Risk and Return Analysis
Exercise # 01
Suppose that two investment proposals, X and Z have the following discrete probability distributions of expected cash flows in each of the next 3 years. Each of them requires same amount of investment of Tk 120000.
Proposal X  Proposal Y  
Probability  Cash Flow (Tk)  Probability  Cash Flow (Tk) 
0.10  30000  0.10  32000 
0.20  43000  0.25  43000 
0.40  54000  0.30  45000 
0.20  24000  0.25  45000 
0.10  35000  0.10  46000 
Which project is more risky? Which project should be accepted and why? Assume 12 percent cost of capital.
Solution
We know that project with more coefficient of variation (CV) is more risky and we should accept lower risky project. We have to find out the CV as follows:
Let, Cash Flow for X =X , Cash Flow for Y = Y
Probability = P
Here,
Expected value for project – X: ( ) =
Expected value for project – Y: ( ) =
Standard deviation for project X: s_{x} =
Standard deviation for project Y: s_{y} =
Coefficient of variation for project X: CV_{x} =
Coefficient of variation for project Y: CV_{y} =
Expected value for project X & Y:
Project – X  Project – Y  
Cash Flow (Tk) (X)  Probability (P)  X P  Cash Flow (Tk) (Y)  Probability (P)  Y P 
30000  0.10  3000  32000  0.10  3200 
43000  0.20  8600  43000  0.25  2750 
54000  0.40  21,600  45000  0.30  13500 
24000  0.20  4,800  45000  0.25  11250 
35000  0.10  3,500  46000  0.10  4600 
∑XP = 41,500 

 ∑XP = 43,200 
Calculation of Standard deviation for project X & Y:
Project – X  Project – Y  
Cash Flow (Tk) (X)  Probability (P)  Cash Flow (Tk) (Y)  Probability (P)  
30000  0.10  13225000  32000  0.10  12544000 
43000  0.20  450000  43000  0.25  10000 
54000  0.40  62500000  45000  0.30  972000 
24000  0.20  61250000  45000  0.25  810000 
35000  0.10  4225000  46000  0.10  784000 
= 141250000 

 = 15120000 
Standard deviation for project X:
s_{x} =
=
= 11884.86
Standard deviation for project Y:
s_{y} =
=
= 3888.44
Or,
Standard deviation for project X:
s_{x} =
=
=
= 11884.86
Standard deviation for project Y:
s_{y} =
=
=
= 3888.44
Coefficient of variation for project X:
CV_{x} = = = 28.64%
Coefficient of variation for project Y:
CV_{y} = = = 9%
Comment/Decision: Since Coefficient of Variation of Project – X (V_{x}) is greater than Coefficient of Variation of Project – Y (CV_{y}), so project X is more risky. So we should accept lower risky project X.
Exercise # 02
A project costing Tk 100000 has the following information:
Period 1  Period 2  Period 3  
Probability  Net Cash Flow (Tk) 
Probability  Net Cash Flow (Tk) 
Probability  Net Cash Flow (Tk) 
0.10  30000  0.10  20000  0.10  20000 
0.25  40000  0.25  30000  0.25  30000 
0.30  50000  0.30  40000  0.30  40000 
0.25  60000  0.25  50000  0.25  50000 
0.10  70000  0.10  60000  0.10  60000 
You are required to calculate the expected value, standard deviation and coefficient of variation of the project. You are also required to calculate the NPV of the project assuming 10 percent cost of capital.
Solution
Here,
Expected value (E / ) =
Standard deviation (s) = ,
Project Standard deviation (s) =
Coefficient of variation (CV) =
Net present value (NPV) = – Project cost
The Expected values of periods 1, 2 and 3:
E_{1} =
= (30000 × 0.10) + (40000 × 0.25) + (50000 × 0.30) + (60000 × 0.25) + (70000 × 0.10)
= Tk 50000 [Ans.]
E_{2} =
= (20000 × 0.10) + (30000 × 0.25) + (40000 × 0.30) + (50000 × 0.25) + (60000 × 0.10)
= Tk 40000 [Ans.]
E_{3} =
= (20000 × 0.10) + (30000 × 0.25) + (40000 × 0.30) + (50000 × 0.25) + (60000 × 0.10)
= Tk 40000 [Ans.]
Net present value (NPV):
NPV = – Project cost
=
=
= 108565 – 100000
= Tk 8565 [Ans.]
Standard deviations of periods 1, 2 and 3:
s_{1} =
=
=
= Tk 11402
s_{2} =
=
=
= Tk 11402
s_{3} =
=
=
= Tk 11402
Standard deviation of the project:
s =
=
=
=
= Tk 16420
Coefficient of variation (CV) of the project:
CV = = = 191.71%
Exercise # 03
Ali & Sons Industries has constructed a table, shown below that gives expected cash inflows and certainty equivalent factors for these cash inflows. These measures are for a new machine with a 5year life that requires an initial investment of Tk 95000. The firm has a 15% cost of capital, and the riskfree rate is 10%.
