, obviously risk objects based explicitly on moments remain so, but other distributionally-based measures are not always obviously reducible to moment functions. Life will be easier when we can do so! We will quickly survey some risk functions of interest, so we all are clear on terminology.

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Portfolio Mean, Variance and Covariance In the previous note we defined mean, variance, skewness and kurtosis associated with the distribution of a single random variable. Now we summarize the multivariate extension relevant to a portfolio. Suppose an investor with cash P0 distributes her investment at time t = 0 into N assets with cash proportions P0 wi . If the price of each individual asset (e.g. the share price) at time t is Si t , then the number of units of each asset bought is ni =

P0 wi

(1)

Si 0

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The value of this portfolio at a later time t is then simply N

N

Pt = ‚ni Sit = P0 ‚ i=1

i=1

wi Si0

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N

Sit = P0 ‚wi Rit

(2)

i=1

where the return Ri t = Si t ê Si 0 for each asset. Dividing by P0 we have that RPt =

Pt P0

N

= ‚wi Rit

(3)

i=1

So we work with portfolio returns as the weighted sum of asset returns. Note that this is a single-period view. The returns are treated as random variables. First we take expectations. By linearity, we have N

N

E@RPt D = RPt = ‚wi E@ Rit D = ‚wi Rit i=1

i=1

(4)

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Slide 7 of 40 For the portfolio variance we do a slightly more complicated but very standard manipulation. 2

N

N

Variance@RPt D ! EBIRPt - RPt M F ! ‚‚Ci j wi w j

(5)

i=1 j=1

where the covariance matrix Ci j is given by Ci j ! EAIRi t - Ri t M IR j t - R j t ME

(6)

This can also be rewritten as Ci j ! Si S j ri j

(7)

where the individual asset return variances are 2

S2i ! EBIRi t - Ri t M F

(8)

and ri j gives the correlation of asset returns.

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Slide 8 of 40 Be careful not to confuse these things with their counterparts in Brownian motions under log-normal assumptions. If the assets, in our by now old-fashioned view, evolve according to dSi Si

= mi dt + si dWt

(9)

then, for example, there are various transformations to be made, including, for example, if we stay exact, Ri t ! ‰ mi t S2i

!‰

2 mi t

(10) H‰

si2 t

- 1L

(11)

Clearly for small times, S2i ~ si2 t

(12)

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Sharpe Ratio An old portfolio (or asset) performance indicator introduced by William Sharpe is the ratio S=

EAR - R f E

(13)

s

where Rf is the return on a benchmark asset, such as the deposit rate. R is the return and s the standard deviation of returns. This is manifestly a simple function of the first two moments.

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Value at Risk - VaR Let the cumulative distribution function of the portfolio returns be F HxL. The signed Value at Risk, or sVaR is defined as a function of a percentile u, 0 § u § 1 by the equation FHsVaRL = u i.e. , in terms of the quantile function, or inverse CDF, Q (u) , of the distribution. sVaR = QHuL

(14)

If the portfolio distribution is Gaussian, mean m, std dev s, sVaR = QHuL = m + s F-1 HuL where F is the Gaussian CDF, F-1 the Gaussian quantile fn.

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ü Gaussian VaR Think about this - to make a sample from a normal distrbution we make a sample from a standard one, multiply by s, then add m. We usually take u small and certainly u < 1 ê 2, so the coefficient of s is negative. The minimization of loss is then the maximization of sVaR= QHuL (we "push it to the right"). It can confuse people when you talk about trying to maximize a figure associated with a loss, so we swap signs and minimize -sVaR which we will call the VaR, Value at Risk. So we are trying to minimize the loss, (the positive quantity) VaR = I- F-1 HuLM s - m

(16)

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ü Some numbers for VaR and Moments Link We would perhaps be interested in the values of u corresponding to percentiles 10%, 5%, 2.5%, 1%, 0.1%. uvals = 80.1, 0.05, 0.025, 0.01, 0.001<; The values of F-1 are obtained by applying the function QuantileN@u_D := Sqrt@2D InverseErf@2 u - 1D Map@QuantileN, uvalsD 8- 1.28155, - 1.64485, - 1.95996, - 2.32635, - 3.09023<

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Slide 13 of 40 „ The VaR controversy You should be aware that there is no consensus on the usefulness of Var. There are papers that discuss the matter, but you will even find the Wikipedia summary helpful for non-mathematical context: http://en.wikipedia.org/wiki/VaR In particular you might be interested in comments by Nassim Taleb to the effect that VaR gives false confidence and is charlatanism because it claims to estimate the risks of rare events, which is impossible. Others have described VaR as "an airbag that works all the time, except when you have a car accident." My own view is that the problem in part has more to do with too much dependence on the use of the normal distribution, with its very low probabilities of extreme events. There are other issues of a more mathematical nature, that we shall return to, on adding VaR for more than one portfolio.

