Practice Problems
FIN 5210: Investments
Fall 2014
Problem Set 1: Security Valuation
1.
A bond is offered with face value of $1,000, a 10% coupon rate with semi-annual
payments, and twenty years to maturity. If the market interest rate is 12%, what
should be the price?
2.
If the market interest rate rises to 15%, what will be the new price for the bond in
question 1?
3.
If the market interest rate drops to 9% instead, what will be the new price for the
bond in question 1?
4.
If the market interest rate changes to 8%, what will be the new price for the bond in
question 1?
5.
If the market interest rate rises to 10%, what will be the new price for the bond in
question 1?
Note that when the market rate equals the coupon rate, the price equals the face value.
6.
If the market interest rate drops to 6% instead, what will be the new price for the
bond in question 1?
7.
If the market interest rate changes to 17%, what will be the new price for the bond in
question 1?
8.
If the market interest rate rises to 18%, what will be the new price for the bond in
question 1?
9.
If the market interest rate drops to 16% instead, what will be the new price for the
bond in question 1?
In the first three problems the market rate varied around 12%.1 In the next three
problems it varied around 8% (changing by 2% both up and down). In the final three
it varied around 17% (changing by 1% each way). Note that the capital gain from a
decrease of x% in the market interest rate would be greater that the capital loss from
an increase of x%. This relationship is known as convexity.
10. A bond is offered with face value of $1,000, a 10% coupon rate with semi-annual
payments, and ten years to maturity. If the market interest rate is 12%, what should
be the price?
11. If the market interest rate rises to 15%, what will be the new price for the bond in
question 10?
1It
changed by 3%, first going up from 12% to 15% and then going down from 12% to 9%.
Page 1 of 6
Practice Problems
FIN 5210: Investments
Fall 2014
12. If the market interest rate drops to 9% instead, what will be the new price for the
bond in question 10?
13. If the market interest rate changes to 8%, what will be the new price for the bond in
question 10?
14. If the market interest rate rises to 10%, what will be the new price for the bond in
question 10?
15. If the market interest rate drops to 6% instead, what will be the new price for the
bond in question 10?
16. If the market interest rate changes to 17%, what will be the new price for the bond in
question 10?
17. If the market interest rate rises to 18%, what will be the new price for the bond in
question 10?
18. If the market interest rate drops to 16% instead, what will be the new price for the
bond in question 10?
Note that the capital gains or losses from changes in market interest rates are greater the
longer the time to maturity. That is, bonds with longer maturities are riskier to hold.
19. A bond is offered with face value of $1,000, a 5% coupon rate with semi-annual
payments, and twenty years to maturity. If the market interest rate is 12%, what
should be the price?
20. If the market interest rate rises to 15%, what will be the new price for the bond in
question 19?
21. If the market interest rate drops to 9% instead, what will be the new price for the
bond in question 19?
22. If the market interest rate changes to 8%, what will be the new price for the bond in
question 19?
23. If the market interest rate rises to 10%, what will be the new price for the bond in
question 19?
24. If the market interest rate drops to 6% instead, what will be the new price for the
bond in question 19?
25. If the market interest rate changes to 17%, what will be the new price for the bond in
question 19?
26. If the market interest rate rises to 18%, what will be the new price for the bond in
question 19?
Page 2 of 6
Practice Problems
FIN 5210: Investments
Fall 2014
27. If the market interest rate drops to 16% instead, what will be the new price for the
bond in question 19?
28. A bond is offered with face value of $1,000, a 5% coupon rate with semi-annual
payments, and ten years to maturity. If the market interest rate is 12%, what should
be the price?
29. If the market interest rate rises to 15%, what will be the new price for the bond in
question 28?
30. If the market interest rate drops to 9% instead, what will be the new price for the
bond in question 28?
31. If the market interest rate changes to 8%, what will be the new price for the bond in
question 28?
32. If the market interest rate rises to 10%, what will be the new price for the bond in
question 28?
33. If the market interest rate drops to 6% instead, what will be the new price for the
bond in question 28?
