Question

Investment Theory

4. Your current portfolio of $750,000 is fully diversified, yielding an expected monthly return of 0.67% with a monthly return volatility of 2.37%. You expect to inherit $250,000 of company A stock, which has an expected return of 1.25% a month and a monthly return standard deviation of 2.95%. The correlation of stock A with your current portfolio is 0.4. a. Assuming you keep the stock, compute the monthly expected return and standard deviation of the new portfolio (i.e. the total portfolio that includes your current portfolio and the new stock)? b. What is the annualized expected return of the portfolio from part a? What is the annualized volatility? C. A friend suggests selling stock A and investing the proceeds in stock of firm B instead. Stock B has the same expected return and volatility level as A. Are you indifferent between these two alternatives? d. You discuss your alternatives with your financial adviser, commenting “I am more concerned about the downside of losing money than I am about overall total risk”. What other risk measures might your financial adviser suggest (in addition to standard deviation)? How would those affect your reward-for-volatility ratio? Explain.

Answer 1

Given in question :

1. Current portfolio return is 0.67% per month
2. Current portfolio standard deviation (SD) is 37% per month
3. Current portfolio amount is $750,000
4. New stock return is 1.25% per month
5. New stock standard deviation (SD) is 2.95%
6. New stock amount is $250,000

Answer – a

New portfolio shall consist of old portfolio and new stock inherited

Monthly expected return of new portfolio = Rp * W1 + Rs * W2

Where-

Rp is the return from current portfolio – 0.67%

Rs is the return from new stock – 1.25%

W1 & W2 are the weights of investment ratio in current portfolio and new stock respectively

Putting the above figures in the formula, we get –

Monthly expected return of new portfolio = Rp * W1 + Rs * W2

                                                                  = 0.67 * (7.5/10) + 1.25 * (2.5/10)

                                                                   = 0.67 * 0.75 + 1.25 * 0.25

                                                                   = 0.5025 + 0.3125

                                                                   = 0.815

or 0.815%

Monthly expected standard deviation of new portfolio =

Variance of new portfolio

Variance of new portfolio = (SDp)↑2 * (Rp)↑2 + (SDs)↑2 * (Rs)↑2 + 2 * SDp * SDs * Rp * Rs * r

Where –

SDp is the standard deviation from current portfolio – 2.37%

SDs is the standard deviation from new stock – 2.95%

Rp is the return from current portfolio – 0.67%

Rs is the return from new stock – 1.25%

R is the correlation of new stock with current portfolio – 0.4

Putting ths above figure in the formula, we get –

Variance of new portfolio = (SDp)↑2 * (Rp)↑2 + (SDs)↑2 * (Rs)↑2 + 2 * SDp * SDs * Rp * Rs * r

                                         = 2.37↑2 * 0.67↑2 + 2.95↑2 * 1.25↑2 + 2 * 2.37 * 2.95 * 0.67 * 1.25 * 0.4

                                         = 5.6169*0.4489 + 8.7025*1.5625 + 4.6843

                                         = 2.5214 + 13.5976 + 4.6843

Variance of new portfolio = 20.8033

Put the variance into below :

Monthly expected standard deviation of new portfolio =
Variance of new portfolio

                                                                                              =

20.8033

                                                                                               = 4.56

                                                                                             Or 4.56%

Answer – b

Annual expected return of the new portfolio = Monthly expected return of new portfolio * 12

                                                                      = 0.815 * 12

                                                                       = 9.78%

Annualized volatility of new portfolio = Monthly expected standard deviation of new portfolio * 12

                                                          = 4.56 * 12

                                                          = 54.72%

Answer – c

In the given case if investment is made in stock B instead of stock A with same returns and volatility then we are not indifferent between these two alternatives because we have to first consider the correlation between stock B and current portfolio and thereafter we can conclude the whether these two alternatives are indifferent or not.

Answer –d

In addition to standard deviation one must also consider the Beta of a particular portfolio which is also a measurement of risk. It would affect the reward-for-volatility ratio significantly because a standard deviation is the risk related to a particular security but beta is the measurement of systematic risk which will also include not only security volatility but also market volatility.