Question

Your uncle holds just one stock, East Coast Bank (ECB). You agree that this stock is relatively safe, but you want to demonstrate that his risk would be even lower if he were more diversified. You obtain the following returns data for West Coast Bank (WCB). Measured by the standard deviation of returns, by how much would your uncle's risk have been reduced if he had held a portfolio consisting of 70% in ECB and the remainder in WCB?

Answer 1

Complete question:

Assume that your uncle holds just one stock, East Coast Bank (ECB), which he thinks has very little risk.  You agree that the stock is relatively safe, but you want to demonstrate that his risk would be even lower if he were more diversified.  You obtain the following returns data for West Coast Bank (WCB).  Both banks have had less variability than most other stocks over the past 5 years.  

                   Year               ECB                WCB  

               2004             40.00%            40.00%

               2005            -10.00%            15.00%

               2006             35.00%            -5.00%

               2007             -5.00%           -10.00%

               2008             15.00%            35.00%

a. What is the expected return and risk of each stock?

b. Measured by the standard deviation of returns, by how much would your uncle's risk have been reduced if he had held a portfolio consisting of 60% in ECB and the remainder in WCB?  In other words, what is the difference between portfolio's standard deviation and weighted average of components' standard deviations? (Hint: check the example on page 11-12 on my note).

Solution:

The estimated return of the stock is the average profit.

So the average of ECB is (40-10+35-5+15)/5

=  $\frac{75 percent}{5}$

= 15% expected return

WCB expected return = 40+15-5-10+35  

= $\frac{75 percent}{5}$

= 15%

They've had the same planned return.

This is generally defined in the Greek letter Mu, (U) A weighted average may also be used to calculate portfolio volatility.

Standard deviation of ECB is $\sqrt{{ sum [(x-U)^2]/5}}$

so for ECB:

$(40-15)^2$= $25^2$ =6.25%

$(-10-15)^2$= $-35^2$ = 0.1225

$(35-15)^2$= $20^2$ = 0.04

$(-5-15)^2$= $-20^2$ = 0.04

$(15-15)^2$=0

now 0.0625+0.1225+0.04+0.04+0=0.265

stdev= $\sqrt{(0.265/5)}$ = 0.23

So WCB is the same except in a different order to make things quick I'm only going to add the median again WCB=0.23

Then the 60/40 portfolio will be the "weighted average" of the returns.

portfolio returns

2004: (60%*40%)+(40%*40%) = 40%

2005: (60%*-10%)+(40%*15%) = 0%

2006: (60%*35%)+(40%*-5%) = 19%

2007: (60%*-5%)+(40%*-10%) = -7%

2008:(60%*15%)+(40%*35%) = 23%

we have an average return of (40+19-7+23)/5 = 75/5 =15%  

The estimated return of all combined stocks is a better way to do so.

we knew they both had expected returns of 15% so we can say  

(60%*15%)+(40%*15%)=15%  so the portfolio has an expected return of 15%

Now we do the standard deviation for the whole portfolio and get

$(40-15)^2$= $25^2$ =6.25%

$(0-15)^2$ = - $25^2$ =6.25%

$(19-15)^2$= $4^2$ = 0.16%

$(-7-15)^2$ = $-22^2$ = -4.84%

$(23-15)^2$= $8^2$ = 0.64%

now add them up and get 9.78%

$\sqrt{(9.78%/5)}$ = 13.98%

Therefore, the normal portfolio variance is 13.98 per cent and the predicted portfolio return is 15 per cent.

Every stock has a standard deviation of 23 per cent and an average return of 15 per cent, meaning that the fund has the same estimated return but with less standard deviation. This ensures that the same gain is less costly. It's stronger than any of these products.