Assignment 1: (due November 6)
Latest Revision: October 30, 1995 (to add Eurodollar yields)
Note: This problem set is to be completed separately by each student -- which means that it is not acceptable to compare solutions. In some of the questions, it will greatly
help if you use EXCEL for the calculations.
Make sure to print the calculations and explain what you are doing.
Salomon Bros. quotes (stated rate) the yields to maturity compounded semi-annually on zero-coupon Treasury securities (STRIPs) as:
1 year 7.87
2 year 8.20
3 year 8.41
5 year 8.90
7 year 9.22
10 year 9.50
20 year 9.97
30 year 8.92
The interest is compounded semi-annually. (Hint: To find semi-annual interest rates, divide given interest rates by 2.)
(a) At these yields, what are the actual prices of all of these zero coupon bonds with maturities from 1 to 30 years?
(b) What are the durations and modified durations of these STRIPs?
(c) If interest rates moved up 2% on all of these, what would be the new set of zero coupon prices? What would the holding period return be if you bought at the above quoted price and sold after the rates moved up 2%?
(d) Answer question (c) for a drop of 2% in interest rates.
(e ) What are the predicted percentage changes using modified duration for these plus or minus 2% rate moves? Describe the errors made by the duration approximation to the gains and losses, i.e., how do the errors behave across the maturities, and can you account for the pattern in these errors.
(f) Assume that forward interest rates are constant between maturities listed in the part (a). For example, assume that the four semi-annual intervals between the 5 year and 7 year have the same forward rate. Calculate the (semi-annual) forward rates implied between the following maturities:
(i)Year 0 and 1
(ii)Year 1 and 2
(iii)Year 2 and 3
(iv)Year 3 and 5
(v)Year 5 and 7
(vi)Year 7 and 10
(vii)Year 10 and 20
(viii)Year 20 and 30
Graph these rates and the yield curve of part (a) on the same graph. Comment on them.
(a)Compute the prices of Treasury securities with 8% coupons (annual stated rate paid semi-annually) and maturities of 5, 10 and 20 years, using three different interest rate scenarios: 8%, 9% and 10%. (Be sure to show all the steps, i.e., breaking the bond into an annuity part and a principal part. Also assume that the par value of the securities are $100.)
(b) From those prices, what modified durations would you infer for each of those maturities given the 9% market interest rate scenario. (Hint: Price elasticity is a good estimate for modified duration. Calculate two different price elasticities for each maturity, depending on the direction of the interest rates. Approximate the modified duration for each maturity by averaging these two results.)
(c) Compare the price volatilities (modified durations) of the coupon Treasuries to those of the STRIPs in question 1. Why are they different?
A standard futures contract for Treasury bills calls for delivery of a T-bill with 90-92 days to maturity and a par amount of $1,000,000. Assuming delivery of a 91 day bill:
(a) Compute the value of the bill at delivery if the annual `discount' rates are (i) 7% and (ii) 8%. (Hint: Use the banker's discount formula to get the prices.)
(b) What is the bond price elasticity if the bond price moves (i) from 7% to 8% and (ii) from 8% to 7%.
(a ) A government bond dealer holds a portfolio of $100 million par of 10 year Treasuries, with 8% coupon (stated) rate (semi-annual payments). Using the modified duration calculations from question 2, find the gain or loss that the dealer experiences if current market rates increase by 1% from 9%
(b) Treasury bond futures have as standard delivery an 8%, 20
year bond, such as that priced in question 2. The par amount
in 1 contract is $100,000. A hedge position attempts to
create (equal but opposite) gains in futures markets from
those that occur in the cash market when rates move.
Calculate the amount the government bond dealer would gain
(or lose) on each shorted T-bond contract if the rates moved
from 9% to 10%?
(c) How many T-bond futures should the dealer short so that the
gain (or loss) of part (a) is offset by the loss (or gain)
by the position in futures?}
(d) How does the hedge perform if the interest rate drops from
9% to 8%? Calculate the gain or loss of the cash and futures
positions and show the net effect. Why would the bond dealer
perform the hedge specified in part (c)?
You borrow $100,000 to buy a house at an annual (stated) rate of 10%,
monthly compounding. You will make constant payments for 15 years
will fully amortize the loan. However, there is an up-front fee
6 points or $6,000 that effectively reduces your borrowing
$94,000 without reducing you payments. What is your real
rate, i.e., your internal rate of return, monthly
compounded, based on the $94,000? I want you to show that you can
use more than the EXCEL IRR function.
Graph the present value of the loan payments, less
the present value of the usable funds, versus the interest rate.
The internal rate of return is the rate at which the
present value of the investment is zero. Since the rate specified
is 10%, but there are 6 points, a graph of present values at
rates between 10% and 12% by .05% or .10% increments should
deliver the IRR. Be sure to show your work.
Merrill Lynch earned $1.24 per share in 1971 and earned $3.56
1983. What compound annual growth rate of earnings does that
You have an extraordinary opportunity to buy only one of two
each for a price of $100. Bond A has a coupon rate of 14% per
(compounded semiannually) for
2 years and then matures. Bond B pays a coupon of 13%
five years and then matures. Current market
interest rates are given in the table in question 1.
What are the internal rates of return for each bond? What are
the present values of the cash flows from each bond?
Which bond should you buy?
What is convexity, as it is used in bond trading? Do
investors like to buy an asset that has convex payoffs, or
would they rather short it? Explain. How is convexity related
to modified duration?
Consider the Eurodollar futures quotations. Assume that the "settle
yields" are the stated annualized forward rates (compounded quarterly),
i.e. the APR rate. Suppose you had quarterly cash flows through to
September 2005 that were equal to $1.75 million per quarter with the
first payment beginning in three months from now (use the closest
contract) and the last payment is $101.75 million.
The risk of the cash flows is roughly equal to the risk of a
Eurodollar deposit. What is the present value of the cash flows?
Below is a set of Eurodollar quotations. You can these into
a spread sheet. Alternatively, you can use another day's quotations.
for Tuesday October 24, 1995
*Optional. Not required for