Year  Cash inflows (Tk)  Certainty equivalent factors 
1  35000  1.0 
2  35000  .8 
3  35000  .6 
4  35000  .6 
5  35000  .2 
(A) What is the net present value (ignoring risk)?
(B) What is the certainty equivalent net present value?
(C) Should the firm accept the project? Explain.
(D) Management has some doubts about the estimate of the certainty equivalent factor for year 5. There is some evidence that it may not be any lower than that for year 4. What impact might this have on the decision you recommended in C? Explain.
Solution
Requirement (A):
Calculation of Net Present Value (NPV) ignoring risk:
NPV = – Project cost
=
=
= 117325 – 95000
= Tk 22325 [Ans.]
Requirement (B):
Calculation of Certainty Equivalent Net Present Value (NPV):
NPV = – Project cost
= Here, CCF = Certain Cash Flow
= = CF ´ Certainty Equivalent Factor
= 89426 – 95000
= – Tk 5574 [Ans.]
Requirement (C):
Since NPV is negative, so the project should not be accepted. Because in this case the project bears a loss of Tk 5574.
Requirement (D):
In this case, the calculation of certainty equivalent factory of year 5 should be revised to reinsure the NPV. It may occur that the NPV would be positive and the investing decision may be changed.
Exercise # 04
A manufacturing company is considering investing in either of two mutually exclusive projects, X and Y. The firm has a 14% cost of capital, and the riskfree rate is currently 9%. The initial investment, expected cash inflows, and certainty equivalent factors associated with each of the projects are shown in the following table:
Year  Project X Initial investment = Tk 40000  Project Y Initial investment = Tk 56000  
Cash inflows (Tk)  Certainty equivalent factors 
Cash inflows (Tk)  Certainty equivalent factors  
1  20000  .90  20000  .95 
2  16000  .80  25000  .90 
3  12000  .60  15000  .85 
4  10000  .50  20000  .80 
5  10000  .40  10000  .80 
(A) Calculate the net present value (unadjusted for risk) for each project. Which is preferred using this measure?
(B) Calculate the certainty equivalent net present value for each project. Which is preferred using this riskadjustment technique?
(C) Compare and discuss your findings in A and B. Which, if either, of the projects do you recommend? Explain.
Solution
Requirement (A):
Calculation of Net Present Value (NPV) ignoring risk (unadjusted for risk):
For Project X:
NPV = – Project cost
=
=
= 49069 – 40000
= Tk 9069 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 63940 – 56000
= Tk 7940 [Ans.]
Comment: Since NPV (Project X) > NPV (Project Y), project X should be preferred.
Requirement (B):
Calculation of Certainty Equivalent Net Present Value (NPV):
For Project X:
NPV = – Project cost
=
=
= 38989 – 40000
= – Tk 1011 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 62749 – 56000
= Tk 6749 [Ans.]
Comment: Since certainty equivalent NPV of Project Y is positive, so project Y should be preferred.
Requirement (C):
Comment: Under NPV of unadjusted risk, Project X should be preferred, but under NPV of certainty equivalent, Project Y should be preferred. We know, risk adjusted project is better than risk unadjusted project. That’s why Project Y should be preferred finally.
Exercise # 05
Considering investment in one of three mutually exclusive projects X, Y, and Z. The firm’s cost of capital, R, is 15%, and the riskfree RF, is 10%. The firm has gathered the following basic cash flow and risk index data for each project.
 Projects  
X  Y  Z  
Initial investments  Tk 15000  Tk 11000  Tk 19000 
Year  Cash inflows  
1  6000  6000  4000 
2  6000  4000  6000 
3  6000  5000  8000 
4  6000  2000  12000 
Risk (b)  1.80  1.20  0.80 
(A) Find the net present value (NPV) of each project using the firm’s cost of capital. Which project is preferred in this situation?
(B) The firm uses the following equation to determine the riskadjusted discount rate, RADR, for each project:
R = R_{F} + (R_{m} – R_{F})b
Where,
R_{F} = riskfree rate of return
b = risk index of the project
R_{m} = cost of capital of the market
(C) Use the risk adjusted discount rate for each project to determine its riskadjusted NPV. Which projects is preferable in this situation?
(D) Compare and discuss your findings in A and C. Which project do you recommend?
Solution
Requirement (A):
Calculation of Net Present Value (NPV) ignoring risk (unadjusted for risk):
For Project X:
NPV = – Project cost
=
=
= 17136 – 15000
= Tk 2136 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 12678 – 11000
= Tk 1678 [Ans.]
For Project Z:
NPV = – Project cost
=
=
= 20144 – 19000
= Tk 1144 [Ans.]
Comment: Since NPV (Project X) > NPV (Project Y) > NPV (Project Z), project X should be preferred.
