MØA 155 - Fall 2011
PROBLEM SET: Hand in 1
Exercise 1.
An investor buys a share for $100 and sells it five years later, at the end of the year, at the price of $120.23.
Each year the stock pays dividends of $12 per share. The annual average rate of inflation is 2%.
What is the average effective real return on the investment?
1. 10.30%
2. 12.50%
3. 12.75%
4. 13.00%
5. I choose not to answer.
Exercise 2. NPV (BM 3.4) [2]
A factory costs $800,000. You reckon that it will produce an inflow after operating costs of $170,000 a year
for 10 years.
1. If the opportunity cost of capital is 14 percent, what is the net present value of the factory?
2. What will the factory be worth at the end of 5 years?
Exercise 3. House Sale (RWJ 4.7) [3]
You are selling your house. The Smiths have offered you $115,000. They will pay you immediately. The
Joneses have offered you $150,000, but they cannot pay you until three years from today. The prevailing
interest rate is 10%.
1. Which offer should you choose?
Exercise 4. Projects [3]
A project costs 100 today. The project has positive cash flows of 100 in years one and two. At the end of
the life of the project there are large environmental costs resulting in a negative cash flow in year 3 of −95.
Determine the internal rate(s) of return for the project.
Exercise 5. Machine [4]
A company is considering its options for a machine to use in production. At a cost of 47 they can make some
small repairs on their current machine which will make it last for 2 more years. At a higher cost of 90 they
can make some more extensive repairs on their current machine which will make it last for 4 more years. A
new machine costs 300 and will last for 8 years. The company is facing an interest rate of 10%. Determine
the best action.
Exercise 6. Q [2]
Equity in the company Q has an expected return of 12%, a beta of 1.4 and a standard deviation of 20%.
The current risk free interest is 10%.
1. What is the current expected market return?
Exercise 7.
You are considering paying 2 million NOK to buy a new house. You expect to sell your old house for 800
thousand, and need to finance the remainder by borrowing. The bank quotes you a 30 year loan with an
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interest rate of 6% to be paid as a monthly annuity, i.e. in monthly installments of equal size. The first
amount is due at the end of the first month. Interest is calculated using monthly compounding.
What is the amount you will be paying each month?
(a) NOK 6,982
(b) NOK 7,159
(c) NOK 7,195
(d) NOK 11,990
(e) I choose not to answer.
Exercise 8.
A corporation’s Annual Report contains the following information:
Sales: 2,000,000 kr.
Variable costs: 850,000 kr.
Overhead costs: 395,000 kr.
Depreciation: 248,000 kr.
Corporate tax rate: 34%
Calculate the corporation’s after-tax cash flows
1. 582,620 kr.
2. 724,620 kr.
3. 755,000 kr.
4. 977,620 kr.
5. I choose not to answer.
Exercise 9. Lucky Pierre [5]
Hi-Tech industries is involved in genetic research. It has enjoyed the astounding dividend growth rate of
25% recently. Next period’s dividend is expected to be $20 and the current share price is $200.
1. Your stockbroker, Lucky Pierre, believes that the stock is under-priced. He tells you that it has an
expected return of 35% per year, but once “Wall Street” sees its mistake, the stock price will shoot up.
What assumptions did Pierre make to reach his conclusion?
2. If Pierre is correct (and the expected return is 35%), what is the expected stock price next period?
3. You, having taken a course in Finance, do not really trust Pierre’s work (remember, he gets paid on
a commission basis.) You feel that the dividend growth will slow down after 1 more year of 25% to a
modest 10% per year. What are the expected dividend payments for the firm for the next three years?
4. Suppose you expect the stock return to be 20% once the growth rate slows down to 10% (after next
year). What will the expected stock price be in year 1? What is the current expected return on the
stock this year?
Exercise 10. [3]
A bond promises the following sequence of payments:
t
Cashflow Xt
= 1
= 10
2
2
10
3
10
4
110
The interest rates rt and prices dt of future risk free cash flows are as follows
t
rt
dt
=
=
=
1
5.3%
0.95
2
5.4%
0.9
3
5.6%
0.85
4
5.7%
0.80
Interest rates are compounded annually.
1. Calculate the bond’s price
Exercise 11. Bond [4]
The appropriate discount rate for cash flows received one year from now is 7.5%. The appropriate discount
rate for cash flows received two years from now is 11%. The appropriate discount rate for cash flows received
three years from now is 14%. Interest rates are compounded annually.