34. If the market interest rate changes to 17%, what will be the new price for the bond in
question 28?
35. If the market interest rate rises to 18%, what will be the new price for the bond in
question 28?
36. If the market interest rate drops to 16% instead, what will be the new price for the
bond in question 28?
There is something else you should notice from the data you have generated in the above
problems. The relative price changes2 resulting from a change in the market interest
rate are not the same for all bonds of the same maturity. Price volatility is higher
when the coupon rate is lower. This is known as coupon bias.
37. Calculate the duration for each of the bonds in problems 1, 10, 19, and 28. The
duration is the weighted-average maturity of the package of payments, where the
weights are determined by each payment’s portion of total present value.
Do not try to calculate duration with the calculator. You would need to use a spreadsheet
to calculate the PV of each payment and then get the average maturity, so you may
want to use the answers given in the answer sheet. See how the duration reveals the
effects of convexity and coupon bias. The longer the duration, the more sensitive the
bond price is to interest rate fluctuations.
2That
is, (Pnew – Pold) / Pold.
Page 3 of 6
Practice Problems
FIN 5210: Investments
Fall 2014
38. A bond is offered with face value of $1,000, a 10% coupon rate with semi-annual
payments, and twenty years to maturity. If the market price is $818.18, what market
interest rate is implied? That is, what is the yield to maturity?
39. What is the yield to maturity for the above bond if the market price is $950?
40. What is the yield to maturity for the above bond if the market price is $850?
41. What is the yield to maturity for the above bond if the market price is $1,050?
42. What is the yield to maturity for the above bond if the market price is $1,150?
43. What is the yield to maturity for the above bond if the market price is $1,100?
44. What is the yield to maturity for the above bond if the market price is $1,025?
45. A high tax bracket investor is comparing two bond issues for possible inclusion in
her portfolio. She expects market interest rates to be declining slowly but steadily
over the foreseeable future. Bond A is a 6% ten-year Aaa-rated bond selling at
$750.76. Bond B is an 8% ten-year Aaa-rated bond selling at $875.38. What is the
yield to maturity for each bond, assuming semi-annual payment of coupons? Which
would you recommend, based on the investor's expectations about future interest
rates? She is not interested in current income, but rather in capital appreciation.
46. If the market interest rate is 15%, what should be the price of a share of 8% preferred
stock with par value of $100?
47. If the market interest rate is 10%, what should be the price of a share of 8% preferred
stock with par value of $100?
48. If the market interest rate is 12%, what should be the price of a share of 8% preferred
stock with par value of $100?
49. If the market interest rate is 8%, what should be the price of a share of 8% preferred
stock with par value of $100?
50. If a share of 8% preferred stock with par value of $100 is selling for $75, what is the
market interest rate?
51. If a share of 8% preferred stock with par value of $100 is selling for $85, what is the
market interest rate?
52. If a share of 8% preferred stock with par value of $100 is selling for $65, what is the
market interest rate?
53. If a share of 8% preferred stock with par value of $100 is selling for $80, what is the
market interest rate?
Page 4 of 6
Practice Problems
FIN 5210: Investments
Fall 2014
54. Suppose an established family business is for sale to the employees. The family has
already removed all the assets that can possible be taken from the business without
killing its viability, so the only thing left of value would be the cash flows from
future operations. The amount that can be committed for debt service is $10 million
per year. The employee stock ownership trust has negotiated a loan with 15%
interest that would be repaid with equal annual installments over a period of eight
years. Calculate the price the employee ownership trust can pay for the business.
55. Art Grunnion wants to start a company that would build open-ocean speedboats of
the highest possible quality. He would put $1 million into the company on
September 18, 2014. Even with all its earnings plowed back into the company, he
expects that he will have to follow up his initial investment with additional $1
million investments on the company's anniversary date in each of the next four years
(2015, 2016, 2017 and 2018). In 2019 (assume that it will be on the 18th of
September), he expects to be able to sell the company for $10 million. Calculate the
expected IRR for this bit of “venture capital financing.” Is it a good investment, if
the opportunity cost of capital is 15%?