Requirement (B):
Calculation of Risk Adjusted Discount Rate (RADR):
For Project X:
RADR = R_{F} + (R_{m} – R_{F})b
= 10% + (15% – 10%)1.8
= 10% + 5% × 1.8
= 10% + 9%
= 19% [Ans.]
For Project Y:
RADR = R_{F} + (R_{m} – R_{F})b
= 10% + (15% – 10%)1.2
= 16% [Ans.]
For Project Z:
RADR = R_{F} + (R_{m} – R_{F})b
= 10% + (15% – 10%)0.8
= 14% [Ans.]
Requirement (C):
Calculation of Net Present Value (NPV) (adjusted for risk):
For Project X:
NPV = – Project cost
=
=
= 15828 – 15000
= Tk 828 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 12453 – 11000
= Tk 1453 [Ans.]
For Project Z:
NPV = – Project cost
=
=
= 20630 – 19000
= Tk 1630 [Ans.]
Comment: Since NPV (Project Z) > NPV (Project Y) > NPV (Project X), project Z should be preferred.
Requirement (D):
Comment: Under NPV of unadjusted risk, Project X should be preferred, but under NPV of adjusted risk, Project Z should be preferred. We know, risk adjusted project is better than risk unadjusted project. That’s why Project Z should be preferred finally.
Exercise # 06:
A company is considering two mutually exclusive investments. The company wishes to use two different evaluation methods – certainty equivalents and riskadjusted rate of return – in its analysis. Cost of capital is 12% and the current riskfree rate of return is 7%. Cash flows associated with the two projects are as follows:
 Project X  Project Y 
Initial investments  Tk 70000  Tk 78000 
Year  Cash inflows (Tk)  
1  30000  22000 
2  30000  32000 
3  30000  38000 
4  30000  46000 
(A) Use a certainty equivalent approach to calculate the net present value of each project given the following certainty equivalent factors:
Year  Project X  Project Y 
1  .85  .95 
2  .90  .90 
3  .95  .85 
4  .95  .80 
(B) Use a riskadjusted rate of return approach to calculate the net present value of each project given that project X has a RADR factor of 1.20 and project Y has a RADR factor of 1.40.
(C) Explain why the results of the two approaches may differ from one another. Which project would you choose? Justify your choice.
Solution
Requirement (A):
Calculation of Certainty Equivalent Net Present Value (NPV):
For Project X:
NPV = – Project cost
=
=
= 92422 – 70000
= Tk 22422 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 99129 – 70000
= Tk 21129 [Ans.]
Comment: NPV (Project X) > NPV (Project Y), project X should be preferred.
Requirement (B):
Calculation of Risk Adjusted Discount Rate (RADR):
For Project X:
RADR = R_{F} + (R_{m} – R_{F})b
= 7% + (12% – 7%)1.2
= 7% + 5% × 1.2
= 7% + 6%
= 13% [Ans.]
For Project Y:
RADR = R_{F} + (R_{m} – R_{F})b
= 7% + (12% – 7%)1.4
= 14% [Ans.]
Calculation of Net Present Value (NPV) (adjusted for risk):
For Project X:
NPV = – Project cost
=
=
= 89234 – 70000
= Tk 19234 [Ans.]
For Project Y:
NPV = – Project cost
=
=
= 96806 – 78000
= Tk 18806 [Ans.]
Comment: Since NPV (Project X) > NPV (Project Y), project X should be preferred.
Requirement (D):
Comment: Since under both approaches, NPV (Project X) > NPV (Project Y), so project X should be preferred finally.
Exercise # 07
The market portfolio has an expected return of 18 percent and standard deviation of 30 percent. The standard deviation of Beximco Pharmaceutical Ltd.’s stocks is 25 percent and its correlation coefficient with the market portfolio is 60 percent.
(A) What is the beta of Beximco Pharmaceuticals?
(B) What would be happen to the beta if standard deviation were 40 percent? If the correlation coefficient were 75 percent?
(C) Of the total variance of the stock in the original example, what portion is accounted for by systematic risk? By unsystematic risk?
(D) What, in general, would happen to the total variance of a portfolio if a portion of the stock were sold and the proceeds were invested to unrelated stocks having the same beta?
Solution
Requirement (A):
Given The market portfolio expected return, E(r) = 18 percent
Standard deviation of market portfolio, s_{m} = 30 percent
Standard deviation of Beximco, s_{B} = 25 percent
Correlation coefficient with market portfolio, r = 0.60
Requirement: Beta of Beximco (b_{B}) = ?
Beta of Beximco (b_{B}) = = = 0.50 [Ans.]
Requirement (B):
Given Standard deviation of market portfolio, s_{m} = 40 percent
Standard deviation of Beximco, s_{B} = 25 percent
Correlation coefficient with market portfolio, r = 0.75
Requirement: Beta of Beximco (b_{B}) = ?
Beta of Beximco (b_{B}) = = = 0.47 [Ans.]
Requirement (C):
According to Markowitz, no portion is eliminated under systematic risk and the entire portion can be eliminated under unsystematic risk.
