Question

Question 1 (16 marks)

Bond XYZ is a $1,000, 4% semi-annual coupon bond issued today on 1 Jan 2023, maturing on 31 December 2032. Coupons are paid on every 30 June and 31 December.

(a) If you bought Bond XYZ today at a yield (YTM) of 3% compounded semiannually, what’s your purchase price? Did you buy the bond at a premium, par or discount? (4 marks)

(b) One year later Bond XYZ’s yield went up to 5% compounded semiannually and you sold it immediately after receiving the coupon.
i) Compute the bond’s current yield. (2 marks)
ii) Compute the bond’s one-year capital gains yield. (4 marks)
iii) Assume that you reinvested all coupons at 5% compounded semi-annually. Calculate the total amount of coupon income (coupon payments and reinvestment of coupon payments) at the end of the 1-year holding period. (3 marks)
iv) Based on the results from parts (bii) and (biii), calculate the Holding Period Yield for the investment horizon of 1 year (HPY1-year). (3 marks)
[Hint: In computing the 1-year HPY for the bond investment, you figured that the correct answer could not be found by simply adding up the current yield and the 1- year capital gains yield because the current yield, by definition, would fail to consider the reinvestment of the semi-annual coupons received during the year.]

Answer 1

(a) Using the bond pricing formula:

PV = (C/2) / (1 + r/2) + (C/2) / (1 + r/2)^2 + … + (C/2) / (1 + r/2)^20 + 1000 / (1 + r/2)^20

where C = $20 (4% of $1,000/2), r = 3%/2 = 1.5%, and PV is the present value of the bond, we can solve for PV:

PV = (20/1.015) + (20/1.015^2) + … + (20/1.015^20) + 1000/1.015^20

PV = $1,118.57

Since the bond was purchased at a price higher than its face value, it was bought at a premium.

(b)(i) The current yield is the annual coupon payment divided by the bond's market price. The annual coupon payment is $40 ($1,000 x 4%), and the market price is the selling price, which we need to compute. Using the same formula as in part (a), with r = 5%/2 = 2.5%:

Selling price = (20/1.025) + (20/1.025^2) + … + (20/1.025^19) + 1020/1.025^19

Selling price = $1,048.90

Therefore, the current yield is $40 / $1,048.90 = 3.81%.

(b)(ii) The one-year capital gains yield is the percentage change in price from the original purchase price to the selling price, plus any coupon payments received during the year. The coupon payment received during the year is $20, since it is a semi-annual coupon bond. The original purchase price was $1,118.57, and the selling price is $1,048.90. Therefore, the capital gains yield is:

(1048.90 - 1118.57 + 20) / 1118.57 = -5.68%

Note that the negative sign indicates a loss.

(b)(iii) To calculate the total amount of coupon income, we need to first calculate the total coupon payments received during the year, which is $40. We then need to compute the future value of the reinvested coupon payments at the end of the year, using a rate of 2.5%/2 = 1.25% for each semi-annual reinvestment. Thus, the future value of the first coupon payment of $20 received on June 30, 2023, at the end of the year is:

FV = 20 x (1 + 0.0125)^2 = $20.50

Similarly, the future value of the second coupon payment of $20 received on December 31, 2023, at the end of the year is:

FV = 20 x (1 + 0.0125) = $20.25

Therefore, the total coupon income at the end of the year is $40 + $20.50 + $20.25 = $80.75.

(b)(iv) The holding period yield for the investment horizon of one year (HPY1-year) is the total return on the investment over the one-year period, expressed as a percentage of the initial investment. Using the formula:

HPY1-year = (Ending value - Beginning value + Income) / Beginning value

where Beginning value is the original purchase price of $1,118.57, Ending value is the selling price of $1,048.