56. A venture capital partnership is considering whether to fund a new pharmaceuticals
research company. The initial investment to be paid immediately would be $5
million. Then the expectations for future investment are $10 million a year from
now, $20 million two years from now, $50 million three years from now, and $100
million four years from now. Five years from now the venture capitalists expect to
sell the company for $1 billion. Is it a good investment, if the opportunity cost of
capital is 15%? Calculate NPV and IRR.
57. Common stock for Red River Enterprises paid a dividend of $1 per share this year,
and dividends are expected to grow at the rate of 8% per year for the foreseeable
future. If an investor assumes that the risk level of Red River warrants a 12% rate of
return, what would be the intrinsic value per share, using the normal growth model
developed by Myron Gordon?
58. Recompute the value of Red River common stock assuming an expected growth rate
of 10%.
59. Recompute the value of Red River common stock assuming an expected growth rate
of 11%.
60. Recompute the value of Red River common stock assuming an expected growth rate
of 11.5%.
61. Recompute the value of Red River common stock assuming an expected growth rate
of 11.9%.
Note that the Gordon Model has a tendency to “explode” when the growth rate is close to
the required return. What else is wrong with it?
Page 5 of 6
Practice Problems
FIN 5210: Investments
Fall 2014
62. Assume Rf = 10% and Rm = 15%. What is the required rate of return for an
investment with b = 2?
63. Assume Rf = 10% and Rm = 15%. A stock with b = 2 has expected return of 17%.
Is it overvalued, undervalued, or correctly valued?
64. Assume Rf = 10% and Rm = 15%. A stock with b = .5 has expected return of 14%.
Is it overvalued, undervalued, or correctly valued?
65. Assume Rf = 10% and Rm = 15%. A stock with b = 1.6 has expected return of 18%.
Is it overvalued, undervalued, or correctly valued?
66. An investment is twice as risky as average, in terms of relevant risk. What is its
beta?
67. An investment is half as risky as average, in terms of relevant risk. What is its beta?
Page 6 of 6
Practice Problems
FIN 5210: Investments
Solutions: Set 1
1.
FV is 1000, PMT is 50, interest per year is 12, P/YR is 2, N is 40, mode is END, calculate
PV. The answer is $849.54. (The negative sign in the display is due to the sign
convention.)
2.
The only thing that needs to be changed in the calculator entries would be the interest per
year, which now is 15. (Here are the other entries: FV is 1000, PMT is 50, P/YR is 2, N is
40, mode is END). When you calculate PV, the answer is $685.14. (The negative sign in
the display is due to the sign convention.)
3.
Once again, the only thing that needs to be changed in the calculator entries would be the
interest per year, which now is 9. The PV is $1,092.01.
4.
Once again, the only thing that needs to be changed in the calculator entries would be the
interest per year, which now is 8. The PV is $1,197.93.
5.
Interest per year is 10 (other entries remain unchanged from the above problems). PV is
$1,000.
6.
Interest per year is 6 (other entries remain unchanged from the above problems). PV is
$1,462.30
7.
Interest per year is 17 (other entries remain unchanged from the above problems). PV is
$603.99.
8.
Interest per year is 18 (other entries remain unchanged from the above problems). PV is
$569.71
9.
Interest per year is 16 (other entries remain unchanged from the above problems). PV is
$642.26
10.
FV is 1000, PMT is 50, interest per year is 12, P/YR is 2, N is 20, mode is END, calculate
PV. The answer is $885.30. (The negative sign in the display is due to the sign
convention.)
11.
The only thing that needs to be changed in the calculator entries would be the interest per
year, which now is 15. (Here are the other entries: FV is 1000, PMT is 50, P/YR is 2, N is
20, mode is END). When you calculate PV, the answer is $745.14. (The negative sign in
the display is due to the sign convention.)
12.
Interest per year is 9 (other entries remain unchanged from the above problems). PV is
$1,065.04
13.
Interest per year is 8 (other entries remain unchanged from the above problems). PV is
$1,135.90
14.
Interest per year is 10 (other entries remain unchanged from the above problems). PV is
$1,000.00
15.