Requirement (D):
Under same beta, total variance remains same in the market portfolio.
Exercise # 08
The common stocks of Bata Shoe Ltd and Apex Footwere Ltd have expected returns of 15 percent and 20 percent respectively, while the standard deviations are 20 percent and 50 percent respectively. The correlation coefficient between the two stocks is 46 percent. What is the expected value of return and standard deviation of a portfolio consisting of (i) 40 percent and 60 percent and (ii) 50 percent and 50 percent respectively?
Solution
Requirement (A):
Given E(r)_{Bata} = 15 percent, E(r)_{Apex} = 20 percent,
s_{Bata} = 20 percent, s_{Apex} = 50 percent,
Correlation coefficient, r = 0.46
Requirements: (a) Portfolio (Combined) expected return, E(r_{p}) = ?
(b) Portfolio (Combined) standard deviation, s_{p} = ?
(i) When weights are 40%:60%:
(a) E(r_{p}) = W_{Bata} ´ E(r)_{Bata} + W_{Apex} ´ E(r)_{Apex}
= 0.40 ´ 0.15 + 0.60 ´ 0.20
= 0.18
= 18% [Ans.]
(b) s_{p} =
=
= 0.3442
= 34.42% [Ans.]
(ii) When weights are 50%:50%:
(a) E(r_{p}) = W_{Bata} ´ E(r)_{Bata} + W_{Apex} ´ E(r)_{Apex}
= 0.50 ´ 0.15 + 0.50 ´ 0.20
= 0.175
= 17.5% [Ans.]
(b) s_{p} =
=
= 0.309
= 30.9% [Ans.]
Exercise # 09
Securities A, B, and C have the following characteristics with respect to expected return, standard deviation, and correlation between them:
Expected Return (R)  Standard Deviation (s)  Correlation Coefficients  
AB  AC  BC  
Company A  .10  .02  .50  .75  .60 
Company B  .15  .16  –  –  – 
Company C  .14  .08  –  –  – 
What is the expected value of return and standard deviation of a portfolio comprised of (i) equal investments in each and (ii) 40%, 40% and 20% respectively.
Solution
(i) When weights are 1:1:1
(a) E(r_{p}) = W_{A} ´ E(r)_{A} + W_{B} ´ E(r)_{B} + W_{C} ´ E(r)_{C}
= (1/3)(0.10) + (1/3)(0.15) + (1/3)(0.14)
= 13% [Ans.]
(b) s_{p} =
=
= 8.67% [Ans.]
(ii) When weights are 40:40:20
(a) E(r_{p}) = W_{A} ´ E(r)_{A} + W_{B} ´ E(r)_{B} + W_{C} ´ E(r)_{C}
= (0.40)(0.10) + (0.40)(0.15) + (0.20)(0.14)
= 12.8% [Ans.]
(b) s_{p} =
=
= 8.32% [Ans.]
Exercise # 10
The securities of company A and B have the following expected returns and standard deviations. The correlation between the two stocks is 0.35.
 R  s 
Company A  .16  .20 
Company B  .12  .15 
(A) Compute the risk and return for the following portfolios:
(i) 70 percent A, 30 percent B.
(ii) 30 percent A, 70 percent B.
(iii) 50 percent A, 50 percent B.
(iv) 80 percent B, 20 percent A.
(v) 60 percent A, 40 percent B.
(vi) 60 percent B, 40 percent A.
(vii) 75 percent A, 25 percent B.
(B) Which of these portfolios is optimum? Why? Show in a graph.
(C) Suppose that the investor could borrow or lend at 8 percent. How would this affect your graph.
Solution
Requirement (A):
Given R_{A} = .16, R_{B} = .12,
s_{A} = .20, s_{B} = .15,
Correlation coefficient, r = 0.35
Requirements: (a) Portfolio (Combined) expected return, R_{p} = ?
(b) Portfolio (Combined) standard deviation, s_{p} = ?
(i) When weights are 70%:30%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.70 ´ 0.16 + 0.30 ´ 0.12
= 14.8% [Ans.]
(b) s_{p} =
=
= 16.14% [Ans.]
(ii) When weights are 30%:70%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.30 ´ 0.16 + 0.70 ´ 0.12
= 13.2% [Ans.]
(b) s_{p} =
=
= 13.80% [Ans.]
(iii) When weights are 50%:50%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.50 ´ 0.16 + 0.50 ´ 0.12
= 14% [Ans.]
(b) s_{p} =
=
= 14.45% [Ans.]
(iv) When weights are 20%:80%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.20 ´ 0.16 + 0.80 ´ 0.12
= 12.8% [Ans.]
(b) s_{p} =
=
= 13.91% [Ans.]
(v) When weights are 60%:40%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.60 ´ 0.16 + 0.40 ´ 0.12
= 14.4% [Ans.]
(b) s_{p} =
=
= 15.18% [Ans.]
(vi) When weights are 40%:60%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.40 ´ 0.16 + 0.60 ´ 0.12
= 13.6% [Ans.]
