Question

1.You have a 4-year investment holding horizon, would like to earn a 5% 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 bonds or another. Find the Duration and Modified Duration for each bond

Bond 1 has a 5% annual coupon rate, $1000 maturity value, n = 4 years, YTM =5% (pays a $50 annual coupon at the end of each year and $1,000 maturity

payment at maturity).

Bond 2 is a zero coupon bond with a $1000 maturity value, and n = 4 years; YTM= 5%. (pays no coupons); only a $1,000 maturity payment at maturity).

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 4-year investment horizon to duration match to ensure your desired 5% annual compound return if you hold either bond to the end of 4 years? Explain why. (assume the same default risk for each bond)

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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

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2. For the Zero Coupon Bond 2 above that has a $1,000 maturity value and 4 years to maturity, what will be your annual compound yield for your 4-year holding

period if you hold the bond to maturity, receiving the $1000 maturity value at the end of Year 4?

Annual Compound Yield for Bond 2 at End of Year 4 __________________

3 Suppose for the Coupon Bond 1 above that has a 5% annual coupon rate, $1,000 maturity value and 4 years to maturity, rates go down to 3% after you purchase the

bond for the life of the bond. Thus, you have to invest each of your coupon payments at the 3% rate, and you hold the bond to maturity.

What will be your annual compound yield if the bond is held to maturity?

Hint: Recall FV of Bond Coupons Reinvested for 4years = Coupon Payment (FVIFA 3%, 4)

ACY ={[(FV of Coupons +Maturity Value)] / (Bond’s Price)] ^1/n } - 1, where n = 4 years

Annual Compound Yield for Bond 1 at the End of Year 4

____________

c.Explain why you didn’t received your desired annual compound yield for Bond 1.

Answer 1

1) a. Finding the Price of both bonds:

Price = Coupon amount x PVIFA (YTM%, n years) + Maturity value x PVIF (YTM%, nth year)

Bond 1 = $ 50 x PVIFA (5% , 4 years) + $ 1000 x PVIF (5% , 4th year) = 50 x 3.546 + 1000 x 0.8227 = 177.30 + 822.70 = $ 1,000

Bond 2 (Zero coupon bond) = Maturity Value x [1 / (1 + YTM%)^{n years}] = $ 1000 x [1 / (1 + 0.05)⁴] = 1000 x 0.8227 = $ 822.70

Price Bond 1 - $ 1,000 Price Bond 2 - $ 822.70

b. Duration : (Duration is the average time in which the price of the bond (our initial investment) gets recovered by the periodic cash flows received from the bond)

Bond 1

Year (A)	Cash Flows - $ (B)	PV Factor @ 5% (C)	PV of Cash flows (D) = A x B x C
1	50	0.9524	47.62
2	50	0.9070	90.70
3	50	0.8638	129.57
4	1000 + 50 = 1050	0.8227	863.835

Total of PV of cash flows - column (d) = 1131.725

Duration = Total of PV of cash flows / Price of Bond = 1131.725 / 1000 = 1.132 years (approximately)

Bond 2 : Since Bond 2 is a Zero Coupon bond (ZCB), its Duration will be equal to its Years to maturity = 4 years. This is beacuse a ZCB does not pay any coupon during its investment period, but rather only one lumpsum maturity value, which is equal to the par value of bond. Hence, the bond's price will be recovered only on maturity.

Duration Bond 1 - 1.132 years Duration Bond 2 - 4 years

c. Modified duration or Volatility = Duration / 1 + YTM

Bond 1 = 1.132 years / 1 + 0.05 = 1.08%

Bond 2 = 4 years / 1.05 = 3.81%

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.

e. % Change in Price for Bond 1 = - 1.08 x 0.01 = - 0.0108%

    % Change in Price for Bond 2 = - 3.81 x 0.01 = -0.0381%