Interest per year is 6 (other entries remain unchanged from the above problems). PV is
$1,297.55
16.
Interest per year is 17 (other entries remain unchanged from the above problems). PV is
$668.78
17.
Interest per year is 18 (other entries remain unchanged from the above problems). PV is
$634.86
Prof. Kensinger
Fall 2014
page 1
Practice Problems
FIN 5210: Investments
Solutions: Set 1
18.
Interest per year is 16 (other entries remain unchanged from the above problems). PV is
$705.46
19.
FV is 1000, PMT is 25, interest per year is 12, P/YR is 2, N is 40, mode is END, calculate
PV. The answer is $473.38. (The negative sign in the display is due to the sign
convention.)
20.
The only thing that needs to be changed in the calculator entries would be the interest per
year, which now is 15. (Here are the other entries: FV is 1000, PMT is 25, P/YR is 2, N is
40, mode is END). When you calculate PV, the answer is $370.28. (The negative sign in
the display is due to the sign convention.)
21.
Interest per year is 9 (other entries remain unchanged from the above problems). PV is
$631.97
22.
Interest per year is 8 (other entries remain unchanged from the above problems). PV is
$703.11
23.
Interest per year is 10 (other entries remain unchanged from the above problems). PV is
$571.02
24.
Interest per year is 6 (other entries remain unchanged from the above problems). PV is
$884.43
25.
Interest per year is 17 (other entries remain unchanged from the above problems). PV is
$321.13
26.
Interest per year is 18 (other entries remain unchanged from the above problems). PV is
$300.77
27.
Interest per year is 16 (other entries remain unchanged from the above problems). PV is
$344.15
28.
FV is 1000, PMT is 25, interest per year is 12, P/YR is 2, N is 20, mode is END, calculate
PV. The answer is $598.55. (The negative sign in the display is due to the sign
convention.)
29.
The only thing that needs to be changed in the calculator entries would be the interest per
year, which now is 15. (Here are the other entries: FV is 1000, PMT is 25, P/YR is 2, N is
20, mode is END). When you calculate PV, the answer is $490.28. (The negative sign in
the display is due to the sign convention.)
30.
Interest per year is 9 (other entries remain unchanged from the above problems). PV is
$739.84
31.
Interest per year is 8 (other entries remain unchanged from the above problems). PV is
$796.15
32.
Interest per year is 10 (other entries remain unchanged from the above problems). PV is
$688.44
33.
Interest per year is 6 (other entries remain unchanged from the above problems). PV is
$925.61
34.
Interest per year is 17 (other entries remain unchanged from the above problems). PV is
$432.20
Prof. Kensinger
Fall 2014
page 2
Practice Problems
FIN 5210: Investments
Solutions: Set 1
35.
Interest per year is 18 (other entries remain unchanged from the above problems). PV is
$406.64
36.
Interest per year is 16 (other entries remain unchanged from the above problems). PV is
$460.00
37.
8.20 years, 6.31 years, 9.42 years, 7.27 years
38.
P/YR is 2, mode is END, FV is 1000, PMT is 50, N is 40, PV is –818.18, calculate the
interest per year. (The negative sign for PV is due to the sign convention.) Answer is
12.49%
39.
PV is –950 (other entries remain unchanged from problem 38). Interest per year is 10.61%
40.
PV is –850 (other entries remain unchanged from problem 38). Interest per year 11.99%
41.
PV is –1050 (other entries remain unchanged from problem 38). Interest per year 9.44%
42.
PV is –1150 (other entries remain unchanged from problem 38). Interest per year 8.43%
43.
PV is –1100 (other entries remain unchanged from problem 38). Interest per year 8.92%
44.
PV is –1025 (other entries remain unchanged from problem 38). Interest per year 9.71%
45.
Both bonds yield 10%; but she would pick Bond A because it gives lower reinvestment risk
and a smaller income tax burden, with the same yield. Given her expectations for declining
interest rates, bond A would also offer a higher expected capital gain.
46.
P0 = $8 / 0.15 = $53.33
47.
