Problem 1: Portfolio Required Return You are the money manager of a $10 million investment fund, which consists of four stocks. This fund has the following investments and betas:
| Stock | Investment | Beta |
| A | $3,000,000 | 1.50 |
| B | $1,000,000 | (0.50) |
| C | $2,000,000 | 1.25 |
| D | 4,000,000 | 0.75 |
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If the market's required rate of return is 12 percent, and the risk-free rate is 4 percent, what is the fund's required rate of return?
Weights of the investments
W A =3000000/10000000 =0.3
W B =1000000/10000000 =0.1
W C =2000000/10000000 =0.2
W D =4000000/10000000 =0.4
Portfolio beta = 0.3*1.50 + 0.1*-0.50 + 0.2*1.25 + 0.4*0.75
= 0.45 + -0.05 + 0.25 + 0.3
= 0.95
R p = rf +(Rm – rf)(b p )
=4% + (12% -4%)(0.95)
= 4% +7.6%
=11.6%
Problem 2: Required Rate of Return Stock R's beta = 1.5 Stock S's beta = 0.75 Consider that the required return on an average stock is 14 percent. The risk-free rate of return is 6 percent. If this is so, the required return on the riskier stock exceeds the required return on the less risky stock by how much?
R i = rf +(Rm – rf)(b i )
R r = 6 +(14-6)1.5 and R s = 6 +0.75 *(14 – 6)
R r = 18% and R s 12%
Difference = 18 -12 =6% (Pratt & Grabowski, 2010).
Problem 3: CAPM and Required Return Calculate the required rate of return for XYZ Inc. using the following information: The investors expect a 3.0 percent rate of inflation. The real risk-free rate is 2.0 percent. The market risk premium is 6.0 percent. XYZ Inc. has a beta of 1.7. Over the past five years, the realized rate of return has averaged 13.0 percent.
Rf = 2% + 3% = 5%
RM = 13%
Beta = 1.7
R i = rf +ᵝ(RM – Rf)
=5% + 1.7(13- 5)
=5% + 13.6%
=18.6
(Pratt & Grabowski, 2010; Sherman, 2011).
Problem 4: Bond Valuation You have two bonds in your portfolio. Each bond has a face value of $1000 and pays an 8 percent annual coupon. Bond X matures in 1 year, and Bond Y matures in 15 years. If the going interest rate is 4 percent, 9 percent, and 14 percent, what will the value of each bond be? Assume Bond X only has one more interest payment to be made at maturity. Assume there are 15 more payments to be made on Bond Y. The longer-term bond's price varies more than the shorter-term bond's price when interest rates change. Explain why. The long term bonds are affected by inflation and changes in the interest rates due to the long period
Bond X
(80/4%) * [1-1/((1+0.04)^1)] + 1000/[(1+0.04)^1]
2000*0.0385 + 961.54
=1038.54
(80/9%) * [1-1/((1+0.09)^1)] + 1000/[(1+0.09)^1
888.89 * 0.083 + 917.43
=73.78 + 917.43
= 991.21
(80/13%) * [1-1/((1+0.13)^1)] + 1000/[(1+0.13)^1
615.38 *0.115 + 884.96
955.76
Bond Y
(80/4%) * [1-1/((1+0.04)^15)] + 1000/[(1+0.04)^15
2000*-0.4447 + 555.26
889.47+555.26
1444.73
(80/9%) * [1-1/((1+0.09)^15)] + 1000/[(1+0.09)^15
888.89 * 0.7255 + 274.54
=919.43
(80/13%) * [1-1/((1+0.13)^15)] + 1000/[(1+0.13)^15
615.38 * 0.8401 + 159.89
516.98 +159.89
676.87 (Parameswaran, 2011).
Problem 5: Yield to Call Five years ago, XYZ Inc. issued 20-year bonds with a 12 percent annual coupon rate at their $1,000 par value. The bonds had five years of call protection and an 8 percent call premium. Yesterday, XYZ Inc. called the bonds. For this problem, imagine that the investor who purchased the bonds when they were issued held them until they were called. Considering this, compute the realized rate of return. Should the investor be happy with XYZ Inc. calling the bonds? Why or why not?
