PSTAT 171 Mathematics of Fixed Income Markets Questions

PSTAT 171 – MATHEMATICS OF FIXED INCOME MARKETSFINAL EXAM
JUNE 8, 2022
HAL W. PEDERSEN
You exam must be uploaded to GauchoSpace by 11am on Friday June 10, 2022.
Provide sufficient reasoning to back up your answer but do not write more than
necessary. Write down key formulas that you are using if applicable. Please make
your final results easy to read.
This quiz consists of 12 questions. Good luck!
Problem 1. Olena loans Amy $20,000. Amy’s loan is to be repaid in equal quarterly
installments over 8 years at a nominal annual quarterly rate of 8%. Bob also borrows
$20,000 from Olena for 8 years but only pays interest as it is due each year at the
annual effective rate of 8.5%. Bob will repay the entire loan balance at the end of the
8-year period. How much more interest will Bob pay than Amy pays over the life of
the loan?
Problem 2. In an account, interest is credited using a simple interest rate of 5%
every year for the first four years. After four years, interest is credited at the force of
t+ 12
interest δt = 1+t+t
2 , t ≥ 4. The following values are numerically equal:
• The current value at time t = 4 of payments of 1,000 at time t = 2 and 400
at time t = 7.
• The present value at time t = 0 of payment of X at time t = 10.
Calculate X.
Problem 3. A loan is repaid by twenty end-of-quarter payments of $1,000. The
interest for the first three years is 6% convertible quarterly and for the last two years
is 8% convertible quarterly. Find
a the outstanding loan balance after the fourteenth payment;
b the amount of interest in the seventh payment.
Problem 4. Consider two following two perpetuities.
• A perpetuity-immediate with effective interest rate of i. The payment is constant and equal to 100 in the first 2n years. Starting from time 2n + 1 and
for all future years, the payment increases by K% each year, i.e. payment is
10(1 + K%) for year 2n + 1, then 10(1 + K%)2 and so on. The present value
is $3,600.
• At the same interest rate i, the present value of a 2n-year deferred increasing perpetuity-immediate, with first payment equal to 20(1 + K%) and then
increasing by K% each year, is equal to $3,800.
1
2
PSTAT 171
Known that the quantity v n = (1 + i)−n = 0.7, calculate K%.
Problem 5. Amy buys a 10-year $ 10,000 par value bond with 10% semiannual
coupons. During the first five years, the coupons are reinvested into a fund A which
earns 5% annual effective interest rate over the whole 10-year period. In the second
five years, the coupons are reinvested into a fund B which earns 9% annual effective
rate.
a Find the total value in bond, fund A and fund B at the end of 10 years.
b Find the overall annual yield rate earned by Amy.
Problem 6. Olga buys a 8-year annuity for X. Each year, Olga will receive 2 at the
end of the first month, 4 at the end of the second month, and for each month thereafter
the payment increases by 2. The annual nominal interest rate is 9% convertible
quarterly. Calculate X.
Problem 7. The fund balance in an account on January 1 is 1000. On April 1, the
value has increased to 1010 and 10 is withdrawn. Then the balance on June 30 is
1030 and the balance on December 31 becomes 1050. (Show all your answers in the
form of xx.xx%.)
a Calculate the dollar-weighted yield rate and the time-weighted yield rate.
b If there was a deposit of 10 immediately after June 30 balance was calculated,
calculate the dollar-weighted yield rate and the time-weighted yield rate.
c Redo [b] if the deposit was immediately before the June 30 balance was calculated.
d Verbally explain what you find from dollar-weighted yield rates from [b] and
[c] and interpret it.
Problem 8. A company has liability of $2000 due at the end of the second year.
The company tries to buy two assets of A1 at time 1 and A9 at time 9 to immunize
the portfolio. The interest rate is 8%.
a Determine A1, A9.
b Determine whether or not the portfolio is immunized.
Problem 9. Matt purchased a 30-year par value bond with semi-annual coupons at
a nominal annual rate of 10% convertible semi-annually. The bond was purchased at
a price of 1895.12. The bond can be called at its par value X on any coupon date
starting at the end of year 25 after the coupon is paid. The price guarantees that
Matt will receive a nominal annual rate of interest convertible semi-annually of at
least 8%. Calculate X.
Problem 10. Bill Buys a 15-year 1000 par value 8% bond with semi-annual coupons.
The price assumes a nominal yield of 8% convertible semi-annually. As Bill receives
each coupon payment, he immediately puts the money into a savings account earning
interest at an annual effective rate of i.
At the end of 15 years, immediately after Bill receives the final coupon payment and
PSTAT 171
3
the redemption payment, Bill has earned an annual effective yield of 7% on his overall
investment. Calculate i.
Problem 11. Consider a yield curve defined by the following equation:
ik = 0.08 + 0.003k − 0.001k 2
where ik is the annual effective rate of return for zero-coupon bonds with maturity of
k years.
Let j be the one-year effective rate during the 5th year that is implied by this yield
curve. Calculate j.
Problem 12. You are given the following n-year forward rates:
Calculate the price of a 4-year, 1000 par bond with 6% annual coupons.