Problem set 1 solutions PDF

Preview

Summary

Download Problem set 1 solutions PDF

Description

Q3 (Essential to cover) Consider a coupon bond with a face value of $100, a coupon rate of 10%, a time-to-maturity of three years and a price of $119.80. What is its yield-to-maturity? (Hint: Solve numerically using a scientific calculator or Excel) Solution: The cash flow diagram of this bond is:

-P = -119.80

c1 = 10

c2 = 10

C3 = 10 FV = 100

P = c 1 /(1 + y)1 + c 2 /(1 + y)2 + c 3 /(1 + y)3 + FV/(1 + y)3

(1)

So if you know how to solve cubic equations, then you can directly apply the cubic equation solver to obtain the value of y, which is the yield-to-maturity under question. Alternatively, you can use a numeric solution (try-and-error) to obtain the approximate value of y. Step 1: take an initial value of y Since P > FV (traded at premium), so y < C = 10%. Let’s take the starting value of y as 9% and calculate the price when y = 9%. P(9%) = 102.53 Step2: adjust y until the calculated price crossing the actual price $119.80 Since P(9%) = 102.53 < 119.80, we should further decrease y, until the calculated price is close or below the actual price. P(3%) = 119.8003 And P(2%) = 123.07 So we conclude that the actual y should be 3% (close enough to the true value). Step3 (if necessary): Interpolation to get the answer For demonstration purpose, assume the actual price is 121.44, and you are asked to get an answer of y in percentage (rounded to first decimal). Since P(2%) = 123.07 > 121.4, and P(3%) = 119.80 < 121.4, so the actual y should lie between 2% and 3%. So we can obtain the approximate value of y by interpolation.

y1 = 2% P1 = 123.07

y=? P = 121.44

y2 = 3% P2 = 119.80

Hence y ~=2%*[(121.44-119.80)/(123.07-119.80)] + 3%*[(123.07-121.44)/(123.07-119.80)] = 2.5%. As you can see, the weight on a rate (e.g., 2%) is actually the relative distance between the actual price ($121.44) and the price calculated from the other rate (P(3%) = 119.80). The idea is that the further away the price of 3% is from the actual price, the closer the actual y to 2%.

Additional Note: With hindsight, this question is not well designed – you need to try too many times with hypothetical ys until the calculated price crosses the actual price. In the exam, if I decide to include YTM calculation (I have not decided yet), I will choose numbers you don’t have to try too many times. And as I indicated in the lecture note, at most I will set the maximum time periods up to T = 3. And if I require you to do the interpolation to get the answer, I will put a note and specify to what decimals the answer should be.

Q9 (Essential to cover) A 2-year bond with par value of $1000 making annual coupon payments of $100 is priced at $1000. What is the yield to maturity of the bond (without any calculation, are you able to answer this question?)? What will be the realized compound yield to maturity if the 1-year interest rate next year turns out to be: a) 8%, b) 10%, c) 12%? Solution: The cash flow diagram of this bond is:

-P = -1000

c1 = 100

c2 = 100 FV = 1000

P = c 1 /(1 + y)1 + c 2 /(1 + y)2 + FV/(1 + y)2

(1)

[1] yield to maturity Since P = FV, so y = C = 10%. [2] Realized compound yield – let’s take 8% 1-year interest rate next year for illustration Step1: To calculate realized compound yield, you will first need to reinvest all interim cash flows (coupon payments in this question) to the maturity at the interest rate at the time of coupon payment.

-P = -1000

c1 = 100

c2 = 100 FV = 1000

So the cash flow at the end of maturity (time 2) from reinvesting c1 = 100*(1+8%) = 108. Step 2: calculate the total cash flow (with reinvestment) at the end of maturity So the total cash flow at the end of maturity (time 2) of this bond = 108+100+1000 = 1208. Step 3: calculate total realized return and annualize it Total (realized) return (R) = total cash flows at maturity/price – 1 = 1208/1000 – 1 Annualize total return to annual return: r = (1+R)1/T – 1 = (1208/1000) ½ - 1 = 9.9% You can replicate the calculation above to 10% and 12% of 1-year interest rate next year. You will find that realized compound yield = 10% if the reinvestment rate = 10%, and above 10% if the reinvestment rate = 12%. So YTM (10% in this question) equals to realized compound yield if all interim cash flows could be reinvested at YTM (which is unlikely the case). Additional Note: The calculation is very simple for the two-year bonds in our example. But the same calculation process could be easily applied to more general case. For example, in the general bond example below (T-year maturity), to calculate the cash flow at the maturity (time T) from reinvesting the

