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of the random variables. In particular, notice that covariance comes up as a term

when computing the variance of the sum of two (not independent) random variables.

27

Hence, the variance operator is not, in general, a linear operator. That is, variance,

in general, is not additive.

It is worthwhile to go through the proofs of these results, at least for the case of

discrete random variables. Let X and Y be discrete random variables. Then,

XX

E[aX + bY ] = (ax + by) Pr(X = x, Y = y)

xâˆˆSX yâˆˆSy

XX XX

= ax Pr(X = x, Y = y) + bx Pr(X = x, Y = y)

xâˆˆSX yâˆˆSy xâˆˆSX yâˆˆSy

X X X X

=a x Pr(X = x, Y = y) + b y Pr(X = x, Y = y)

xâˆˆSX yâˆˆSy yâˆˆSy xâˆˆSX

X X

=a x Pr(X = x) + b y Pr(Y = y)

xâˆˆSX yâˆˆSy

= aE[X] + bE[Y ] = aÂµX + bÂµY .

Furthermore,

E[(aX + bY âˆ’ E[aX + bY ])2 ]

var(aX + bY ) =

E[(aX + bY âˆ’ aÂµX âˆ’ bÂµY )2 ]

=

E[(a(X âˆ’ ÂµX ) + b(Y âˆ’ ÂµY ))2 ]

=

a2 E[(X âˆ’ ÂµX )2 ] + b2 E[(Y âˆ’ ÂµY )2 ] + 2 Â· a Â· b Â· E[(X âˆ’ ÂµX )(Y âˆ’ ÂµY )]

=

a2 var(X) + b2 var(Y ) + 2 Â· a Â· b Â· cov(X, Y ).

=

2.5.3 Linear Combination of two Normal random variables

The following proposition gives an important result concerning a linear combination

of normal random variables.

Proposition 39 Let X âˆ¼ N (ÂµX , Ïƒ 2 ), Y âˆ¼ N(ÂµY , Ïƒ 2 ), Ïƒ XY = cov(X, Y ) and a and

X Y

b be constants. De& the new random variable Z as

ne

Z = aX + bY.

Then

Z âˆ¼ N(ÂµZ , Ïƒ 2 )

Z

where

ÂµZ = aÂµX + bÂµY

Ïƒ 2 = a2 Ïƒ 2 + b2 Ïƒ 2 + 2abÏƒ XY

Z X Y

28

This important result states that a linear combination of two normally distributed

random variables is itself a normally distributed random variable. The proof of the

result relies on the change of variables theorem from calculus and is omitted. Not all

random variables have the property that their distributions are closed under addition.

3 Multivariate Distributions

The results for bivariate distributions generalize to the case of more than two random

variables. The details of the generalizations are not important for our purposes.

However, the following results will be used repeatedly.

Linear Combinations of N Random Variables

3.1

Let X1 , X2 , . . . , XN denote a collection of N random variables with means Âµi ,variances

Ïƒ 2 and covariances Ïƒ ij . De& the new random variable Z as a linear combination

ne

i

Z = a1 X1 + a2 X2 + Â· Â· Â· + aN XN

where a1 , a2 , . . . , aN are constants. Then the following results hold

ÂµZ = E[Z] = a1 E[X1 ] + a2 E[X2 ] + Â· Â· Â· + aN E[XN ]

N N

X X

= ai E[Xi ] = ai Âµi .

i=1 i=1

Ïƒ 2 = var(Z) = a2 Ïƒ 2 + a2 Ïƒ 2 + Â· Â· Â· + a2 Ïƒ 2

Z 11 22 NN

+2a1 a2 Ïƒ 12 + 2a1 a3 Ïƒ 13 + Â· Â· Â· + a1 aN Ïƒ 1N

+2a2 a3 Ïƒ 23 + 2a2 a4 Ïƒ 24 + Â· Â· Â· + a2 aN Ïƒ 2N

+Â·Â·Â· +

+2aNâˆ’1 aN Ïƒ (N âˆ’1)N

In addition, if all of the Xi are normally distributed then Z is normally distributed

with mean ÂµZ and variance Ïƒ 2 as described above.

Z

3.1.1 Application: Distribution of Continuously Compounded Returns

Let Rt denote the continuously compounded monthly return on an asset at time t.

Assume that Rt Ëœ iid N (Âµ, Ïƒ 2 ). The annual continuously compounded return is equal

the sum of twelve monthly continuously compounded returns. That is,

11

X

Rt (12) = Rtâˆ’j .

j=0

29

Since each monthly return is normally distributed, the annual return is also normally

distributed. In addition,

" 11 #

X

E[Rt (12)] = E Rtâˆ’j

j=0

11

X

E[Rtâˆ’j ] (by linearity of expectation)

=

j=0

11

X

Âµ (by identical distributions)

=

j=0

= 12 Â· Âµ,

so that the expected annual return is equal to 12 times the expected monthly return.

