Chapter 5-risk.pdf

8/16/2019 Chapter 5-risk.pdf

http://slidepdf.com/reader/full/chapter-5-riskpdf 1/37

1

Risk

• This class: Probability refresher• Some empirical properties of asset returns

Kay Giesecke



2

We live in an uncertain world

• We will relax the assumption of certain cash ows• We will consider a single period and assume that the cash ow at

the end of that period is uncertain

• We need to model this uncertainty

– Probability and statistics come into play

Kay Giesecke



3

Probability

A probability space (Ω

, F , P ) is a mathematical model for uncertainty• Ω is a set representing the possible states of the world

– A state or scenario is an element ω ∈Ω

– An event is a subset of Ω

• F is the set of events whose probability can be assessed

• P (E ) gives the probability of any event E ∈F

Kay Giesecke



4

Probability

Example: Flipping a coin• Ω = H, T

• F = Ω ,∅, H , T , the power set of Ω

• If the coin is fair, P [H ] = P [T ] = 12 (the uniform probability)

Kay Giesecke



5

Probability

Example: Rolling a die• Ω = 1, 2, 3, 4, 5, 6

• F is the power set of Ω (which has 26 elements)

– The event 2, 4, 6 is the event that an even number occurs

– The event 1, 2 is the event that a number < 3 occurs

• If the die is fair, P [i] = 16 for all i = 1 , . . . , 6 (again the uniform

probability)

Kay Giesecke



6

Probability

Random variable• A real-valued random variable X is a function from Ω into R

– We sometimes write X (ω) to make the dependence of X onthe state of the world ω ∈Ω explicit

• Indicator variable 1E of an event E ∈F :

1E (ω) =1 , ω ∈E (the event occurs)

0 , else (the event does not occur)

Kay Giesecke



7

Probability

Distribution function• X ∈R has distribution function

F X (x) = P [X ≤x], x∈R

• Properties of a distribution function– F X is non-decreasing

– limx ↓−∞ F X (x) = 0 and limx ↑∞ F X (x) = 1

– F X is right-continuous: F X (a) = lim x ↓ a F X (x)

Kay Giesecke



8

Probability

Distribution function

Kay Giesecke



9

Probability

Density function• X ∈R has density function f X if F X (x) = R

x−∞ f X (y)dy for

some function f (y) ≥0 such that R ∞

−∞ f (y)dy = 1

Probability function

• X ∈I has probability function pX if F X (x) = Pi ≤ x pX (i) where pX (i) = P [X = i] for i∈I

Kay Giesecke



10

Probability

Mean• The mean or expectation of X is a number denoted by E [X ]

– X has density f X : E [X ] = R xf X (x)dx

– X has probability function pX (i): E [X ] =

Pi ipX (i)

• Some properties– If X (ω) = c (that is, X is non-random), then E [X ] = c

– If X (ω) = aX 1(ω) + bX 2(ω) for a and b constants, then

E [X ] = aE [X 1] + bE [X 2]

Kay Giesecke



11

Probability

Variance• The variance of X is

Var [X ] = E [(X −E [X ])2] = E [X 2]−E [X ]2 ;

this is a measure of the dispersion of X around its mean

• The standard deviation is SD [X ] = p Var [X ]

• In nance, the standard deviation of a price or rate variable isoften called volatility

Kay Giesecke



12

Probability

Normal (Gaussian) distribution• A random variable X ∼N (µ, σ 2) is normally distributed with

mean µ∈R and standard deviation σ > 0 if it has density φµ, σ

given by

φµ, σ (x) = 1√ 2πσ 2 e− 1

2 σ 2 (x − µ)2

so the distribution is completely characterized by µ and σ2

• If µ = 0 and σ2 = 1 we speak of a standard normal variable

• The standard normal distribution function

Φ (y) = Z y

−∞φ0,1(x)dx

Kay Giesecke



13

Risk

• Probability refresher• This class: Some empirical properties of asset returns

Kay Giesecke



14

Asset return

• A security (or “asset”) costs X 0 today. The price of the asset atthe end of a xed holding period (a day, week, year etc.) is X 1(ω).

