<<

. 2
( 2)



linear combination of random variables is not a linear combination of the variances
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)
x∈SX y∈Sy
XX XX
= ax Pr(X = x, Y = y) + bx Pr(X = x, Y = y)
x∈SX y∈Sy x∈SX y∈Sy
X X X X
=a x Pr(X = x, Y = y) + b y Pr(X = x, Y = y)
x∈SX y∈Sy y∈Sy x∈SX
X X
=a x Pr(X = x) + b y Pr(Y = y)
x∈SX y∈Sy

= aE[X] + bE[Y ] = aµX + bµY .
Furthermore,


E[(aX + bY ’ E[aX + bY ])2 ]
var(aX + bY ) =
E[(aX + bY ’ aµX ’ bµY )2 ]
=
E[(a(X ’ µX ) + b(Y ’ µY ))2 ]
=
a2 E[(X ’ µX )2 ] + b2 E[(Y ’ µY )2 ] + 2 · a · b · E[(X ’ µX )(Y ’ µY )]
=
a2 var(X) + b2 var(Y ) + 2 · a · b · 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 ∼ N (µX , σ 2 ), Y ∼ N(µY , σ 2 ), σ 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 ∼ N(µZ , σ 2 )
Z

where
µZ = aµX + bµY
σ 2 = a2 σ 2 + b2 σ 2 + 2abσ 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 µi ,variances
σ 2 and covariances σ ij . De& the new random variable Z as a linear combination
ne
i

Z = a1 X1 + a2 X2 + · · · + aN XN
where a1 , a2 , . . . , aN are constants. Then the following results hold


µZ = E[Z] = a1 E[X1 ] + a2 E[X2 ] + · · · + aN E[XN ]
N N
X X
= ai E[Xi ] = ai µi .
i=1 i=1



σ 2 = var(Z) = a2 σ 2 + a2 σ 2 + · · · + a2 σ 2
Z 11 22 NN
+2a1 a2 σ 12 + 2a1 a3 σ 13 + · · · + a1 aN σ 1N
+2a2 a3 σ 23 + 2a2 a4 σ 24 + · · · + a2 aN σ 2N
+··· +
+2aN’1 aN σ (N ’1)N
In addition, if all of the Xi are normally distributed then Z is normally distributed
with mean µZ and variance σ 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 ˜ iid N (µ, σ 2 ). The annual continuously compounded return is equal
the sum of twelve monthly continuously compounded returns. That is,
11
X
Rt (12) = Rt’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 Rt’j
j=0
11
X
E[Rt’j ] (by linearity of expectation)
=
j=0
11
X
µ (by identical distributions)
=
j=0
= 12 · µ,
so that the expected annual return is equal to 12 times the expected monthly return.
Furthermore,
à 11 !
X
var(Rt (12)) = var Rt’j
j=0
11
X
var(Rt’j ) (by independence)
=
j=0

X
11
σ 2 (by identical distributions)
=
j=0

= 12 · σ 2 ,
so that the annual variance is also equal to 12 times the monthly variance2 . For the
annual standard deviation, we have

SD(Rt (12)) = 12σ.


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
• Plot the distributions for each random variable (make a bar chart). Comment
on any di¬erences or similarities between the distributions.
• 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.
• Find Pr(X > 14)
• Find Pr(8 < X < 20)
• Find the probability that X takes a value that is at least 6 away from its mean.
• Suppose y is a constant de&ned such that Pr(X > y) = 0.10. What is the value
of y?
• 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 X˜N (0.05, (0.10)2 ) and Y ˜N(0.025, (0.05)2 ).
• Plot the normal curves for X and Y
• Comment on the risk-return trade-o¬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
R˜N (0.07, (0.12)2 ) and that W0 = $25, 000.
• What is the distribution of end of month wealth W1 = W0 (1 + R)?
• What is the probability that end of month wealth is less than $20,000?
• 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 Mi¬„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.




32

<<

. 2
( 2)