BA 443
Final practice questions and solutions
To prepare for the quantitative material for the final, please study these questions, in-class examples and in-class questions.
Textbook questions and problems:
Chapter 18: Questions 2, 7, 8 (ignore the call feature, state two factors),
Problem 7, 9(a), 10 (a, b), 11(a), 12, Appendix Exhibit 18A.1
19: Problems 1, 3 (a, b), 4 (a, b), 6, 7
25: Problems 1, 7
Extra questions:
1. As a portfolio manager, you use a constant proportion (CPPI) strategy to rebalance your portfolio. The portfolios are rebalanced once a year with m = 1.2. One client has target portfolio weights of 60% equity and 40% cash. This client has $2 million invested at the start of the year. If in each of the next two years the equity component has returns of 8% and 7%, respectively, and the cash component remains unchanged, what is the value of the portfolio at the end of the second year? What would the value be with a buy-and-hold strategy?
2. What is the return for the month for the following two portfolios?
Portfolio 1: Beginning value= $500,000; cash inflow at the start of the month = $25,000; Ending value = $530,000
Portfolio 2: Beginning value= $500,000; cash inflow in day 10 of the month = $20,000; portfolio value at day 10 = $519,000; cash inflow on day 25 = $5,000; portfolio value at day 25 = $529,000; Portfolio value at the end of the month = $530,000. Portfolio values include cash inflows.
3. An analyst wants to evaluate Portfolio X, consisting entirely of U.S. common stocks, using both the Treynor and Sharpe measures. The following table provides the average annual rate of return for Portfolio X, the market portfolio (S&P 500) and U.S. T-bills during the past 8 years.
Avg. Return Std. Dev. Beta
Portfolio X 10% 18% 0.6
S&P 500 12% 13%
T-bills 6%
a. Calculate both the Treynor and Sharpe measure for Portfolio X and the market. Did Portfolio X overperform or underperform based on each of the measures?
b. Why do we find conflicting results using the two measures?
4. Based on the information provided below, provide a complete evaluation of the performance of the equity portfolio. The period of evaluation is one year. The ending portfolio value, including a $100,000 cash inflow at the end of the year was $1.2 million. The starting portfolio value was $1 million. The portfolio’s standard deviation is 20% and the portfolio beta is 1.2.
The portfolio is currently equally invested in two sectors although it is benchmarked against a portfolio that has an equal weighting in four sectors. The benchmark portfolio return over the past year was 8%. The benchmark portfolio’s standard deviation is 25% and its beta is 1.3.
The portfolio’s tracking error was 10%. The S&P 500 return over the year was 6% with a standard deviation of 18% and the risk-free rate was 4.5%.
Weightings Returns
Sector Portfolio Benchmark Portfolio Benchmark
1 50% 25% 11% 8%
2 50% 25% 9% 7%
3 25% 10%
4 25% 7%
1. Comment on the firm’s composite performance by calculating its Treynor, Sharpe and Information Ratios.
2. Using attribution analysis, how much of the manager’s value added was due to the difference in sector weighting compared to the benchmark and how much of the value added was due to security selection?
Solutions
Chapter 18 questions:
18.2
2. The most crucial assumption the investor makes is that cash flows will be received in full and reinvested at the promised yield. This assumption is crucial because it is implicit in the mathematical equation that solves for promised yield. If the assumption is not valid, an alternative method must be used, or the calculations will yield invalid solutions.
18.7
7(a). Given that you expect interest rates to decline during the next six months, you should choose bonds that will have the largest price increase, that is, bonds with long durations.
7(b). Case 1: Given a choice between bonds A and B, you should select bond B, since duration is inversely related to both coupon and yield to maturity.
Case 2: Given a choice between bonds C and D, you should select bond C, since duration is positively related to maturity and inversely related to coupon.
Case 3: Given a choice between bonds E and F, you should select bond F, since duration is positively related to maturity and inversely related to yield to maturity.
18.8
You should select portfolio A because it has a longer duration (5.7 versus 4.9 years) and greater convexity (125.18 versus 40.30), thereby offering greater price appreciation. Portfolio A is also noncallable, therefore there is no danger of the bonds being called in by the issuer when interest rates decline (as you expect they will).
Chapter 18 problems:
18.7
7. CFA Examination I (1993)
7(a). Modified duration is Macaulay duration divided by 1 plus the yield to maturity divided by the number of coupons per year:
Where k is the number of coupons per year. If the Macaulay duration is 10 years and the yield to maturity is 8 percent, then modified duration equals 10/(1+ (.08/2)) = 9.62.
