Investment Science
Problem Session
(Chapter 2-Chapter 4)
MS&E242
Summer 2014
Chapter 4:
1. Spot rates and forward rates
2
Chapter 4:
1. Spot rates and forward rates
Problem 1: Chapter 4, Exercise 2
Given the (yearly) spot rate curve s=(5.0, 5.3, 5.6, 5.8, 6.0, 6.1),
find the spot rate curve for next year.
3
Chapter 4:
2. Discount factor and running PV
4
Running present value
5
Running present value
6
Present value updating
7
Example: present value updating
Year k 0 1 2 3 4 5 6 7 Cash flow 20 25 30 35 40 30 20 10 Short rate 0.060 0.069 0.075 0.080 0.084 0.086 0.090 Discount factor PV(k) 0.943 0.935 0.930 0.926 0.922 0.921 0.917 169.06 158.01 142.19 120.61 92.46 56.87 29.17 10.00 8
Present value updating (cont.)
9
Example: floating rate bond
10
Chapter 4:
2. Discount factor and running PV
Problem 2: Chapter 4, Exercise 11
A (yearly) cash flow stream is x=(-40, 10, 10, 10, 10, 10, 10). The
spot rate are those of Exercise 2.
(a) Find the current discount factor d0,k and use them to determine
the (net) present value of the stream.
(b) Find the series of expectations dynamics short-rate discount
factors, and use the running present value method to evaluate
them.
11
Chapter 4:
2. Discount factor and running PV
Problem 2: Chapter 4, Exercise 11
12
Duration in the context of the term structure
12% 10% Rate 8% 6% 4% 2% 0% 0 2 4 6 8 10 12
14
16
18
20 Years 13
Fisher-Weil duration
14
Fisher-Weil duration (cont.)
15
Fisher-Weil formulas
16
Chapter 4:
3. Construction of zero coupon bonds
Problem 3: Chapter 4, Exercise 7
17
Chapter 4:
3. Construction of zero coupon bonds
Problem 3: Chapter 4, Exercise 7
18
Chapter 3:
1. Amortization
19
Problem 4: Chapter 3, Exercise 5
(Mortgage restructuring) An investor purchased a small
apartment building for $250,000. She made a down payment of
$50,000 and financed the balance with a 30-year, fixed rate
mortgage at 12% annual interest rate, compounded monthly. For
exactly 20 years she has equal-sized monthly payments as
required by the terms of the loan. Now she has the opportunity
to restructure the mortgage by refinancing the balance. She
could borrow the current balance, payoff the original loan, and
assume a new loan for the balance. (No points for any other
charges are involved in the transaction.) The new loan is a 20year, fixed-rate loan at 9%, compounded monthly, to be paid in
equal monthly installments. Suppose she has a risk-free savings
account that pays 5%, compounded monthly. Should she
restructuring the mortgage?
20
Problem 4: Chapter 3, Exercise 5
21
Chapter 3:
2. Bond Formulas
• Yield
• Duration
22
Problem 5: Chapter 3, Exercise 7
The Z corporation issues a 10%, 20-year bond at a time when yields are
10%. The bond has a call provision that allows the corporation to force a
bond holder to redeem his or her bond at face value plus 5%. After 5
years the corporation finds that exercise of this call provision is
advantageous. What can you deduce about the yield at that time?
(Assume one coupon payment per year)
23
Problem 6: 2013 Midterm, problem 5
Duration (10 points). Rank the following securities according to
their interest rate sensitivity in declining order:
1) A 30-year zero coupon bond at 6%
2) A 30-year fixed-rate mortgage with an interest rate of 6%
3) A 30-year bond with a coupon rate of 6%
4) A 30-year bond with a coupon rate of 3%
Solution: 1) - 4) - 3) - 2)
24
Chapter 3:
3. Immunization
25
Duration of a portfolio
26
Immunization
27
Example: Immunization
28
Example: Immunization (cont.)
29
Shortcomings of immunization
30
Problem 7: Chapter 3, Exercise 16
Consider the four bonds having annual payments as shown in Table 3.9.
They are traded to produce a 15% yield.
(a) Determine the price of each bond.