P0 = $8 / 0.10 = $80.00
48.
P0 = $8 / 0.12 = $66.67
49.
P0 = $8 / 0.08 = $100.00
50.
R = 8 / 75 = 10.67%
51.
R = 8 / 85 = 9.41%
52.
R = 8 / 65 = 12.31%
53.
R = 8 / 80 = 10.00%
54.
See Example 7, Topic 3, slide 58. Value is $44.9 million
55.
See Example 8, Topic 3, slide 59. IRR is 24.07%
56.
See Example 9, Topic 3, slide 60. IRR is 108.99%
57.
P0 = (1 x 1.08) / (0.12 – 0.08) = $27.00
58.
P0 = (1 x 1.10) / (0.12 – 0.10) = $55.00
59.
P0 = (1 x 1.11) / (0.12 – 0.11) = $111.00
60.
P0 = (1 x 1.115) / (0.12 – 0.115) = $223.00
61.
P0 = (1 x 1.119) / (0.12 – 0.119) = $1,119.00
62.
Rrequired = 10% + 2(15% – 10%) = 20%
63.
Since the required return is 20%, the expected return at the current price is too low. The
price will have to come down. The stock is currently overvalued.
Prof. Kensinger
Fall 2014
page 3
Practice Problems
FIN 5210: Investments
Solutions: Set 1
64.
Rrequired = 10% + 0.5(15% – 10%) = 12.5%. Since the required return is 12.5%, the
expected return at the current price is too high. The price will have to rise. The stock is
currently undervalued.
65.
. Rrequired = 10% + 1.6(15% – 10%) = 18% Since the required return is 18%, the expected
return at the current price is just right. The stock is priced at equilibrium.
66.
β=2
67.
β = 0.5
Prof. Kensinger
Fall 2014
page 4
FIN 5210: Investments
Practice Problems
Solutions: Set 1
Table Illustrating Coupon Bias and Convexity
old
rate
new
rate
old price
20-year, 10% bonds
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
10-year, 10% bonds
12%
12%
8%
8%
17%
17%
20-year, 5% bonds
10-year, 5% bonds
Prof. Kensinger
new price
capital gain
(loss)
relative
change
$849.54
$849.54
$1,197.93
$1,197.93
$603.99
$603.99
$685.14
$1,092.01
$1,000.00
$1,462.30
$569.71
$642.26
($164.40)
$242.47
($197.93)
$264.37
($34.28)
$38.27
-19.35%
+28.54%
-16.52%
+22.07%
-5.68%
+6.34%
15%
9%
10%
6%
18%
16%
$885.30
$885.30
$1,135.90
$1,135.90
$668.78
$668.78
$745.14
$1,065.04
$1,000.00
$1,297.55
$634.86
$705.46
($140.16)
$179.74
($135.90)
$161.65
($33.92)
$36.68
-15.83%
+20.30%
-11.96%
+14.23%
-5.07%
+5.48%
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$473.38
$473.38
$703.11
$703.11
$321.13
$321.13
$370.28
$631.97
$571.02
$884.43
$300.77
$344.15
($103.10)
$158.59
($132.09)
$181.32
($20.36)
$23.02
-21.78%
+33.50%
-18.79%
+25.79%
-6.34%
+7.17%
12%
12%
8%
8%
17%
17%
15%
9%
10%
6%
18%
16%
$598.55
$598.55
$796.15
$796.15
$432.20
$432.20
$490.28
$739.84
$688.44
$925.61
$406.64
$460.00
($108.27)
$141.29
($107.71)
$129.46
($25.56)
$27.80
-18.09%
+23.61%
-13.53%
+16.26%
-5.91%
+6.43%
Fall 2014
page 5
FIN 5210: Investments
Practice Problems
Solutions: Set 1
Illustration of Convexity
$1,200.00
$1,000.00
Price
$800.00
$600.00
$400.00
20-year
$200.00
10-year
5-year
1-year
$0.00
1
3
5
7
9
11
13
15
Rate
Prof. Kensinger
Fall 2014
17
19
21
23
25
27
29
(%)
page 6