7(b). For option-free coupon bonds, modified duration is a better measure of the bond’s sensitivity to changes in interest rates. Maturity considers only the final cash flow, while modified duration includes other factors. These factors are the size of coupon payments, the timing of coupon payments, and the level of interest rates (yield-to-maturity).
7(c). Modified duration increases as the coupon decreases. Modified duration decreases as maturity decreases.
7(d). Convexity measures the rate of change in modified duration as yields change. Convexity refers to the shape of the price-yield relationship and can be used to refine the modified duration approximation of the sensitivity of prices to interest rate changes. Convexity shows the extent to which bond prices rise at a greater rate (as yields fall) than they fall (as yields rise). The effect of duration on price and the effect of convexity on price should be added together to obtain an improved approximation of the change in price for a given change in yield.
18.9
9(a). If yieldtomaturity (YTM) on Bond B falls 75 basis points:
75 basis points = 75/100 = .75D in YTM
Pro.Price D = (modified duration) (D inYTM)+ (l/2)(convexity)(D in YTM)2
= (6.8)(.75) + (1/2)(.6)(.75)2
= 5.1+.16875
= 5.27
So the projected price will rise to $105.27 from its current $100 price.
18.10
10(a). Final Spot Rate:
70 70 70 70 1070
1000 = + + + +
(1 + y1)1 (1 + y2)2 (1 + y3)3 (1 + y4)4 (1 + y5)5
70 70 70 70 1070
1000 = + + + +
(1.05)1 (1.0521)2 (1.0605)3 (1.0716)4 (1 + y5)5
1000 = 66.67 + 63.24 + 58.69 + 53.08 + 1070/ (1 + y5)5
1000 241.68 = 1070/(1 + y5)5
758.32 = 1070/(1 + y5)5
(1 + y5)5 = 1070/758.32
y5 = (1.411)1/5 – 1 = 7.13%
Final Forward Rate:
(1.0713)5 _ 1 = 1.411 _ 1 = 1.0699 1 = 7.00%
(1.0716)4 1.3187
10(b). Yieldtomaturity is a single discounting rate for a series of cash flows to equate these flows to a current price. It is the internal rate of return.
Spot rates are the unique set of individual discounting rates for each period. They are used to discount each cash flow to equate to a current price. Spot rates are the theoretical rates for zero coupon bonds.
Spot rates can be determined from a series of yieldstomaturity in an internally consistent method such that the cash flows from coupons and principal will be discounted individually to equate to the series of yieldto-maturity rates.
Yieldtomaturity is not unique for any particular maturity, whereas spot rates and forward rates are unique.
Forward rates are the implicit rates that link any two spot rates. They are a unique set of rates that represent the marginal interest rate in a future period. They are directly related to spot rates, and therefore yieldto-maturity. Some would argue (expectations theory) that forward rates are the market expectations of future interest rates. Regardless, forward rates represent a breakeven or rate of indifference that link two spot rates. It is important to note that forward rates link spot rates, not yieldtomaturity rates.
18.11
11(a). Calculation of OneYear Forward Rate for January 1, 1996:
Date Calculation of Forward Rate
1/1/93 3.5%
1/1/94 (1.045)2 = (1.035) (1 + f)
f = 0.0551 or 5.51%
1/1/95 (1.05)3 = (1.035) x (1.0551) x (1 + f)
f = 0.0601 or 6.01%
1/1/96 (1.055)4 = (1.035) x (1.0551) x (1.0601) x (1 + f)
f = 0.0701 or 7.01%
18.12
12(a). Current yield = Annual dollar coupon interest / Price = 70/960 = 7.3%.
The annual yield to maturity (YTM) is
n C M
P = S +
t=1 (1 + y)t (1 + y)n
where: P = price of the bond
C = semiannual coupon
M = maturity value
n = number of periods, and
y = semiannual yield to maturity.
That is,
10 35 1000
960 = S +
t=1 (1 + y)t (1 + y)10
y = 4.0% semiannual return, YTM=8% per year.
Horizon yield (also called total return) accounts for coupon interest, interest on interest, and proceeds from sale of the bond.