– X 1(ω) is a random variable

– At the end of the period, all uncertainty is resolved and thevalue of X 1(ω) is observed

• Returns over the holding period, assuming X 0 > 0

– The total return is the random variable R(ω) = X 1 ( ω )X 0

– The rate of return is the random variable r(ω) = X 1 ( ω )− X 0

X 0

For an asset with limited liability X 1(ω) ≥0 so thatr (ω) ≥ −1

Kay Giesecke



15

Asset return

Short selling (“shorting”)• This is a cash sale of an asset that you don’t own

– Borrow the asset for a xed period (broker)

– Sell asset on the market

– Buy back the asset from the market at then current prices atthe end of the lending period and return to the lender plus fee(and dividend, if any), or extend lending arrangement

• You bet that the asset price falls over the lending period

• If prices rise and you cannot extend your lending agreement, youhave to buy back at possibly unfavorable prices, causing you tomake a loss (no limited liability!). This is called a short squeeze .

Kay Giesecke



16

Asset return

Short sale

Kay Giesecke



17

Asset return

The compounding convention matters• If r t is the annual rate of return over [t, t + 1] , we have

X t +1 = (1 + r t )X t for t = 0 , 1, . . . : this corresponds to an accountthat grows at an annual rate of r t over [t, t + 1] withcompounding cycle 1 year

• We are interested in the return r t over [t, t + 1] under continuouscompounding, which satises X t +1 = er t X t . This leads to theconcept of the log-return

r t = logX t +1

X t ∈R

Kay Giesecke



18

Asset return

Diff

erent time periods• Partition your planning horizon (year, quarter etc.) in n periods

– A year has n = 52 · 5 −10 = 250 trading days

• Suppose the period log-returns r i are iid

• The log-return r over the planning horizon satiseser = er 1 · · · er n so r = P

ni=1 r i and

E [r ] =n

Xi=1

E [r i ] = nE [r 1]

The variance over the horizon satises Var [r ] = nVar [r 1] so

SD [r ] = √ n · SD [r 1]

Kay Giesecke



19

Asset return

Daily S&P 500 log-returns Jan 1950 to Oct 2011

!"#$ !"%$ !"&$ !"'$ !""$ ($$$ ($!$− (#

− ($

− !#

− !$

−#

$

#

!$

!#

) * +

− , - . / 0 1 3 4 - 0 5 - 1 . 6

789-

Kay Giesecke



20

Asset return

Histogram of daily S&P 500 log-returns Jan 1950 to Oct 2011, 463bins

− !" −!# −$" −$# −" # " $# $"#

$##

!##

%##

&##

"##

'##

(##

)##

Kay Giesecke



21

Asset return

Histogram: choosing the number of bins b• We have used the Diaconis-Freedman formula:

b =max( r i ) −min( r i )

h

where r i is the return and

h = 2 IQRn1/ 3

for n = 15547 , the number of samples

Kay Giesecke



22

Asset returnHistogram of daily S&P 500 log-returns Jan 1950 to Oct 2011, 463bins, tted normal distribution N (µ, σ 2)

− !" −!# −$" −$# −" # " $# $"#

$##

!##

%##

&##

"##

'##

(##

)##

Kay Giesecke



23

Asset returnHistogram of daily S&P 500 log-returns Jan 1950 to Oct 2011, 463bins, tted t-distribution t(µ, σ 2 , ν )

− !" −!# −$" −$# −" # " $# $"#

$##

!##

%##

&##

"##

'##

(##

)##

Kay Giesecke



24

Asset return

Daily S&P 500 log-returns Jan 1950 to Oct 2011• Assuming that the realized daily log-returns r i (ω) are samples

from a common return distribution, we estimate the mean µ of that distribution by the sample mean

µ(ω) = 1n

n

Xi =1r i (ω) = 0 .0279%

• We estimate the variance σ2 of the return distribution by thesample variance

σ 2(ω) = 1n −1

n

Xi=1

(r i (ω) −µ(ω))2 = 0 .00982

• These estimators are unbiased: E [µ] = µ and E [σ 2] = σ2 (check!)

Kay Giesecke



25

Asset returnMean blur: mean asset return estimates are noisy

• The sample mean is based on the two endpoints of the sample only

µ = 1n

n

Xi=1

r i = 1n

n − 1

Xi =0

logX i +1

X i=

1n

logn − 1

Yi =0

X i+1

X i != 1n

logX nX 0

• A measure for the accuracy of an estimator ˆ p of a parameter p isthe mean square error

MSE[ˆ p] = E [(ˆ p − p)2] = Var[ˆ p] + Bias[ˆ p]2

• For the unbiased estimate µ of µ, we get

MSE[µ] = Var[µ] = σ2

n ,

the square root of which is also known as the standard error

Kay Giesecke



26

Asset return

Standard error for the daily S&P returns• We have n = 15547 data points, so the estimated standard error is