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ü TVaR, CVaR, ETL This is the expected loss given that one is in the tail region to the left of the VaR a. If a = Q -1 HuL, we have a

TVaR =

Ÿ-¶ x f HxL „ x

(17)

a

Ÿ-¶ f HxL „ x

Some integration by parts gives a

TVaR = a -

Ÿ-¶ FHxL „ x FHaL

§ a = sVaR

This is also called the conditional value at risk or expected tail loss.

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Slide 15 of 40 In the particular case of a Gaussian model some further integration leads to (the signed quantity) TVar = m - gHuL s

(19)

where this time fIF-1 HuLM gHuL =

(20)

u

So the risk minimization problem this time is to minimize gHuL s - m

(21)

It is evident that in the Gaussian case the TVaR and VaR calculations are almost identical, differing only in the relative weights weight of s and m. The TVaR associated with a percentile u1 is equal to the VaR at a percentile u2 , where u2 ! F -

fHQHu1 LL

(22)

u1

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Slide 16 of 40 F@x_D := 1 ê 2 H1 + Erf@x ê Sqrt@2DDL; f@x_D := 1 ê Sqrt@2 PiD Exp@- x ^ 2 ê 2D; vtv@u_D := F@- 1 ê u f@QuantileN@uDDD [email protected] 0.01957 In the Gaussian case "the 5% tail VaR equals the 2% VaR". Don't confuse this type of equivalence with theoretical properties (see later). [email protected] 0.0019143

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ü One-sided variability ratios Farinelli and Tibiletti (2002) define a one-sided performance ratio as a function of a benchmark b as follows: FHb, p, qL =

EHHHX - bL+ L p L1êp

(23)

EHHHX - bL- Lq L1êq

This is a nice idea. We are looking at the probability of exceeding our target b divided by the probability of falling lower. We allow various powers to be taken of the two sides.

S. Farinelli and L. Tibiletti, Sharpe thinking with asymmetrical preferences, working paper, 2002.

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ü Omega First we set p = q = 1 FHb, 1, 1L =

EHHX - bL+ L EHHX - bL- L

(24)

Some integration by parts assuming a distribution function F HxL gives the expression ¶

FHb, 1, 1L =

Ÿb H1 - FHxLL „ x b

Ÿ-¶ FHxL „ x

(25)

which is then recognizable as the W function W(b), (see e.g. Cascon et al, 2002). This is often known as "Keating's Omega" due to its extensive development by Keating, one of its co-definers.

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ü Sortino ratio Similarly, the "Sortino ratio" is obtained by setting p = 1, q = 2. EHHX - bL+ L

RS HbL = FHb, 1, 2L =

(26)

E IHHX - bL- L2 M which is the mean upside divided by the downside standard deviation. A Cascon, C. Keating and W. Shadwick, 2002, An introduction to Omega, Finance Development Centre.

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ü Gaussian Omega Let X ~N(m, s). Let the normal density be f and CDF F , with F ' = f . Then some integration (left as an exercise for you!) WHbL =

f HzL + z FHzL - z f HzL + z FHzL

,

z = Hb - mL ê s

(27)

Check that RHS is a strictly decreasing function of z. Hence maximization of W is minimization of b-m s

(28)

for the portfolio, for each choice of threshold b. This is just the minimization over allowed choices of w, of b - r.w wT .cov.w r is the expected return and cov is the covariance matrix.

(29)

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Going non-Gaussian again... We have discussed a few risk measures and identified some of the simplifications that arise when the situation is Gaussian. So what happens when our distributions are non-Gaussian? We need a distributional model for single assets, portfolios and entire positions. I will focus on the Student T model. This is not to say I think it is universally the right distribution. It is a better approximation to reality than Gaussian, close to the most likely one, with power-law tail. We can look at analytical models and simulation. I will consider the analytics first as you get a lot of insight. Analogous questions for VG, Johnson-SU....

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Some Student T technology We are going to look at

ü The CDF ü The inverse CDF or Quantile function ü General inverse beta functions ü Special fast closed-form quantiles ü VaR ü CVaR There is a lot one can do analytically in this particular fat-tailed world.

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ü Theoretical backup Shaw, 2006, Journal of Computational Finance, “Sampling Studentʼs T distribution....” Steinbrecher and Shaw, 2008, European Journal of Applied Mathematics, “Quantile Mechanics” Study of quantiles pioneered by Hill many years ago - basis of some NAG algorithms. We will consider Baileyʼs 1990s extension of Box-Muller later.

