Question

Ah Cheung is the portfolio manager for large pension fund and is considering investing $100 million to purchase the following 3 choices of ABC Ltd issued bonds:

•ABC Bond A matures in 5 years and pays a 9%p.a. coupon on a $1,000 face value bond).

•ABC Bond B matures in 10 years, pays an 8%p.a. coupon yield and is being offered at PAR.

•ABC Bond C is a zero-coupon bond but will pay the face amount of $1,000 per bond at maturity in 10 years.

Note: All bonds pay coupon semi-annually and have market prices that imply a YTM of 8%p.a. Each of the 3 bonds is based on a $1,000 face value (par value) to be repayable at maturity.

(A) Without performing any calculations, can you roughly guess the price of the 3 bonds compared to PAR? (Hint: Observe the relationship between coupon and coupon.)

(B) Calculate each bond price?

(C) In the following table, it is the prices of the three bonds at 5% & 11% yield (YTM) comparing to their valuations when the required yield (YTM) was 8%p.a.:

YTM	ABC Bond A	ABC Bond B	ABC Bond C
5%	$1,175.04	$1,233.84	$610.27
11%	$924.62	$820.74	$342.73

Explain why each of these bonds has a different degree of sensitivity to a +/- 3% change in required yield (YTM). Give your reasoning in terms of volatility based on maturity and coupon effect.

(D) Calculate the duration of ABC Bond A, B, C?

(E) As an alternative, Ah Cheung has been invited to invest $1 million in a 10-year bond of a second firm, XYZ. XYZ bonds are similar in risk to “ABC Bond B”: they both pay 8%p.a. coupon for 10 years, but XYZ bonds pay coupon annually, not semi-annually as in the case of ABC. The XYZ bonds are priced 99% (or $990 per $1,000 face value).

What yield to maturity (YTM) is implied by the XYZ bond? Compare this yield to the 8%p.a. of the ABC semi-annual coupon bond (Bond B) above. In which bond should Ah Cheung invest?

Answer 1

A
Security	Coupon	Yield	Relation	Price
Bond A	9%	8%	Coupon > par	greater than par
Bond B	8%	8%	Coupon = par	equal to par
Bond C	0%	8%	Coupon < par	less than par

The price of bond compared to PAR can be judged based on the coupon rate of the bond and the yield. If the coupon pays the coupon at a rate higher than the yield, it is expected to be traded at a price higher than the par. If the coupon payment is less than the yield, the expected price is lower than the par. If the two rates are equal, then the bond is trading at par. a zero coupon bond always trades at a discount.

B
Security	Coupon	Yield	Settlement	Maturity	Period	Payment frequency	Price	Price formula
Bond A	9%	8%	4/2/2019	4/2/2024	5	semi annual	1,040.55	=PRICE(4/2/19,4/2/24,9%,8%,1000,2(semi-annual),1(actual-actual))
Bond B	8%	8%	4/2/2019	4/2/2029	10	semi annual	1,000.00	=PRICE(4/2/19,4/2/29,8%,8%,1000,2(semi-annual),1(actual-actual))
Bond C	0%	8%	4/2/2019	4/2/2029	10	at maturity	456.39	=PRICE(4/2/19,4/2/29,0%,8%,1000,2(semi-annual),1(actual-actual)) or FV(4%/2,10*2,0,-1000,0)

Note- coupon payment frequency of a zero coupon bond is zero. However, we have to choose one of the standard parameters in the price formula. Hence, we can choose any payment frequency but the answer wont changes as the coupon payment rate is 0.

C
YTM	ABC Bond A	ABC Bond B	ABC Bond C
5%	1,175.04	1,233.84	610.27
11%	924.62	820.74	342.73
8%	1,040.55	1,000.00	456.39
% change
5%	12.92%	23.38%	33.72%
11%	-11.14%	-17.93%	-24.90%

YTM is the expected rate of return realized from holding the bond till maturity. It can also be understood as the IRR of the cash flows of the bond payments. When the maturity is longer for a bond, the expected rate of return i.e. the yield from that bod is lower. The largest payment from a bond is usually the principal repayment. Hence, the longer it takes to receive the principal back, the lower is the yield. Any changes in the interest rate or yield will impact the bond with longer maturity more than an otherwise similar bond with shorter maturity. Similarly, when the coupon payments are higher, higher portion of the total bond returns are realized earlier. Hence, the impact of changes in interest rate or yield impact a bond with higher coupon payments less than an otherwise similar bond with lower coupon payments. These results can be seen in the above table. The bond with the highest coupon payment and shortest maturity is affected the least by the changes in the yield. Whereas, the bond with the lowest coupon (zero-coupon bond) is impacted the most by the changes in the yield. Note- the sensitivity of a bond to the changes in yield/ interest rates is called as the bond duration.

D
Security	Coupon	Yield	Settlement	Maturity	Period	Payment frequency	Duration	Duration formula
Bond A	9%	8%	4/2/2019	4/2/2024	5	semi annual	4.154176	=DURATION(4/2/19,4/2/24,9%,8%,1000,2(semi-annual),1(actual-actual))
Bond B	8%	8%	4/2/2019	4/2/2029	10	semi annual	7.0669697	=DURATION(4/2/19,4/2/29,8%,8%,1000,2(semi-annual),1(actual-actual))
Bond C	0%	8%	4/2/2019	4/2/2029	10	at maturity	10	=DURATION(4/2/19,4/2/29,0%,8%,1000,2(semi-annual),1(actual-actual))

Note- duration of a zero-coupon bond is equal to its maturity as no intermediate payments are made.

E
Security	Coupon	Yield	Settlement	Maturity	Period	Payment frequency	Price	Yield	Yield formula
Bond XYZ	8%	-	4/2/2019	4/2/2029	10	annual	990.00	8.15%	=YIELD(4/2/19,4/2/29,8%,990,1000,1(annual),1(actual-actual))
Bond B	8%	8%	4/2/2019	4/2/2029	10	semi annual	1,000.00	8.00%	=YIELD(4/2/19,4/2/29,8%,1000,1000,2(semi-annual),1(actual-actual))

Since the yield of XYZ bond is higher than the otherwise similar bond B of ABC, Cheung should invest in bond XYZ.