1. What is the price of a two–year bond with a 6% (annual) coupon and a face value of 1000?
2. What is the yield–to–maturity of this bond?
Exercise 12. [4]
A 3 year bond with a face value of $100 makes annual coupon payments of 10%. The current interest rate
(with annual compounding) is 9%.
1. Find the bond’s current price.
2. Suppose the interest rate changes to 10%, determine the new price of the bond by direct calculation.
3. Instead of direct calculation, use duration to estimate the new price and compare it to the correct
price.
4. Use convexity to improve on your estimation using duration.
Exercise 13. Compounding [2]
A bank quotes an interest rate of 10% with quarterly compounding.
1. What is the equivalent interest rate with continous compounding?
Exercise 14. Doubling [2]
How long does it take to double a $100 initial investment when investing at a 5% continously compounded
interest rate?
Exercise 15. Hilda Hornbill (BM 8.7)
Hilda Hornbill has invested 60% of her money in share A and the remainder in share B. She assesses the
prospects as follows.
A
B
Expected return
15% 20%
Standard deviation
20% 22%
Correlation between returns
0.5
1. What is the expected return and standard deviation of returns on her portfolio?
2. How would your answer be changed if the correlation coefficient were 0 or -0.5?
3. Is Ms. Hornbill’s portfolio better or worse than one invested entirely in share A or is it not possible to
say?
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Empirical
Solutions
MØA 155 - Fall 2011
PROBLEM SET: Hand in 1
Exercise 1.
The return rn on the five year investment in nominal terms:
0 = −100 +
12
12
12 + 120.34
12
12
+
+
+
+
2
3
4
(1 + rn ) (1 + rn )
(1 + rn )
(1 + rn )
(1 + rn )2
The IRR is: rn = 15%.
In real terms, the exact calculation is
r=
1 + rn
1.15
−1=
− 1 = 0.1275 = 12.75%
1+i
1.02
It is an approximation only to estimate
r ≈ rn − i = 0.15 − 0.02 = 0.13
The correct answer is 3, 12.75%
Exercise 2. NPV (BM 3.4) [2]
1. All numbers are in thousands:
PV of annuity = 170 · 5.216 = 886.72
N P V = 886.72 − 800 = 86.72
2. After 5 years, the value of is equal the PV of getting $170 per year for 5 years:
170 · 3.433 = 583.61
Exercise 3. House Sale (RWJ 4.7) [3]
1. You need to compare the present values of the two offers. The PV of the Smiths’ offer is $115,000.
The PV of the Joneses is:
150000
= 112, 697
PV =
(1 + r)3
Thus, the Smiths offer the best terms (highest PV), accept their offer.
Exercise 4. Projects [3]
The following picture shows the NPV as a function of the interest rate, and illustrates the fact that there
are two solutions y to the problem of solving
0 = −100 +
100
100
−95
+
+
2
1 + y (1 + y)
(1 + y)3
4
10
5
0
-5
NPV-10
-15
-20
-25
-30
-30
-20
-10
0
interest
10
20
30
Exercise 5. Machine [4]
One way to go about this is to find the net present value of each alternative by repeating each to make it
last 8 years.
Small repairs
t=
0 2 4 6
Ct = 47 47 47 47
N P V = 47 +
47
47
47
+
+
= 144.475
2
4
(1 + 0.1)
(1 + 0.1)
(1 + 0.1)6
Larger repairs
t
Ct
=
=
0
90
N P V = 90 +
1 2 3 4
0 0 0 90
90
= 151.471
(1 + 0.1)4
The cheapest is the small repairs.
Exercise 6. Q [2]
E[r] = rf + (E[rm ] − rf )β
E[rm ] =
0.12 − 0.1
E[r] − rf
+ rf =
+ 0.1 = 11.4%
β
1.4
Exercise 7.
Annuity factor(r=0.005 n=360) = 166.792
Amount = 1.2 mill / 166.792 = 7194.58
(c) is correct
Exercise 8.
(2000 − 850 − 395 − 248)(1 − 0.34) + 248 = 582.62
5
(Numbers in thousands)
1. is the correct answer
Exercise 9. Lucky Pierre [5]
1. He has used the constant dividend growth assumption.
P0 =
D1
20
=
= 200
r−g
r − 0.25
20
= 0.10
200
r = 0.35
r − 0.25 =
2.