(b) s_{p} =
=
= 13.98% [Ans.]
(vii) When weights are 75%:25%
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.75 ´ 0.16 + 0.25 ´ 0.12
= 15% [Ans.]
(b) s_{p} =
=
= 16.69% [Ans.]
Requirement (B):
Comment: Since portfolio (iv) has the highest expected rate of return, so it is the optimum market portfolio.
Requirement (C):
Comment: When the investment requires a cost of capital 8 percent, then the net return affected by cost and standard deviation of market portfolio.
Exercise # 11
The following portfolios are available in the market. In the context of the capital asset pricing model what is the expected return of security A if it has the following characteristics and if the following information holds for the market portfolio?
Standard deviation of Security A .30
Standard deviation of market portfolio .20
Expected return of market portfolio .15
Correlation between possible returns of
Security A and the market portfolio .75
Risk free return .09
(A) What would happen to the required return if the standard deviation of security A were higher?
(B) What would happen if the correlation were less or more?
(C) What is the functional relationship between the required return for a security and market risk?
Solution
Given Standard deviation of Security A (s_{A}) .30
Standard deviation of market portfolio (s_{m}) .20
Expected return of market portfolio (r_{m}) .15
Correlation between possible returns of
Security A and the market portfolio (r) .75
Risk free return (r_{f}) .09
Requirement: Expected rate of return (r_{e}) = ?
Here, beta of A (b_{A}) = = = 1.125 [Ans.]
Under capital asset pricing model (CAPM):
Expected rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.09 + (0.15 – 0.09)1.125
= 15.75% [Ans.]
Requirement (A):
When the standard deviation of security A were higher, then the required return would be higher.
Requirement (B):
When the correlation were less, then the required return would be lower, and when the correlation were more, then the required return would be higher.
Requirement (C):
The required return for a security and market risk are related by beta.
Exercise # 12
At present, suppose the riskfree rate is 12 percent and the expected return on the market portfolio is 16 percent. The expected return for four stocks are listed together with their expected beta.
Stock  Expected Return  Expected Beta 
A  18%  1.35 
B  15%  0.85 
C  16%  1.20 
D  20%  1.75 
(A) On the basis of these expectations, which stocks are overvalued?
(B) If the riskfree rate were to rise to 13 percent and the expected return on the market portfolio rose to 18 percent, which stocks would be overvalued? Which would be undervalued?
Solution
Requirement (A):
Given Risk free return (r_{f}) .12
Expected return of market portfolio (r_{m}) .16
Requirement: Required rate of return (r_{e}) = ?
For Stock A:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.12 + (0.16 – 0.12)1.35
= 17.4%
Since required return (17.4%) < expected return (18%), the stock A is undervalued.
For Stock B:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.12 + (0.16 – 0.12)0.85
= 15.4%
Since required return (15.4%) > expected return (15%), the stock A is overvalued.
For Stock C:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.12 + (0.16 – 0.12)1.2
= 16.8%
Since required return (16.8%) > expected return (16%), the stock A is overvalued.
For Stock D:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.12 + (0.16 – 0.12)1.75
= 19%
Since required return (19%) < expected return (20%), the stock A is undervalued.
Requirement (B):
Given Risk free return (r_{f}) .13
Expected return of market portfolio (r_{m}) .18
Requirement: Required rate of return (r_{e}) = ?
For Stock A:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.13 + (0.18 – 0.13)1.35
= 19.75%
Since required return (19.75%) > expected return (18%), the stock A is overvalued.
For Stock B:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.13 + (0.18 – 0.13)0.85
= 17.25%
Since required return (17.25%) > expected return (15%), the stock A is overvalued.
For Stock C:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.13 + (0.18 – 0.13)1.2
= 19%
Since required return (19%) > expected return (16%), the stock A is overvalued.
For Stock D:
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.13 + (0.18 – 0.13)1.75
= 21.75%
Since required return (21.75%) > expected return (20%), the stock A is overvalued.
Exercise # 13
Assuming that you are able to both borrow and lend at the risk free rate of 10 percent. The market portfolio of securities has an expected return of 15 percent and a standard deviation of 28 percent. Determine the expected return and standard deviation of the following portfolios:
(A) All fund is invested in the risk free asset.
(B) Onethird is invested in the risk free asset and twothird in the market portfolio.
(C) All fund is invested in the market portfolio. Additionally you borrow an additional onethird of your wealth to invest in the market portfolio.
Solution
Given R_{f} = .10, R_{m} = .15,
s_{A} = 0, s_{m} = .28,
Correlation coefficient, r = 0
Requirements: (i) Portfolio (Combined) expected return, R_{p} = ?
(ii) Portfolio (Combined) standard deviation, s_{p} = ?
(A) When weights are 1:0
(i) E(r_{p}) = W_{r} ´ R_{f} + W_{m} ´ R_{m}
= 1 ´ 0.10 + 0 ´ 0.15
= 10% [Ans.]
