10 Mathematics for Finance
Suppose that you have $10,000 to invest in a portfolio. You decide to buy
x= 50 shares, which fixes the risk-free investment aty= 60. Then
V(1) =
{
11 ,600 if stock goes up,
9 ,600 if stock goes down,KV={
0 .16 if stock goes up,
− 0 .04 if stock goes down.Theexpected return, that is, the mathematical expectation of the return on the
portfolio is
E(KV)=0. 16 × 0. 8 − 0. 04 × 0 .2=0. 12 ,
that is, 12%. Theriskof this investment is defined to be the standard deviation
of the random variableKV:
σV=√
(0. 16 − 0 .12)^2 × 0 .8+(− 0. 04 − 0 .12)^2 × 0 .2=0. 08 ,
that is 8%. Let us compare this with investments in just one type of security.
Ifx=0,theny= 100,that is, the whole amount is invested risk-free. In
this case the return is known with certainty to beKA=0.1, that is, 10% and
the risk as measured by the standard deviation is zero,σA=0.
On the other hand, ifx= 125 andy=0,theentireamountbeinginvested
in stock, then
V(1) ={
12 ,500 if stock goes up,
7 ,500 if stock goes down,andE(KS)=0.15 withσS=0.20, that is, 15% and 20%, respectively.
Given the choice between two portfolios with the same expected return, any
investor would obviously prefer that involving lower risk. Similarly, if the risk
levels were the same, any investor would opt for higher return. However, in the
case in hand higher return is associated with higher risk. In such circumstances
the choice depends on individual preferences. These issues will be discussed in
Chapter 5, where we shall also consider portfolios consisting of several risky
securities. The emerging picture will show the power of portfolio selection and
portfolio diversification as tools for reducing risk while maintaining the ex-
pected return.
Exercise 1.4
For the above stock and bond prices, design a portfolio with initial wealth
of $10,000 split fifty-fifty between stock and bonds. Compute the ex-
pected return and risk as measured by standard deviation.