Furthermore,

Ãƒ 11 !

X

var(Rt (12)) = var Rtâˆ’j

j=0

11

X

var(Rtâˆ’j ) (by independence)

=

j=0

X

11

Ïƒ 2 (by identical distributions)

=

j=0

= 12 Â· Ïƒ 2 ,

so that the annual variance is also equal to 12 times the monthly variance2 . For the

annual standard deviation, we have

âˆš

SD(Rt (12)) = 12Ïƒ.

4 Further Reading

Excellent intermediate level treatments of probability theory using calculus are given

in DeGroot (1986), Hoel, Port and Stone (1971) and Hoag and Craig (19xx). Inter-

mediate treatments with an emphasis towards applications in & nance include Ross

(1999) and Watsom and Parramore (1998). Intermediate textbooks with an emphasis

on econometrics include Amemiya (1994), Goldberger (1991), Ramanathan (1995).

Advanced treatments of probability theory applied to & nance are given in Neftci

(1996). Everything you ever wanted to know about probability distributions is given

Johnson and Kotz (19xx).

2

This result often causes some confusion. It is easy to make the mistake and say that the annual

variance is (12)2 = 144 time the monthly variance. This result would occur if RA = 12Rt , so that

var(RA ) = (12)2 var(Rt ) = 144var(Rt ).

30

5 Problems

Let W, X, Y, and Z be random variables describing next year s annual return on

Weyerhauser, Xerox, Yahoo! and Zymogenetics stock. The table below gives discrete

probability distributions for these random variables based on the state of the economy:

State of Economy W p(w) X p(x) Y p(y) Z p(z)

Depression -0.3 0.05 -0.5 0.05 -0.5 0.15 -0.8 0.05

Recession 0.0 0.2 -0.2 0.1 -0.2 0.5 0.0 0.2

Normal 0.1 0.5 0 0.2 0 0.2 0.1 0.5

Mild Boom 0.2 0.2 0.2 0.5 0.2 0.1 0.2 0.2

Major Boom 0.5 0.05 0.5 0.15 0.5 0.05 1 0.05

â€¢ Plot the distributions for each random variable (make a bar chart). Comment

on any diï¬€erences or similarities between the distributions.

â€¢ For each random variable, compute the expected value, variance, standard de-

viation, skewness, kurtosis and brie! y comment.

Suppose X is a normally distributed random variable with mean 10 and variance

24.

â€¢ Find Pr(X > 14)

â€¢ Find Pr(8 < X < 20)

â€¢ Find the probability that X takes a value that is at least 6 away from its mean.

â€¢ Suppose y is a constant de&ned such that Pr(X > y) = 0.10. What is the value

of y?

â€¢ Determine the 1%, 5%, 10%, 25% and 50% quantiles of the distribution of X.

Let X denote the monthly return on Microsoft stock and let Y denote the monthly

return on Starbucks stock. Suppose XËœN (0.05, (0.10)2 ) and Y ËœN(0.025, (0.05)2 ).

â€¢ Plot the normal curves for X and Y

â€¢ Comment on the risk-return trade-oï¬€s for the two stocks

Let R denote the monthly return on Microsoft stock and let W0 denote ini-

tial wealth to be invested in Microsoft stock over the next month. Assume that

RËœN (0.07, (0.12)2 ) and that W0 = $25, 000.

â€¢ What is the distribution of end of month wealth W1 = W0 (1 + R)?

â€¢ What is the probability that end of month wealth is less than $20,000?

â€¢ What is the Value-at-Risk (VaR) on the investment in Microsoft stock over the

next month with 5% probability?

31

References

[1] Amemiya, T. (1994). Introduction to Statistics and Econometrics. Harvard Uni-

versity Press, Cambridge, MA.

[2] Goldberger, A.S. (1991). A Course in Econometrics. Harvard University Press,

Cambridge, MA.

[3] Hoel, P.G., Port, S.C. and Stone, C.J. (1971). Introduction to Probability Theory.

Houghton Miï¬„in, Boston, MA.

[4] Johnson, x, and Kotz, x. Probability Distributions, Wiley.

[5] Neftci, S.N. (1996). An Introduction to the Mathematics of Financial Derivatives.

Academic Press, San Diego, CA.

[6] Ross, S. (1999). An Introduction to Mathematical Finance: Options and Other

Topics. Cambridge University Press, Cambridge, UK.

[7] Watsham, T.J., and Parramore, K. (1998). Quantitative Methods in Finance.

International Thompson Business Press, London, UK.

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