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ü Recall the Student density In[104]:=

In[3]:=

Out[3]=

h[n_, t_] := Gamma[(n+1)/2]/Sqrt[Pi n]/Gamma[n/2]/ ((1+t^2/n)^((1/2)*(n+1))) h@1, tD 1 p It2 + 1M

In[5]:=

Out[5]=

Integrate@h@1, tD, 8t, - ¶, x<, Assumptions Ø Im@xD ã 0D tan-1 HxL p

+

1 2

This Cauchy distribution has easy CDF. Here are the next few:

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In[6]:=

Out[6]=

25

Slide 25 of 40 Integrate@h@2, tD, 8t, - ¶, x<, Assumptions Ø Im@xD ã 0D 1 2

In[7]:=

x

+1

x2 + 2

Integrate@h@3, tD, 8t, - ¶, x<, Assumptions Ø Im@xD ã 0D tan-1 J

3 x

Out[7]=

p Ix2 + 3M

+

p

x 3

N +

1 2

This pattern on the integer values repeats. Even n is algebraic function. Odd n is mixture of arctan and algebraic function. Makes inversion for quantile of odd n very hard.

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Slide 26 of 40

ü Even integer degrees of freedom In[8]:=

Integrate@h@4, tD, 8t, - ¶, x<, Assumptions Ø Im@xD ã 0D 3ê2

Out[8]=

x3 + Ix2 + 4M 2

+6x

3ê2

2 Ix + 4M In[9]:=

Out[9]=

Integrate@h@6, tD, 8t, - ¶, x<, Assumptions Ø Im@xD ã 0D 1 x I2 x4 + 30 x2 + 135M 4

5ê2

+2

Ix2 + 6M

We can write down a recursion (JCF 2006) to define the CDFs for even dof.

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Slide 27 of 40 Clear@aD; a@0, n_D := Gamma@Hn + 1L ê 2D ê Gamma@n ê 2D ê Sqrt@n PiD; a@k_, n_D := a@k, nD = Hn - 2 kL ê n ê H2 k + 1L a@k - 1, nD;

In[10]:=

In[12]:=

TCDF@n_, x_D := 1 ê 2 + x * Sum@a@p, nD x ^ H2 pL, 8p, 0, n ê 2 - 1
In[15]:=

Table@82 k, 1 ê 2 + Simplify@TCDF@2 k, xD - 1 ê 2D<, 8k, 1, 6
2 2

x Ix2 +6M

4

+

1 2

+

1 2

x2 +2 3ê2

2 Ix2 +4M

x I2 x4 +30 x2 +135M

6

5ê2

4 Ix2 +6M

Out[15]=

+

x Ix6 +28 x4 +280 x2 +1120M

8

7ê2

2 Ix2 +8M 8

6

4

1 2

+

1 2

2

x I8 x +360 x +6300 x +52 500 x +196 875M

10

9ê2

16 Ix2 +10M

+

1 2

x Ix10 +66 x8 +1782 x6 +24 948 x4 +187 110 x2 +673 596M

12

11ê2

2 Ix2 +12M

+

1 2

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Slide 28 of 40 In general, for real n, not necessarily an integer, the CDF has to be expressed in terms of hypergeometric or beta functions. In[26]:=

Out[26]=

Integrate@h@n, tD, 8t, - ¶, x<, Assumptions Ø 8Im@xD ã 0, n > 0
n+1 1 n+1 3 x2 M F J , 2 ; 2; - n N 2 2 1 2

p In[27]:=

Simplify@D@%, xDD n

Out[27]=

n

n GI 2 M

J

1

n+x2 - 2 - 2 N n

p

GI

n+1 M 2 n

n GI 2 M

+

1 2

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ü Beta function reresentation I claim (see Abramowitz and Stegun 26.7.1) In[55]:=

In[56]:=

Out[56]=

F@n_, x_D := 1 ê 2 H1 + Sign@xD * H1 - BetaRegularized@n ê Hx ^ 2 + nL, n ê 2, 1 ê 2DLL F@n, xD ê. Sign@xD Ø 1 1 2

In[57]:=

2-I

n 1 , 2 2

n x2 +n

pdfhopeful = PowerExpand@Simplify@D@%, xD DD n

1

2

2

- -

Out[57]=

nnê2 In + x2 M

n 1 BI 2 , 2 M

30

In[58]:=

Out[58]=

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Slide 30 of 40 Integrate@pdfhopeful, 8x, - ¶, ¶<, Assumptions Ø n > 0D 1 So we have a general CDF, albeit at the price of some “special-function” soup. 1

† BHa, bL = GHaL GHbL ê GHa + bL = Ÿ0 t a-1 H1 - tLb-1 dt. † Bz Ha, bL = Ÿ0z t a-1 H1 - tLb-1 dt.