first coupon payment, you will need to reinvest c1 first to end of time 2 (using the 1-year return at time 1), then reinvest the proceedings at the end of time 2 further to time 3 (using the 1-year return at time 2), so on so forth, until the end of time T. Similarly, you will need to reinvest all other cash flows to the end of maturity. Finally, you sum up all reinvestment cash flows and the final payment (coupon and face value) to obtain the aggregate cash flows from this bond – based on which you can calculate total return over these T years and annualize it to realized compound yield. But don’t worry, if I really decide to test on the realized compound yield, I will set T = 2 so that you only need to reinvest one cash flow (for one year), given that the logic is the same and making T bigger just makes things more complex (rather than more challenging).

-P

c1

c2

ct

cT FV

Selected end-of-chapter questions. BKM Chapter 14 3.

Zero coupon bonds provide no coupons to be reinvested. Therefore, the investor's proceeds from the bond are independent of the rate at which coupons could be reinvested (if they were paid). There is no reinvestment rate uncertainty with zeros.

4.

A bond’s coupon interest payments and principal repayment are not affected by changes in market rates. Consequently, if market rates increase, bond investors in the secondary markets are not willing to pay as much for a claim on a given bond’s fixed interest and principal payments as they would if market rates were lower. This relationship is apparent from the inverse relationship between interest rates and present value. An increase in the discount rate (i.e., the market rate. decreases the present value of the future cash flows.

5.

Annual coupon rate: 4.80%  $48 Coupon payments Current yield:  $48  =  $970  4.95%  

8.

The bond price will be lower. As time passes, the bond price, which is now above par value, will approach par.

9.

Yield to maturity: Using a financial calculator, enter the following: n = 3; PV = −953.10; FV = 1000; PMT = 80; COMP i This results in: YTM = 9.88% Realized compound yield: First, find the future value (FV. of reinvested coupons and principal: FV = ($80 * 1.10 *1.12. + $80 * 1.12. + $1,080) = $1,268.16 Then find the rate (y realized . that makes the FV of the purchase price equal to $1,268.16: $953.10 × (1 + y realized .3 = $1,268.16 ⇒ y realized = 9.99% or approximately 10% Using a financial calculator, enter the following: N = 3; PV = −953.10; FV = 1,268.16; PMT = 0; COMP I. Answer is 9.99%. Note: financial calculator is not allowed in the exam. So please solve the equation with numerical method.

13. Price $400.00 500.00 500.00 385.54 463.19 400.00

Maturity (years. 20.00 20.00 10.00 10.00 10.00 11.91

Bond Equivalent YTM 4.688% 3.526 7.177 10.000 8.000 8.000

16.

If the yield to maturity is greater than the current yield, then the bond offers the prospect of price appreciation as it approaches its maturity date. Therefore, the bond must be selling below par value.

17.

The coupon rate is less than 9%. If coupon divided by price equals 9%, and price is less than par, then price divided by par is less than 9%.

23.

The bond is selling at par value. Its yield to maturity equals the coupon rate, 10%. If the first-year coupon is reinvested at an interest rate of r percent, then total proceeds at the end of the second year will be: [$100 * (1 + r)] + $1,100 Therefore, realized compound yield to maturity is a function of r, as shown in the following table:

31.

r

Total proceeds

Realized YTM = Proceeds/1000 – 1

8%

$1,208

1208/1000 – 1 = 0.0991 = 9.91%

10%

$1,210

1210/1000 – 1 = 0.1000 = 10.00%

12%

$1,212

1212/1000 – 1 = 0.1009 = 10.09%

a. Initial price P 0 = $705.46 [n = 20; PMT = 50; FV = 1000; i = 8] Next year's price P 1 = $793.29 [n = 19; PMT = 50; FV = 1000; i = 7] HPR =

$50 + ($793.29 − $705.46) = 0.1954 = 19.54% $705.46...