Question

You have a 5-year investment holding horizon, would like to earn an 5% annual compound return each year, and have a choice between two different bonds. Whatever amount of money you have to invest will be invested in Bond 1 or Bond 2 (with the number of bonds to be purchased to be determined later).

Find the Duration and Modified Duration for each bond (be sure to show your work and answer the questions below for credit)

  

• Bond 1 has a 5% annual coupon rate (i.e., a $50 coupon at the end of each year), and an  $1000 maturity value, n = 5 years, YTM = 5% (pays a $50 annual coupon at the end of each year and $1,000 maturity payment at maturity at the end of year 5).  

• Bond 2 is a zero coupon bond with a $1000 maturity value, and n = 5 years; YTM= 5%. (The zero coupon bond has no coupon payments; only a $1,000 maturity payment paid at maturity at the end of year 5).

a. Price Bond 1 ______________           Price Bond 2 _____________

b. Duration Bond 1 ______________    Duration Bond 2 ____________

c. Modified Duration Bond 1 ______ _   Modified Duration Bond 2 ____________

d. Which of the two bonds should you choose for your 5-year investment horizon to duration match to ensure your desired 10% annual compound return if you hold either bond to maturity (i.e. for 5 years)? Explain why. (assume the same default risk for each bond).  

__________________________________________

e.  If interest rates go up by 1%, what will be the % Change in the market value for each Bond’s Price? (Hint Change in Price % = - Modified Duration x Change in Rate (expressed as a fraction, i.e. .01).

% Change in Price for Bond 1 ________% Change in Price for Bond 2 ____________

f.  Which of the 2 bonds has more price risk, and which has more reinvestment risk? Explain why.

Answer 1

a. The formula for Bond price:

,
where MP = Market Price
CF = cash flow
r = YTM = 5%
n = time = 1,2,3,4,5

Price for Bond 1 = 50/(1+5%)1+50/(1+5%)2+50/(1+5%)3+50/(1+5%)4​​​​​​​+1050/(1+5%)5​​​​​​​ = $1,000
Price for Bond 2 = 1050/(1+5%)5​​​​​​​ = $1,000

b. Duration formula:

,

where MacDur = Macaulay Duration
CF = Cash flows
t = time periods =1,2,3,4,5
r = YTM = 5%
MP = Bond Market Price

Duration for Bond 1 = (50*1/(1+5%)1+50*2/(1+5%)2+50*3/(1+5%)3​​​​​​​+50*4/(1+5%)4​​​​​​​+1050*5/(1+5%)5​​​​​​​)/1000 = 4.54595
Duration for Bond 2 = (1050*5/(1+5%)5​​​​​​​)/1000 = 4.1135

c. Modified Duration Formula:

Modified Duration = Macaulay Duration / (1+r),
where r = interest rate per annum = 5%

Modified Duration for Bond 1 = 4.54595/1.05 = 4.3295
Modified Duration for Bond 2 = 4.1135/1.05 = 3.9176

d. Bond 1 must be chosen to duration match with the investment horizon to earn the required rate of interest, as its duration is closest to the investment horizon. The durations will be matched and interest rate risks will be covered during this period. Moreover, any intermediate cash flows (Coupons) can be reinvested at the prevailing rate of interest. Bond 2 cannot provide this as it is a zero-coupon bond.
Hence, Bond 1 must be chosen

e. Percentage change in price = -Modified duration of the bond * Percentage change in interest rate

Percentage change in price for Bond 1 = -4.3295*0.01 = -0.043295%
Percentage change in price for Bond 2 = -3.9176*0.01 = -0.039176%

f. Bond 1 has more price risk because its modified duration is higher than Bond 2. This means that for a given percentage change in interest rates, the price of Bond 1 will change more than the price of Bond 2.

Bond 1 has more reinvestment risk as it is a coupon bearing bond. When interest rates change, there shall be a risk that the bondholder shall have to bear, of the interest rates falling, and hence, reinvestment income falling.