1. Present Value

(a) Someone offers you a security which pays$nat the end of thenth yearuntil forever (i.e., it pays$1 at the end of the first year,$2 at the end ofthe second year, and so on). If the annually compounded interest rate is10% per year, what is the fair price of such security?

(b) (Rule of 69) People in the banks have a quick way of finding outhow long it takes to double your money. The trick is to divide 69 bythe continuously compounded interest rate (in percentage). For example,if the continuously compounded interest rate is 10% per year, then youknow it takes 69/10 = 6.9 years to double your money. Why 69? Canyou figure out a similar rule to find out the number of years it takes totriple your money? [Note: Continuous compounding means that thecompounding limit is; that is, if one dollar invested at interest rater, continuously compounded, then one year later the balance becomeser= limm(1 +rm)m.]

(c) You just signed a 30-year lease agreement for a business property. Themonthly rent for the first year is$1,000/month, with the first monthly rentdue today. Starting from the second year onward, the monthly rent will beincreased by 10%/year (i.e., the monthly rent for the second year will be$1,100, the monthly rent for the third year will be $1,000(1.10)2= $1,210,1and so on). Assuming the annually compounded interest rate is 15%/year,what is the present value of the 360 rental payments.

2. Bond Valuation and Arbitrage

Suppose that on January 1, 2020 you observe the following market infor-mation regarding the prevailing prices for three zero-coupon bonds:Bond A, which will pay$100 on January 1, 2021, has a price of$95.24;Bond B, which will pay$100 on January 1, 2022, has a price of$85.73;Bond C, which will pay$100 on January 1, 2023, has a price of$75.13.(a) What is the term structure of spot rates (r1,r2, andr3) on January1, 2020?(b) There is a fourth bond, Bond D, which is a level coupon bond, witha coupon rate of 5% per year,$100 face value, and three years remainingto maturity (annual coupon payments). What is its price on January 1,2020?(c) Suppose one year later, that is, on January 1, 2021, you find that theprice of Bond B and Bond C rise to$94.34 and$87.34, respectively. Whatwill be the market price of Bond D? What will be its yield to maturity?(d) Continuing part (c), suppose on January 1, 2021, you observe that theprice of Bond D is$100. Suppose shorting assets is costly. Specifically, ifyou short$Kof an asset, you have to pay a brokerage fee of$0.00005K2.Can you form an arbitrage portfolio to make a profit on January 1, 2021?If yes, what is the maximum amount of arbitraging profit you can makeon January 1, 2021? Please describe the exact arbitrage strategy and thecorresponding cash flows in each year.

3. Duration

Assume the term structure of interest rates is flat and the market interestrate isr= 10% per year, annually compounded.(a) What are the Macaulay duration and modified duration of an annualcoupon bond with a coupon rate of 5%/year, and a maturity of 10 years?(b) What is the Macaulay duration of a perpetuity that pays$10/year?

4. Mortgage

You plan to borrow$500,000 to buy a house. The amortization period ofthe mortgage is 20 years. You obtain a 5-year fixed rate mortgage fromTD bank at 2%/year (using Canadian mortgage convention).(a) What is your monthly payment?(b) How much do you owe the bank after the 60th payment?(c) For the 24th monthly payment, how much of it is for interest, and howmuch of it is for principal repayment?(d) What is the present value of the interest portion of the first 60 payment