120/12%) * [1-1/((1+0.12)^20)] + 1000/[(1+0.12)^20
1000*0.89633 + 103.67
=1000
After five years
120/12%) * [1-1/((1+0.12)^5)] + 1000/[(1+0.12)^5
1000 * 0.4326 + 567.43
432.57 +567.43
=1000 (Sherman, 2011).
Since the coupon rate and the yield to maturity are the same, the returns of the bonds are the same irrespective of the number of years, and therefore the investor would be happy that the bonds were called earlier because the extended period does not consider the time value of money by having higher returns.
Problem 6: Yield to Maturity XYZ Inc. bonds have 5 years left to maturity. Interest is paid annually, and the bonds have a $1,000 par value and a coupon rate of 8 percent. What is the yield to maturity at a current market price of (1) $800 and (2) $1,200? If a "fair" market interest rate for such bonds was 12 percent—that is, is rd=12%—would you pay $800 for each bond? Why or why not?
$800 = 8% [1-1/(1+i)^5]/i + 1000 * (1/(1+i)^5)
$800 = 8% [1 – 1/(1.138)^5]/0.138 + 1000*(1/1.138)^5)
Therefore yield to maturity at $ 8000 = 13.8%
$1200 = 8% [1-1/(1+i)^5]/i + 1000 * (1/(1+i)^5)
$1200 = 8% [1 – 1/(1+0.0356)^5]/0.0.0356 + 1000*(1/1.0356)^5)
At $1200, yield to maturity =3.56%
If the fair market interest rate for the bond was 12% the $800 would be lower because the yield to maturity for such a bond is $13.8% and therefore 12% is lower.
Problem 7: After-Tax Cost of Debt The XYZ Inc.'s currently outstanding bonds have a 10 percent yield to maturity and an 8 percent coupon. It can issue new bonds at par that would provide a similar yield to maturity. If its marginal tax rate is 40 percent, what is XYZ's after-tax cost of debt?
After tax cost of debt = YTM X (1-T)
= 8% X (1-0.4)
=8% X 0.6
= 4.8%
Problem 8: Present Value of an Annuity Find the present values of the following ordinary annuities if discounting occurs once a year: $300 per year for ten years at 10 percent. $150 per year for five years at 5 percent. $350 per year for five years at 0 percent.
PV Ordinary Annuity = C*[1- ((1+i) -n )/i]
=300 * [1 –((1+0.1)^-10)/0.1]
=1,843.37
150 * [1 –((1+0.05)^-5)/0.05]
=649.422
350 * [1 –(1 - (1+0.0)^-5)/0.0]
=350
Problem 9: Uneven Cash Flow Stream Use the table below to answer the following:
What are the present values of the following cash flow streams if they are compounded at 5 percent annually?
What are the PVs of the streams at 0 percent compounded annually?
| 0 | 1 | 2 | 3 | 4 | 5 | |
| Stream A | $0 | $100 | $400 | $400 | $400 | $300 |
| Stream B | $0 | $300 | $400 | $400 | $400 | $100 |
PV =
PV Stream A
= 100/(1+0.05) +400/(1+0.05)^2 + 400/(1+0.05)^3 + 400/(1+0.05)^4 +300/(1+0.05)^5
=95.24 +441 + 463.05 +486.2 + 382.9 =1,868.374 (Konchitchki, 2011).
Stream B
= 300/(1+0.05) +400/(1+0.05)^2 + 400/(1+0.05)^3 + 400/(1+0.05)^4 +100/(1+0.05)^5
=285.71 + 362.81 +463.05 + 486.2 + 78.35 =1,676.12
At zero interest rates, the present values will remain the same i.e. they will not change from whatever is given on the table (Konchitchki, 2011).
References
Konchitchki, Y. (2011). Inflation and nominal financial reporting: Implications for performance and stock prices. The Accounting Review, 86(3), 1045–1085.
Parameswaran, S. (2011). Fundamentals of financial instruments: Stocks, bonds, foreign exchange, and derivatives. Hoboken, NJ: John Wiley & Sons.
Pratt, S. P., & Grabowski, R. J. (2010). Cost of capital: Applications and examples (4th ed.). Hoboken, NJ: John Wiley & Sons
Sherman, E. H. (2011). Finance and accounting for nonfinancial managers (3rd ed.). New York, NY: Amacom.
