SJES3467
Investment and Financial Analysis I
Dec, 2011
Test 2 - Instruction to candidates 1. Answer Answer All Ques Question tions s 2. Time Time Allowe Allowed d : 1 hour
1. Suppose that the universe of available available risky securities consists of a large number number of stocks, identically distributed with E (r) = 15%, σ = 60%, and a common correlation coefficient of ρ = 0.5. (a) What are the expected return and standard standard deviation of an equally equally weighted weighted risky portfolio of 25 stocks? (b) What is the smallest number number of stocks stocks necessary necessary to generate generate an efficient portfolio portfolio with a standard deviation equal to or smaller than 43%? (c) What is the systematic risk in this security security universe? universe? (d) If T-bills are availab available le and yield 10%, what is the slope of the CAL?
Solution:
The parameters are E (r) = 15%, σ = 60%, and the correlation between any pair of stocks is ρ = 0.5. (a) The portfolio expected expected return is invarian invariantt to the size of the portfolio portfolio because all stocks have have identical expected returns. The standard deviation of a portfolio with n stocks is: σP = = =
1 2 σ +2× n
i
j
1 2 1 σ +2× 2 n n 1 2 1 σ + 1− n n
×
11 nn
×
σ × σ × ρ,
n(n − 1) 2
×
×
i < j, n = 1, 2, . . .
σ2 × ρ
(1)
σ2 × ρ
Hence, the standard deviation of a portfolio with n = 25 stocks is:
1 2 24 60 + 25 25 = 43. 43 .27%
σP =
×
602 × 0.5
(b) Becaus Becausee the stocks stocks are identic identical, al, efficien efficientt portfol portfolios ios are equall equally y weigh weighted ted.. To obtain obtain a standard deviation of 43%, we need to solve for n: 602 602(n − 1) + 0. 0.5 × n n n = 36 36..73
432 =
Thus we need 37 stocks and will come in with volatility slightly under the target.
Sam
SJES3467
Investment and Financial Analysis I
Dec, 2011
(c) As n gets very large, the variance of an efficient (equally weighted) portfolio diminishes, leaving only the variance that comes from the covariances among stocks, that is from equation (1) lim σP = ρ × σ 2 = 0.5 × 602 = 42.43% n→∞
Note that with 25 stocks we came within 0.84% of the systematic risk, that is, the nonsystematic risk of a portfolio of 25 stocks is only 0.84%. With 37 stocks the standard deviation is 43%, of which nonsystematic risk is 0.57%. (d) If the risk-free is 10%, then the risk premium on any size portfolio is 15 -10 = 5%. The standard deviation of a well-diversified portfolio is (practically) 42.43%; hence the slope of the CAL is S = 5/42.43 = 0.1178.
Sam
Page 2 of 5
SJES3467
Investment and Financial Analysis I
Dec, 2011
2. Your client has a $900,000 fully diversified portfolio. She is contemplating investing in ABC Company common stock an unspecified amount $X . You have the following information: •
Original Portfolio ABC Company
Expected Monthly Returns 0.67% 1.25%
Standard Deviation of Monthly Returns 2.37% 2.95%
Table 1: Risk and Return Characteristics •
The correlation coefficient of ABC stock returns with the original portfolio returns is 0.40.
•
Risk free T-bills are known to provide monthly returns of 0.42%
(a) Assuming that she invests in the ABC stock, calculate in terms of X the: i. Expected return of her new portfolio which includes the ABC stock. ii. Covariance of ABC stock returns with the original portfolio returns. iii. Standard deviation of her new portfolio which includes the ABC stock. (b) Calculate X which leads to she having i. a minimum variance portfolio; ii. an optimal risky portfolio. (c) Identify an investment strategy which leads to an optimal risky portfolio providing a 10% increase in returns from the original portfolio. (d) Determine whether the systematic risk of her new portfolio, which includes the government T-bill securities, will be higher or lower than that of her original portfolio. Solution:
(a) Let W 0 = 900, 000 be the initial wealth, wABC and wOP denote respectively the weights of investments held in ABC stocks and the Original Portfolio, OP . We then have: X , W 0 + X W 0 = W 0 + X
wABC = wOP Hence,
i. the expected return E [rP ],of the new portfolio is given by: E [rP ] = wABC E [rABC ] + wOP E [rOP ] X W 0 = × 1.25 + × 0.67 W 0 + X W 0 + X ii. the Covariance of ABC stock returns with the original portfolio returns is given by: Cov(rABC , rOP ) = σABC × σOP × ρ(rABC , rOP ) = 2.37 × 2.95 × 0.40 = 2.7966
Sam
Page 3 of 5
SJES3467
Investment and Financial Analysis I
Dec, 2011
iii. the Standard deviation of her new portfolio which includes the ABC stock is given by: σP = =
2 2 2 2 wABC σABC + wOP σOP + 2 × wABC × wOP × Cov(rABC , rOP )
X W 0 + X
+2×
=
2
×
W 0 W 0 + X W 0 2.80 W 0 + X
2.952 +
X W 0 + X
2
×
2.372
1 2
1 2.952 X 2 + 2.372 W 02 + 2 × 2.80W 0X 2 (W 0 + X )
(b) We determine the X for which: i. the portfolio has minimum variance as follows: dσ2 1 2 = × 2 × 2.95 X + 2.80W 0 2 dX (W 0 + X ) 2 2.952X 2 + 2.372 W 02 + 2 × 2.80W 0 X − (W 0 + X )3
Hence, dσ2 = 2.952 X + 2.80W 0 (W 0 + X ) − 2.952X 2 + 2.372 W 02 + 2 × 2.80W 0 X = 0 dX =⇒ 2.952 − 2.80 W 0 X − 2.372 − 2.80 W 02 = 0
2.372 − 2.80 W 0 i.e., X = = 0.4772W 0 2.952 − 2.80 ii. the portfolio is optimal if:
2 E (RABC )σOP − E (ROP )Cov(rABC , rOP ) wABC = 2 2 E (RABC )σOP + E (ROP )σABC − [E (RABC ) + E (ROP )]Cov(rABC , rOP )
wOP = 1 − wABC
(2) where
E (RABC ) = E (rABC ) − rf E (ROP ) = E (rOP ) − rf
with rf = 0.42 as given in this case and from which X is obtained as X =
1 − wOP W 0 wOP
We form a new portfolio including the T-bill with ABC and the Original Portfolio, OP with the following weights: wT bill ,
(1 − wT bill )wOP ,
(1 − wT bill )wABC
where wOP and wABC are determined using equation (2).
The optimal risky portfolio provides an expected return of E [rRP ]: E [rRP ] = wOP × 0.67 + wABC × 1.25
Sam
Page 4 of 5
(3)
SJES3467
Investment and Financial Analysis I
Dec, 2011
We have to determine wT bill such that wT bill × 0.42 + (1 − wT bill ) × E [rRP ] = 1.10 × 0.67
(4)
Using equation (4) the various weights listed in (3) can be calculated to enable the construction of the desired portfolio.
(d) (c) As the original portfolio is said to be fully diversified, the addition of any further assets to her existing portfolio will have no impact on the systematic risk exposures.
Sam
Page 5 of 5