(ii) s_{p} =
=
= 0% [Ans.]
(B) When weights are 1/3:2/3
(i) E(r_{p}) = W_{r} ´ R_{f} + W_{m} ´ R_{m}
= 1/3 ´ 0.10 + 2/3 ´ 0.15
= 13.33% [Ans.]
(ii) s_{p} =
=
= 18.67% [Ans.]
(C) When weights are 0:(1+1/3) = 0:4/3
(i) E(r_{p}) = W_{r} ´ R_{f} + W_{m} ´ R_{m}
= 0 ´ 0.10 + 4/3 ´ 0.15
= 20% [Ans.]
(ii) s_{p} =
=
= 37.33% [Ans.]
Exercise # 14
Maq Enterprises has a beta of 1.25, the risk free rate is 9 percent and the expected return on the market portfolio is 15 percent. The company presently pays a dividend of Tk 20 per share and investors expect to experience a growth in dividends of 10 percent per annum for many years to come.
(A) What is the stock’s required rate of return according to the CAPM?
(B) What is the stock’s present market price per share, assuming this required return?
Solution
Given Risk free return (r_{f}) .09
Expected return of market portfolio (r_{m}) .16
Beta (b) 1.25
Dividend (D) 20
Growth (g) .10
Requirements: (A) Required rate of return (r_{e}) = ?
(B) Market price of share (P_{o}) = ?
Requirement (A):
Required rate of return (r_{e}) = r_{f} + (r_{m} – r_{f}) b
= 0.09 + (0.16 – 0.09)1.25
= 17.75% [Ans.]
Requirement (A):
Market price of share (P_{o}) = = = Tk 258 [Ans.]
Exercise # 15
A food manufacturing company would like to add a new product D. The expected value and standard deviation of the probability distribution of possible net present value for the product are Tk 120000 and Tk 90000 respectively. The company’s existing products A, B, and C. The expected values of net present value and standard deviation for these products are:
Products  Net Present Value  Standard Deviation 
A  Tk 160000  Tk 80000 
B  Tk 200000  Tk 70000 
C  Tk 100000  Tk 40000 
The correlation coefficients between products are:
Products  A  B  C  D 
A  1.00  –  –  – 
B  .90  1.00  –  – 
C  .80  .95  1.00  – 
D  .50  .25  .30  1.00 
You are required to:
(A) Compute the expected value and the standard deviation of the possible net present values for a combination of existing products.
(B) Compute the expected value and the standard deviation for a combination of existing products plus product D. Compare your results in parts (a) and (b). What can you say about the new product?
Solution
Requirement (A):
(i) Expected value of the project is:
E =
= (160000 × 1) + (200000 × 1) + (100000 × 1)
= Tk 460000 [Ans.]
(ii) Standard deviation of the project is:
s =
=
=
= Tk 65596 [Ans]
Requirement (B):
(i) Expected value of the project is:
E =
= (160000 × 1) + (200000 × 1) + (100000 × 1) + (120000 × 1)
= Tk 580000 [Ans.]
(ii) Standard deviation of the project is:
s =
=
=
= Tk 89897 [Ans]
Comment: New product provides higher expected value and higher standard deviation.
Exercise # 16
From the following data calculate the (i) Standard deviation, (ii) Coefficient of variation and (iii) Covariance between the two stocks.
Period  1  2  3  4  5 
Stock A  .10  – .02  .08  – .04  .08 
Stock B  .12  .03  .06  – .04  .04 
You are required to calculate the portfolio risk assuming equal investment in two stocks.
Solution
Requirement (A):
Given
Period  1  2  3  4  5  Mean 
Stock A  .10  – .02  .08  – .04  .08  .04 
Stock B  .12  .03  .06  – .04  .04  .042 
(i) Standard deviation for stock A:
s_{A} =
=
= 0.0583
Standard deviation for stock B:
s_{B} =
=
= 0.0576
(ii) Coefficient of variation for stock A:
CV_{A} = = = 145.75%
Coefficient of variation for stock A:
CV_{B} = = = 137%
(iii) Covariance of two stocks A and B:
Cov_{AB} =
=
= 0.003
Requirement (B):
(i) Portfolio risk for stock A:
Portfolio risk (b_{A}) = = = 0.88
(ii) Portfolio risk for stock B:
Portfolio risk (b_{B}) = = = 0.90
Exercise # 17
Refer to the following information of returns for stocks A, B, and C.
Probability  A  B  C 
.25  .10  .15  – .05 
.25  – .05  .04  .10 
.25  .20  .25  – .15 
.25  .10  .05  .05 
(A) Compute the expected returns of each stock.
(B) Compute the standard deviation of return for each stock.
(C) Compute the covariance between returns on each pair of stock.
(D) If the return on stock B turns out to be greater than expected, would you expect the return on stock C to be greater or lesser than expected?
Solution
Requirement (A):
(i) Expected value for stock A:
E =
= (.10 × .25) + (.05 × .25) + (.20 × .25) + (.10 × .25)
= 0.0875
= 8.75% [Ans.]