The regularized beta-function we use is Bz Ha, bL ê BHa, bL

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Slide 31 of 40

Quantiles and VaR? It is not always straightforward to invert these CDFs. If you have a system armed with inverse beta functions you can write it down! In[87]:=

In[71]:=

TQuantile@u_, n_D := Module@8arg = If@u < 1 ê 2, 2 u, 2 H1 - uLD<, Sign@u - 1 ê 2D Sqrt@n * H1 ê InverseBetaRegularized@arg, n ê 2, 1 ê 2D - 1LDD GaussQuantile@u_D := Sqrt@2D InverseErf@2 Hu - 1 ê 2LD The graphics are illuminating....

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In[94]:=

In[100]:=

In[101]:=

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Slide 32 of 40 tplot = Plot@Evaluate@Table@TQuantile@y, kD, 8k, 1, 8
5

Out[101]=

0.2

-5

0.4

0.6

0.8

1.0

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ü Do we always have to use inverse Beta? This representation is awkward outside specialist math computing environments. Can we simplify it? We can ask for exact closed forms and useful numerical methods. The closed forms that are more accessible are limited to a few simple cases, viz, n=1,2,4. The reason for these numbers is that the odd n CDFs are a mixture of algebraic and arctan functions, unless n=1, when the CDF is just Out[5]=

tan-1 HxL p

+

1 2

with the simple inverse CauchyQuantile@u_D := Tan@Pi * Hu - 1 ê 2LD

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ü Even n? In this case the CDFs do not look too awful, and with a change of variable to p = n + x 2 the equation determining the quantile function, i.e. u = FHn, xL

(30)

can be reproduced to a polynomial equation in p of degree n-1 . The details are in my JCF paper, but the outcome is that these are solvable when n = 2, 4. For higher n, e.g. 6, we get a quintic polynomial equation, and so on, and Galois theory says we cannot write down a simple expression. When n = 2 we get a simple linear equation, and when n = 3 we obtain a cubic which is solvable by Tartagliaʼs method! We can boil these expressions down to simple functions.

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ü The known simple Student T quantiles (1,2,4) QH1, uL = tan p u -

In[102]:=

1

(31)

2

2u-1

QH2, uL =

(32)

2 u H1 - uL QH4, uL =

q - 4 sin u -

1

(33)

2

where 1

4 cosI 3 cos-1 I a MM

q=

(34)

a a = 4 u H1 - uL

(35)

These are very useful, especially n=4 as this was the number found by Fergusson and Platen for world indices.

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Slide 36 of 40

ü What do we do otherwise Hill, 1970, Algorithm 396, Studentʼs t-qauntiles, Comm ACM 13(1)), 619. Steinbrecher & Shaw ODE admits series solutions for middle and tail (truncated expressions in JCF paper). The ODE is w2 n

+1

!2 w 2

!u

!

1 n

+1 w

!w

2

!u

(36)

with conditions at the origin w

w£

1 2 1 2

! 0; n

=

p n GI 2 M n+1 GI 2 M

Useful idea but will not develop it here.....

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ü VaR is just a quantile! Having done all this work we can finally write down the VaR as it is just a quantile. We need to be a little careful as the variance of the standard T is not unity and we want to make like-with-like comparisons with the Gaussian. As with the Gaussian, we want an object which when we multiply by the standard deviation and add the mean, we have the VaR of the distribution. StudentVaRHm, s, n, uL ! -s QTnormHu, nL - m

QTnormHu, nL = In[103]:=

n-2 n

(38) (39)

TquantileHu, nL

QTnorm@u_, n_D := Sqrt@Hn - 2L ê nD Tquantile@u, nD We have the VaR in closed form for unit-variance-T.

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Slide 38 of 40

ü What about CVaR for the T? In fact we can now make a closed-form for the CVaR as well. The PDF is a function of x^2 and when we integrate it against x in the left tail the integral comes out... In[106]:=

Integrate@t * h@n, tD, 8t, - Infinity, x<, Assumptions Ø n > 1D 1

Out[106]=

nnê2 In + x2 M 2 2

-

n 2

GI

n-1 M 2

n

p GI 2 M

To get the expected value in the tail given that we are in the tail, we evaluate this expression for x at given quantile and then divide by u. That is all there is to it, apart from being careful again with normalizing variance to unity .

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We write the tail expectation as the function In[107]:=

Slide 39 of 40

metaf@t_, n_D := - Hn ^ Hn ê 2L * Hn + t ^ 2L ^ H1 ê 2 - n ê 2L * Gamma@H- 1 + nL ê 2DL ê H2 * Sqrt@PiD * Gamma@n ê 2DL Then the unit-variance Student T CVaR is just

In[108]:=

StuCVaR@u_, n_D := Sqrt@Hn - 2L ê nD metaf@TQuantile@u, nD, nD ê u

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Slide 40 of 40

Summary We have introduced some key ideas related to risk and portfolios. We have developed some technology allowing the management of CDFs, Quantiles, VaR and CVaR for the Student T distribution. Next time we will look at the insight into risk to be obtained using this technology.