σ√ n = 0 .0079%

– Compare with the daily mean return µ = 0 .028%• The error when estimating the mean a ff ects the theories we

consider below, such as mean-variance analysis and the CAPM,which require estimates of the mean return of securities

Kay Giesecke



27

Asset return

Sampling from a distribution• Let X be a random variable with distribution function F X (x) and

quantile function (inverse) q u (X )

• Perform the following two steps:

1. Generate a sample U from a standard uniform distribution2. Set Y = q U (X )

• One can show that Y and X have the same distribution:

P [Y

≤y] = F X (y)

• Thus, Y = q U (X ) is a sample of X

Kay Giesecke



28

Asset returnDistribution function F

X(x) and quantile function q

u(X )

Kay Giesecke



29

Asset returnDaily log-returns simulated from N (0.028%, 0.00982)

!"#$ !"%$ !"&$ !"'$ !""$ ($$$ ($!$−(#

−($

−!#

−!$

−#

$

#

!$

!#

) * +

− , - . / 0 1 3 4 - 0 5 - 1 . 6

789-

Kay Giesecke



30

Asset returnQQ plot of empirical vs. simulated returns

−!"#$ −!"# −!"%$ −!"% −!"!$ ! !"!$ !"% !"%$−!"&

−!"'

−!"#

−!"%

!

!"%

!"#

!"'

()*+,+-./ 123 −4567,8

9 + ) 7 / . 6 5 : ; 2 , ) . / 1 2 3

− 4 5

6 7 , 8

Kay Giesecke



31

Asset returnZoomed QQ plot of empirical vs. simulated returns

−!"!# −!"!$ −!"!% −!"!& −!"!' ! !"!' !"!& !"!% !"!$ !"!#−!"!(

−!"!)

−!"!$

−!"!&

!

!"!&

!"!$

!"!)

!"!(

*+,-.-/01 345 −6789.:

; - + 9 1 0 8 7 < = 4 . + 0 1 3 4 5

− 6 7

8 9 . :

Kay Giesecke



32

Asset returnQQ plot of empirical vs. simulated returns

Kay Giesecke



33

Asset returnA normal distribution is a poor t to the data

• The frequency of extreme events is grossly underestimated by thenormal; the tails of the normal are too light relative to the data

• For example, consider the 12% drop that occurred on BlackTuesday, 29 October 1929

• The normal predicts a daily loss of this magnitude once in every7.4 · 1031 years: if r i ∼N (0.028%, 0.00982), then

P [r i ≤ −0.12] = 5.39 · 10− 35

so a drop of at least 12% in the daily return occurs once in every1/ 5.39 · 10− 35 = 1 .86 · 1034 days or once in every 7.43 · 1031 years

• This is suspect in light of the 20% drop that occurred on blackMonday, only 58 years later

Kay Giesecke



34

Asset returnMore evidence: skewness and kurtosis

• With µn = E [(r i −µ)n ] the nth central moment of the iid periodreturn r i , the skewness S of r i is given by

S = µ3

σ 3 = E [(r i −µ)3]E [(r

i −µ)2]3/ 2

This is a measure of the asymmetry of the distribution: in thesymmetric case µ3 = 0 so S = 0

• The kurtosis K of r i is given by

K = µ4σ 4 = E [(r i −µ)4]

E [(r i −µ)2]2

This is a measure for the “peakedness” of the distribution: for anormal variable K = 3

Kay Giesecke



35

Asset returnMore evidence: skewness and kurtosis

• For the S&P 500 data we obtain

S = −1.048, K = 31 .22

• This is further evidence for the non-normality of returns

– The distribution is skewed to the left: more mass on lowerreturn values than implied by the normal

– The distribution is leptokurtic: more of the variance is due toinfrequent extreme deviations, as opposed to frequent

modestly-sized deviations. This is often loosely called a“fat-tailed” distribution.

Kay Giesecke



37

Asset returnCalculate volatility of asset returns over rolling periods of time

• At any point in time n, the realized volatility over a period of length K is

σn,K =

v uut1

K −1

n

Xi= n − K +1

(r i

−µn,K )2

for the rolling mean return

µn,K = 1K

n

Xi= n − K +1

r i

• Between Jan 1950 and Oct 2011, the S&P 500 index hasexperienced periods of low volatility of around 10%, and periods of high volatility of at least 20%

Kay Giesecke

Chapter 5-risk.pdf

Documents

Transcript of Chapter 5-risk.pdf