(b) Determine the duration of each bond (not the modified duration).
(c) Which bond is most sensitive to a change in yield?
Solution:
(a)PA=885.84; PB=771.68; PC=657.52; PD=869.57
(b)DA=2.72; DB=2.84; DC=3.00; DD=1.00
(c)C is most sensitive to a change in yield.
31
Problem 7: Chapter 3, Exercise 16 (cont.)
(d) Suppose you owe $2,000 at the end of 2 years. Concern about interest rate
risk suggests that a portfolio consisting of the bonds and the obligation should
be immunized. If VA, VB, VC, and VD are the total values of bonds purchased of
types A, B, C and D, respectively, what are the necessary constraints to
implement the immunization? [Hint: There are two equations. (Do not solve)]
32
Problem 7: Chapter 3, Exercise 16 (cont.)
(e) In order to immunize the portfolio, you decide to use bond C and one other
bond. Which other bond should you choose? Find the amounts (in total value)
of each of these to purchase.
33
Chapter 2:
1. Compounding and effective interest rate
r: annual interest rate, compounded with m periods in one year
reff : effective yearly interest rate without compounding
Problem 8: Chapter 2, Exercise 3
Find the corresponding effective rates for
(a)3% compounded monthly
(b)18% compounded monthly
(c)18% compounded quarterly
(d)18% compounded continuously (not included)
34
Chapter 2:
1. Compounding and effective interest rate
r: annual interest rate, compounded with m periods in one year
reff : effective yearly interest rate without compounding
Problem 8: Chapter 2, Exercise 3
Find the corresponding effective rates for
(a)3% compounded monthly: r=3%, m=12!reff=3.04%
(b)18% compounded monthly: r=18%, m=12!reff=19.56%
(c)18% compounded quarterly: r=18%, m=4!reff=19.25%
(d)18% compounded continuously: r=18%!reff=19.72%
35
Chapter 2:
2. Project Evaluation: NPV vs. IRR
Problem 9: Chapter 2, Exercise 8
A young couple has made a nonrefundable deposit of the first
month’s rent (equal to $1,000) on a 6-month apartment lease.
The next day they find a different apartment that they like just as
well, but its monthly rent is only $900. They plan to be in the
apartment only 6 months. Should they switch to the new
apartment? What if they plan to stay 1 year? Assume an interest
rate of 12%
36
Chapter 2:
2. Project Evaluation: NPV vs. IRR
Solution： 1. Identify the cash flows of each option
37
Chapter 2:
2. Project Evaluation: NPV vs. IRR
2. Find the NPV for each option
For 6 months：
NPV1>NPV2
They should not switch
For 1 year：
NPV1<NPV2
They should switch
38
Chapter 2:
2. Project Evaluation: NPV vs. IRR
Summary： 1. Identify the cash flows of each project
• Positive or negative?
• Compounding methods?
2. Evaluation criteria:
• NPV>0 for the project to be feasible, the larger the better
• IRR> discount rate for the project to be feasible, the larger
the better
39
Chapter 2:
2. Project Evaluation: NPV vs. IRR
Summary： 3. NPV vs. IRR
• Disadvantages of NPV
• Complex and require assumptions of each stage
• Disadvantages of IRR
• Not account for the changing discount rate over time
• Cash flows may have multiple IRR
• The external discount rate of the project is unknown
• In most cases, the NPV method is superior.
40
Problem 10: Chapter 2, Exercise 11
You are considering the purchase of of a nice home. It is every way
perfect for you and in excellent condition, except for the roof. The roof
has only 5 years of life remaining. A new roof would last 20 years, but
would cost $20,000. The house is expected to last forever. Assuming that
costs will remain constant and that the interest rate is 5%, what value
would you assign to the existing roof?
Solution:)
1) Change)roof)now,)then)every)20)years:)
!
1
!"! = $20,000×
= $32,097)
(1.05)!"!
!!!
2) Use)the)existing)roof)for)5)years)and)then)change)to)a)new)one,)then)every)20)
years:))
!"!
!"! =
= $25,149)
(1.05)!
The)value)of)the)existing)roof)is)the)difference)of)!"! )and)!"! :))
!"! − !"! = $6,948)
41