(ii) Expected value for stock B:
E =
= (.15 × .25) + (.04 × .25) + (.25 × .25) + (.05 × .25)
= 12.25% [Ans.]
(iii) Expected value for stock C:
E =
= (.05 × .25) + (.10 × .25) + (.15 × .25) + (.05 × .25)
= – 1.25% [Ans.]
Requirement (B):
(i) Standard deviation for stock A:
s_{A} =
=
= 0.0699
= 6.99% [Ans.]
(ii) Standard deviation for stock B:
s_{B} =
=
= 8.53% [Ans.]
(iii) Standard deviation for stock C:
s_{C} =
=
= 9.60% [Ans.]
Requirement (C):
(i) Covariance of two stocks A and B:
Cov_{AB} =
= (.10 – .0875)(.15 – .1225).25 + (– .05 – .0875)(.04 – .1225).25
+ (.20 – .0875)(.25 – .1225).25 + (.10 – .0875)(.05 – .1225).25
= 0.00628 [Ans.]
(ii) Covariance of two stocks B and C:
Cov_{BC} =
= (.15 – .1225)( – .05 + .0125).25 + (.04 – .1225)(.10 + .0125).25
+ (.25 – .1225)(– .15 + .0125).25 + (.05 – .1225)(.05 + .0125).25
= – 0.00809 [Ans.]
(iii) Covariance of two stocks A and C:
Cov_{AC} =
= (.10 – .0875)( – .05 + .0125).25 + (– .05 – .0875)(.10 + .0125).25
+ (.20 – .0875)(– .15 + .0125).25 + (.10 – .0875)(.05 + .0125).25
= – 0.00766 [Ans.]
Requirement (D):
Comment: Since all three stocks equal probability, we would expect the return on stock C to be greater or lesser than expected.
Exercise # 18 Refer to the following observations for stock Y and the market portfolio (M).
Time Period  Observed Returns  
Y  M  
1  .10  .04 
2  .14  .02 
3  .10  .08 
4  .08  .14 
(A) Compute the sample mean return for Y and M.
(B) Compute the sample standard deviations for Y and M.
(C) Compute the sample covariance between returns for Y and M.
(D) Compute the sample beta factor of stock Y.
(E) Compute the sample correlation coefficient between the returns of Y and M.
Solution
Requirement (A):
(i) Sample mean for Y:
E =
=
= .105 [Ans.]
(ii) Sample mean for M:
E =
=
= .07 [Ans.]
Requirement (B):
(i) Sample standard deviation for Y:
s_{Y} =
=
= 0.0252
(ii) Sample standard deviation for M:
s_{M} =
=
= 0.0529
Requirement (C):
Covariance of two Y and M:
Cov_{YM} =
=
= – 0.0011 [Ans.]
Requirement (D):
Sample beta factor for stock Y:
b_{Y} = = = – 1.73 [Ans.]
Requirement (E):
Sample correlation coefficient between the returns of Y and M:
r_{YM} = = = – 0.83 [Ans.]
Exercise # 19
Suppose you purchase Tk 10000 of stock A, Tk 5000 of stock B and borrow Tk 5000. Build up your portfolio and also compute the variance and expected return of the portfolio given the following additional information.
 A  B 
Variance (s^{2})  .25  .49 
E (R)  .12  .18 
The correlation of A with B is .75. Borrowing takes place at a riskfree interest rate of .08.
Solution
Given s_{A}^{2} = .25, s_{B}^{2} = .49,
s_{A} = Ö.25 = 0.5, s_{B} = Ö.49 = .7,
R_{A} = .12, R_{B} = .18,
Correlation coefficient, r = 0.75
Borrowing = 5000, borrowing cost = 0.08
Requirements: (a) Portfolio (Combined) expected return, R_{p} = ?
(b) Portfolio (Combined) standard deviation, s_{p} = ?
(c) Portfolio (Combined) variance, s_{p} = ?
(a) E(r_{p}) = W_{A} ´ R_{A} + W_{B} ´ R_{B}
= 0.67 ´ 0.12 + 0.33 ´ 0.18
= 13.98% [Ans.]
(b) s_{p} =
=
= 53.07% [Ans.]
(c) s_{p}^{2} =
= (53.07%)^{2}
= 28.17% [Ans.]
Comment: When the investment requires a cost of capital 8 percent, then the net return affected by cost and standard deviation of market portfolio.
Exercise # 20
Assume the following information:
Stock  E (R)  Standard Deviation  Correlation Coefficient 
1  .05  .20  1 with 2 = .25 
2  .10  .10  1 with 3 = .35 
3  .20  .15  1 with 4 = .55 
4  .15  .30  2 with 3 = .28 
2 with 4 = .55  
3 with 4 = 0 
A portfolio is formed as follows: Sell short Tk 20000 of stock 1 and buy Tk 30000 of stock 2, Tk 80000 of stock 3, and Tk 30000 of stock 4. The cash provided by the owner of the portfolio is Tk 20000 and any additional funds required to finance the portfolio are borrowed at a riskfree interest rate of 8 percent. There are no restrictions on the use of short sale proceeds.