D1 + P1
1+r
P1 = P0 (1 + r) − D1 = 200 · 1.35 − 20 = 250
P0 =
3.
D1 = 20 (given)
D2 = 20 · 1.1 = 22
D3 = 20 · (1.1)2 = 24.2
Dt = 20 · (1.1)t−1
4.
D2
22
=
= 200
r−g
0.20 − 0.10
P1 + D1
220 + 20
P0 =
=
= 200
1 + r1
1 + r1
P1 =
200(1 + r) = 240
40
= 0.20 = 20%
r1 =
200
Exercise 10. [3]
1.
Bond Price =
4
X
Pt Xt = 0.95 · 10 + 0.9 · 10 + 0.85 · 10 + 0.8 · 110 = 115
t=1
Exercise 11. Bond [4]
1. Price
P0 =
60
1060
+
= 916.13
1.075 (1.11)2
2. Yield to maturity is found by calculating the IRR of the following cashflows
t
Ct
=
0
= −916.13
1
60
IRR = 0.108917
The YTM is 10.89%.
6
2
1060
Exercise 12. [4]
1. The bond price:
t
Ct
NPV = 0 +
=
=
0
0
1
10
2
10
3
110
10
10
110
+
+
= 102.531
1
2
(1 + 0.09)
(1 + 0.09)
(1 + 0.09)3
2. If the interest rate increases to 10%, the bond will be selling at par, equal to 100, which can be
confirmed with direct computation:
NPV = 0 +
10
110
10
+
+
= 100
1
2
(1 + 0.1)
(1 + 0.1)
(1 + 0.1)3
3. Calculate the bond’s duration:
t
1
2
3
Sum
Bondprice
Duration
Ct P V (Ct ) tP V (Ct )
10
9.2
9.2
10
8.4
16.8
110
84.9
254.8
102.5
280.8
102.531
2.74
Modified duration:
D
2.74
=
= 2.51
1+r
1.09
Let us now calculate the change in the bond price
D∗ =
∆P
= −D∗ ∆y = −2.51 · 0.01 = −0.0251
P
Which means theat the bond price changes to:
∆P
P + δP = 102.531 +
P = 102.531 − 0.0251 · 102.531 = 99.957
P
4. Calculate the bond’s convexity
t
1
2
3
Sum
Bondprice
Convexity
Ct
10
10
110
P V (Ct ) P V (Ct )(t2 + t)
9.2
18.3
8.4
50.5
84.9
1019.3
102.5
1088.1
102.531
8.93
Recalculating the change in the bond price using convexity:
∆P
1
1
= −D∗ ∆y + (Convexity) = −2.51 · 0.01 + 8.93(0.01)2 = −0.0251 + 0.00044 = −0.02465
P
2
2
Use this to re-estimate the bond price:
∆P
P = 102.531(1 − 0.02465) = 100.0036
P + δP = 102.531 1 +
P
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Exercise 13. Compounding [2]
1. Equivalent annual rate
(1 +
0.1 4
) − 1 = 10.381289063%
4
Continous rate
ln(1 + r) = 9.8770450366%
Exercise 14. Doubling [2]
e0.05t = 2
ln 2
.05 = 13.863
t=
0
It takes 13.863 years
Exercise 15. Hilda Hornbill (BM 8.7)
A
ω
0.6
E[r̃] 0.15
σ
0.20
ρAB
B
0.4
0.20
0.22
0.5
1.
E[r̃p ]
σ 2 (r̃p )
=
ωA E[r̃A ] + ωB E[r̃B ]
=
17%
=
2 2
2 2
ωA
σA + ωB
σB
+2ωA ωB σA σB ρAB
=
0.62 0.22 + 0.42 0.222
+2 · 0.6 · 0.4 · 0.2 · 0.22 · ρAB
=
0.0144 + 0.007744
+2 · 0.6 · 0.4 · 0.2 · 0.22 · ρAB
=
σp
=
0.0327
√
0.0327 = 18.08%
2. If ρAB = 0, then
σ 2 (r̃p )
=
2 2
2 2
ωA
σA + ωB
σB
=
0.0144 + 0.007744
=
0.022144
=
14.88%.
+2ωA ωB σA σB · 0
σ(r̃p )
If ρAB =- 12 , then
σ 2 (r̃p )
σ(r̃p )
=
0.0144 + 0.007744 − 0.01056
=
0.01158
=
10.76%
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3. The portfolio is better, since it has higher expected return and lower variance.
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