1. Coupon interest + interest on interest
(1 + r)n - 1
= C r
where: C = semiannual coupon
r = semiannual reinvestment rate, and
n = number of periods
That is,
(1.03)6 - 1
=35 = 226.39
0.03
2. Projected sale price at the end of three years is $1,000 because bonds that yield the required rate of return always sell at par
3. Sum the results of Steps 1 and 2 to obtain $1.226.39.
4. Semiannual total return =
1226.39 1/6
-1 = 4.166%
960
5. Double the interest rate found in Step 4 for the annual total rate of return of 8.33
percent.
12(b). The shortcomings of the yield measures are as follows: (l) Current yield does not account for interest on interest (compounding) or changes in bond price during the holding period. It also does not allow for a gain or loss from a bond purchased at a discount or premium. (2) Yield to maturity assumes that the bond is held to maturity and that all coupon interest can be reinvested at the yieldtomaturity rate. Because yields change constantly, that assumption is incorrect. However, yield to maturity is the industry standard for comparing one bond with another. (3) Total return (as calculated) assumes that all coupons can be invested at a constant reinvestment rate and requires assumptions about holding period, reinvestment rate, and yield on the bond at the end of the investor's holding period. Although it is a more complete measure of return, it is only as accurate as its inputs.
Chapter 19 problems
19.1
1. Modified duration = 7/(1 + .10/2) = 6.67 years. We divide the market interest rate by two as most U.S.-based bonds pay interest twice a year.
Percentage change in portfolio = +2 x 6.67 = 13.34 percent
Value of portfolio = $50 million x (1 .1334) = $43.33 million
19.3
3(a). Computation of Duration (assuming 10% market yield)
(1) (2) (3) (4) (5) (6)
Year Cash Flow PV@10% PVof Flow PV as % of Price (1) x (5)
1 120 .9091 109.09 .1014 .1014
2 120 .8264 99.17 .0922 .1844
3 120 .7513 90.16 .0838 .2514
4 120 .6830 81.96 .0762 .3047
5 1120 .6209 695.41 .6464 3.2321
1075.79 1.0000 4.0740
Duration = 4.07 years
3(b). Computation of Duration (assuming 10% market yield)
(1) (2) (3) (4) (5) (6)
Year Cash Flow PV@10% PVof Flow PV as % of Price (1) x (5)
1 120 .9091 109.09 .1026 .1026
2 120 .8264 99.17 .0933 .1866
3 120 .7513 90.16 .0848 .2544
4 1120 .6830 764.96 .7194 2.8776
1063.38 1.0000 3.4212
Duration = 3.42 years
19.4
4(a). Computation of Duration (assuming 8% market yield)
(1) (2) (3) (4) (5) (6)
Year Cash Flow PV@8% PVof Flow PV as % of Price (1) x (5)
1 100 .9259 92.59 .0868 .0868
2 100 .8573 85.73 .0804 .1608
3 100 .7938 79.38 .0745 .2234
4 1100 .7350 808.50 .7583 3.0332
1066.24 1.0000 3.5042
Duration = 3.5 years
4(b). Computation of Duration (assuming 12% market yield)
(1) (2) (3) (4) (5) (6)
Year Cash Flow PV@12% PV of Flow PV as % of Price (1) x (5)
1 100 .8929 89.29 .0951 .0951
2 100 .7972 79.72 .0849 .1698
3 100 .7118 71.18 .0758 .2274
4 1100 .6355 699.05 .7442 2.9768
939.24 1.0000 3.4691
Duration = 3.47 years
19.6.
CURRENT CANDIDATE
BOND BOND
Dollar Investment 839.54 961.16
Coupon 90.00 110.00
i on One Coupon 2.59 3.16
Principal Value at Year End 841.95 961.71
Total Accrued 934.54 1,074.87
Realized Compound Yield 11.0125 11.4999
RCY = [SQRT (Total Accrued/Dollar Investment)-1] x 2
Value of swap: 48.6 basis points in one year
P0 = 45 x (16.04612) + 1,000 x (.11746) = 839.54
P0 = 55 x (15.53300) + 1,000 x (.10685) = 961.16
P0 = 45 x (15.80474) + 1,000 x (.13074) = 841.95
P0 = 55 x (15.31315) + 1,000 x (.11949) = 961.71
where:
16.04612 is the PVA factor and .11746 is the PV factor, both for 40 periods at 5.5 percent per period.
15.53300 is the PVA factor and .10685 is the PV factor, both for 40 periods at 5.75 percent per period.
15.80474 is the PVA factor and .13074 is the PV factor, both for 38 periods at 5.5 percent per period.
15.31315 is the PVA factor and .11949 is the PV factor, both for 38 periods at 5.75 percent per period.