(A) Compute the portfolio weights for each component of the portfolio.
(B) Compute the expected return of the portfolio.
(C) Compute the standard deviation of the portfolio.
Solution
Requirement (A):
W_{1} = 20000/160000 = 0.1250
W_{2} = 30000/160000 = 0.1875
W_{3} = 80000/160000 = 0.5000
W_{4} = 30000/160000 = 0.1875
Requirement (B):
E(r_{p}) = W_{1} ´ R_{1} + W_{2} ´ R_{2} + W_{3} ´ R_{3} + W_{4} ´ R_{4}
= 0.1250 ´ 0.05 + 0.1875 ´ 0.10 + 0.5 ´ 0.20 + 0.1875 ´ 0.15
= 15.31% [Ans.]
Requirement (C):
s_{p} =
=
= 17.5% [Ans.]
Exercise # 21
The following information is available regarding expected net cash inflows and their probability distribution of project costing Tk 12000. Assuming 10% discounting rate, calculate net present value, standard deviation and coefficient of variation.
Year 1  Year 2  Year 3  
Tk 4000  .25  Tk 3000  .50  Tk 5000  .25 
5000  .50  6000  .25  7000  .25 
9000  .25  8000  .25  5000  .50 
Solution
The Expected values of years 1, 2 and 3:
E_{1} =
= (4000 × .25) + (5000 × .50) + (9000 × .25)
= Tk 5750
E_{2} =
= (3000 × .50) + (6000 × .25) + (8000 × .25)
= Tk 5000
E_{3} =
= (5000 × .25) + (7000 × .25) + (5000 × .50)
= Tk 5500
Net present value (NPV):
NPV = – Project cost
=
=
= 13492 – 12000
= Tk 1492 [Ans.]
Standard deviations of years 1, 2 and 3 :
s_{1} =
=
= Tk 1920
s_{2} =
=
= Tk 2121
s_{3} =
=
= Tk 866
Standard deviation of the project:
s =
=
=
= Tk 2558 [Ans.]
Coefficient of variation (CV) of the project is:
CV = = = 171.45%
Exercise # 22
After a careful evaluation of investment alternatives and opportunities, Masters School Supplies has developed a CAPMtype relationship linking a risk index to the return (RADR) as shown in the following table.
Risk index  Required return (RADR) 
0.0  7.0% (risk free rate, RF) 
0.4  9.0 
0.6  10.0 
0.8  11.0 
1.0  12.0 
1.4  14.0 
1.6  15.0 
1.8  16.0 
The firm is faced with two mutually exclusive projects, A and B. The following are the data the firm has been able to gather about the projects:
Project A  Project B  
Initial investment  Tk 20000  Tk 30000 
Project life  5 years  5 years 
Annual cash inflow (CF)  Tk 7000  Tk 10000 
Risk index  0.8  1.4 
Certainty equivalent factors  
Year  Project A  Project B 
0  1.00  1.00 
1  .95  .90 
2  .90  .80 
3  .90  .70 
4  .85  .70 
Greater than 4  .80  .60 
All the firm’s cash inflows have already been adjusted for taxes.
(A) Evaluate the projects using certainty equivalents.
(B) Evaluate the projects using riskadjusted discount rates.
(C) Discuss your findings in (A) and (B), and explain why the two approaches are alternative techniques for considering risk in capital budgeting.
Solution
Requirement (A):
Calculation of Certainty Equivalent Net Present Value (NPV) for A:
NPV = – Project cost (A)
=
=
= 25392 – 20000
= Tk 5392 [Ans.]
Calculation of Certainty Equivalent Net Present Value (NPV) for B:
NPV = – Project cost (B)
=
=
= 30731 – 30000
= Tk 731 [Ans.]
Comment: Since NPV(A) > NPV(B), so project A should be accepted.
Requirement (B):
Calculation of Risk Adjusted Discount Rate (RADR):
For Project A:
RADR = R_{F} + (R_{m} – R_{F})b
= 7% + (11% – 7%)0.8
= 10.2% [Ans.]
For Project B:
RADR = R_{F} + (R_{m} – R_{F})b
= 7% + (14% – 7%)1.4
= 16.8% [Ans.]
Calculation of Net Present Value (NPV) (adjusted for risk):
For Project A:
NPV = – Project cost (A)
=
=
= 26400 – 20000
= Tk 6400 [Ans.]
For Project B:
NPV = – Project cost (B)
=
=
= 32141 – 30000
= Tk 2141 [Ans.]
Comment: Since NPV(A) > NPV(B), so project A should be accepted.
Requirement (C):
Our findings (A) and (B), both indicate that project A is fruitful. But both approaches are alternative techniques for considering different type of risks.