Donald has just finished high school and is thinking about two career paths. Regardless

of the path chosen, he will retire $40$ years after leaving high school.

$-$ Plan A: Start to work right away. Assume Donald makes $$25,000$ per year $($paid

out monthly$)$ for the first $10$ years, then receives a $30\%$ raise every $10$ years $(1-10,$

$11-20,21-30,31-40).$

$-$ Plan B : Go to university. Assume Donald incurs loans of $$20,000$ to cover the

costs of living plus tuition each year $($for four years$).$ Donald takes these loans out

at the beginning of each year. These loans incur no interest while he is a student

but re$-$payment is expected to begin one month after the graduation. His starting

salary after graduation is expected to be $$40,000/$year $($paid out monthly$)$ with

a $25\%$ raise every six years $(5-10,11-16,17-22,23-28,29-34,35-40).$ His student

loans will be repaid over $10$ years with monthly payments $($paying interests at the

rate quoted below$).$Over the entire $40-$year period, the interest rate at which Donald can borrow or invest

is quoted $($stated$)$ at $8\%,$ monthly compounded. Which plan should Donald choose?

$2.$ You obtain a loan of $$500,000$ from a bank to buy a house. The mortgage has a $5-$

year term with a fixed rate of $5\%/$year $($using Canadian mortgage convention$).$ The

amortization period of the mortgage is $20$ years.

$($a$)$ What is the monthly mortgage payment?

$($b$)$ How much do you owe the bank after the $60$th monthly payment?

$($c$)$ For the $26$th monthly payment, how much of it is for interest, and how much of

it is for principal repayment?

$($d$)$ Find the expression for the present value of the principal repayment for the month

t mortgage payment $($i$.$e$.,$ a general expression for any given month t$).$ Is this

present value constant over different months? What is the total present value of

the interest portion of the $60$ mortgage payments?

$3.$ You observe the following market prices for risk$-$free bonds

Bond Maturity Coupon rate Price

T$-$bill $1$ Year $0\%$ $$97\mathrm{.}5$

Canada Bond $3$ Years $2\mathrm{.}25\%$ $$88\mathrm{.}53$

Canada bond $3$ Years $5\mathrm{.}25\%$ $$96\mathrm{.}57$

$($a$)$ All bonds have a face value of $$100$ and all coupon bonds have annual coupon

payments. Use the above information to price the following risk$-$free bonds.

$(1)$ A $3$ year zero$-$coupon bond.

$(2)$ A $2$ year $5\mathrm{.}25\%$ Canada Bond.

$(3)$ A $3$ year $7\mathrm{.}25\%$ Canada Bond.

$($b$)$ There is a three$-$year annuity with annual payments of $$1,000($i$.$e$.,$ paid at the

end of each of the first three years$).$ The market price of the annuity is $$2,560.$

Suppose that one can trade $($i$.$e$.,$ buy, sell, or short$-$sell$)$ any amount of any of the

five bonds and the T$-$bill mentioned above $($but nothing else$)$ with no transaction

cost$.$

Can you create a strategy to make an arbitrage profit? If yes, how? Explain

clearly. $($Hint: the solution is not unique.$)$

$4.$ A $10$ year annual coupon bond has a face value of $$100$ and an annual coupon of $$3.$

Assume that the yield to maturity of the bond is $5\%$ and that the yield curve is flat. $($a$)$ Calculate the Macaulay and modified duration of the bond.

$($b$)$ Calculate the approximate and exact changes in price of this bond if the interest

rate change is $0\mathrm{.}02\%($i$.$e$.,$y $=0\mathrm{.}0002).$

$($c$)$ Calculate the approximate and exact changes in price of this bond if the interest

rate change is $2\%($i$.$e$.,$y $=0\mathrm{.}02).$

$($d$)$ Briefly explain the difference between the results from $($b$)$ and $($c$).$ Can you draw

any implication for hedging with duration?

$($e$)$ You just short$-$sold $($one unit of$)$ the $10$ year coupon bond, but now you feel

nervous. You would like to hedge against this short position. Suppose that you

can trade a $2-$year zero coupon bond and a $20-$year zero coupon bond by any

amount without any transaction cost. Using the proceeds from the short sale,

how can you combine these two zero coupon bonds to immunize your position

against interest rate risk?