Question

You have a 5 year investment holding horizon, would like to earn a 4% annual compound return each year, you have a choice between two bonds. Whatever money you have to invest will be invested in one type of bond or another. Find the Duration and Modified Duration for each bond (show your work and answer the questions below)

• Bond 1 has a 4% annual coupon rate, $1000 maturity value, n = 5 years, YTM = 4% (pays a $40 annual coupon at the end of each year for each of the 5 years and $1,000 maturity payment at the end of year 5). • Bond 2 is a zero coupon bond with a $1000 maturity value, and n = 5 years; YTM= 4%. (pays no coupons; only a $1,000 maturity payment 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 ____________ (Be sure to show your work for the bond price and duration calculations for credit).

d. Which of the two bonds should you choose for your 5-year investment horizon to duration match to ensure your desired 4% annual compound return if you hold the bond for 5 years to maturity? 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.____________________________________________ 2.

a. For the Zero Coupon Bond 2 above, what will be your annual compound yield for your 5 year holding period if the bond is held until maturity. (Hint: PV is the price you calculated for Bond 2 and FV is the bond’s maturity value of $1,000 and n is the 5 year holding period; solving for i) (see formulas below). Hint Recall: Annual Compound Yield = {[FV / PV] ^ 1/n} - 1 or In other words {[What you have at the end of 5 Years / What You Paid] ^1/n } - 1 n = your 5-year holding period. Annual Compound Yield for Bond 2 at End of Year 5 __________________

b. Suppose for the Coupon Bond 1 above, rates go down to 2% after you purchase the bond for the life of the bond. Thus, you have to invest each of your $40 coupon payments at a 2% rate, and hold the bond to maturity, receiving your $1,000 maturity value at the end of year 5. What will be your annual compound yield? Hint: Recall FV of Bond Coupons Reinvested for 5 years = Coupon Payment (FVIFA 2%, 5) ACY = { [(FV of Coupons +Maturity Value) / (Price of Bond)] ^1/n } - 1, where n = 5 years Annual Compound Yield for Bond 1 at the End of Year 5 ____________

c. Explain why you received your desired annual compound return for the 5 year holding period for Bond 2 in a., but didn’t receive your desired Annual Compound Return for Bond 1 for your 5 year holding period in b.? 3. Distinction Between Real and Nominal Interest Rates a. Distinguish between a nominal versus a real interest rate. b. If a bond gives you a 4% nominal annual interest rate and the inflation rate over the year is 2%, what is the real ex post rate of return you receive? Real Rate You Receive _______________ c. If an investor wants a real rate of return of 2% and expects inflation to be 2% next year, what nominal rate should the investor demand? Nominal Rate Investor Demands _______________

***Please show me how to calculate***

Answer 1

a. Bond price:

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

Price for Bond 1 =

40/(1+4%)¹

+ 40/(1+4%)²

+ 40/(1+4%)³

+ 40/(1+4%)⁴

+ 1040/(1+4%)⁵​​

​​​​​ = $1,000

Price for Bond 2 = 1000/(1+4%)⁵ = $821.927

b. Duration formula:

,

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

Duration for Bond 1 = (40*1/(1+4%)¹ + 40*2/(1+4%)² + 40*3/(1+4%)³ + 40*4/(1+4%)⁴ + 1040*5/(1+4%)⁵)/1000 = 4.63 years
Duration for Bond 2 = (1000*5/(1+4%)⁵​​​​​​​)/821.927 = 4109.6355/821.927 = 5 years

​​​​​​​c. Modified Duration Formula:

Modified Duration = Macaulay Duration / (1+r),
where r = interest rate per annum = 4%

Modified Duration for Bond 1 = 4.63/1.04 = 4.45
Modified Duration for Bond 2 = 5/1.04 = 4.81

d. Based on the Duration and Volatility of each bond, we should invest in Bond 1 because its duration is lower (which means the price will be recovered earlier) and its price is also less volatile to changes in interest rate.

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