Question

For dollar amounts, give your answer to the nearest cent. For interest rates, give our answer as a percentage rounded to 2 decimal places.

If any parts of the question use values from earlier parts, use the EXACT values from earlier parts.

a) Explain the relationship between the yield to maturity of a premium bond and its coupon rate.

b) ABC Ltd issues two different bonds with the same yield to maturity:

• a 20-year zero coupon bond
• a 15-year semi-annual coupon bond

Explain which bond is subject to less interest rate risk.

c) ABC Ltd is planning to issue 16-year semi-annual coupon bonds with a face value of $1,000 and a coupon rate of 6.5%. The nominal yield to maturity of potential investors is estimated to be 7.6% per annum. Calculate the required number (expressed as an integer) of semi-annual coupon bonds to be issued if the firm aims to raise $15 million.

d) You purchase a bond issued by XYZ Ltd, which is a 9% semi-annual coupon bond with a term to maturity of 12 years, and currently trading at par. 3 years later, immediately after receiving the 6^th coupon payment, you sell the bond to your best friend. You best friend’s nominal yield to maturity is 7% per annum. Write down an equation that can be solved to find your total realised return over the 3-year holding period.

Answer:

Answer 1

Solution a) In case of a premium bond, the price of the bond is greater than the principal of the bond. Thus, the Yield-to-maturity is less than the Coupon rate for a premium bond.

This can be explained by taking an example of a bond with current price of $104 (PV) and principal value of $100 (FV) issued for 5 years (NPER) with a coupon rate of 10%. Thus, the coupon payment is 10%*100 = $10 (PMT)

The yield-to-maturity can be calculated using the Rate function in excel

=RATE(Nper, PMT, PV, FV,Type)

= RATE(5, 10,-104,100,0) = 8.972%

Thus, for a premium bond, the Yield-to-maturity is less than the coupon rate.

Solution b) Out of the two bonds issued, the 20 years the zero-coupon bond is more sensitive to the interest rate risk. This is because, in the case of a zero-coupon bond, no coupon payments are received in between and all the amount is received at the maturity. Hence, these payments are more sensitive to the changes in interest rates.

Solution c) 16-year semi-annual coupon bond

Thus, Number of periods (NPER) = 16*2 = 32

Face value = $1,000

Coupon rate = 6.5%

Since these are semi-annual payments, so, coupon payments (PMT) = 6.5%/2*1000 = $32.5

Yield-to-maturity (YTM) = 7.6%'

Since these are semiannual payments, hence, the applicable discount rate (RATE) = 7.6%/2 = 3.8%

Price of the bond can be calculated using the PV function = PV(Rate, Nper, PMT, FV,Type)

= PV(3.8%, 32, 32.5,1000,0)

= $899.14

Hence, bond is issued at a price of $899.14

To raise $15 million, the number of bonds to be issued by the company = 15,000,000/899.14

= 16,682.61 = 16,683 bonds

Solution d) The bond offers 9% semi-annual coupon with a term to maturity of 12 years, and currently trading at par.

Suppose Par value = $100

Since the bond is trading at par, hence, the purchase price of the bond = Par value = $100

Since the coupons are offered on a semi-annual basis, thus, coupon payments offered = 9%/2 = 4.5%*100 = $4.5

Also, the total number of coupon payments =  12*2 = 24

After the 6th payments, the remaining number of payments (NPER) = 24 - 6 = 18

Coupon payments (PMT) = $4.5

The yield-to-maturity for his friend = 7%

On a semi-annual basis, the interest rate (Rate) = 7%/2 = 3.5%

Thus, the price of the bond at the end of 3rd year is calculated using the PV function

= PV(Rate, NPER, PMT, FV,Type)

= PV(3.5%, 18, 4.5,100,0)

= $113.19

Hence, the bond is sold at the end of 3rd year at a price of = $113.19

Holding period yield = (Selling Price + Coupon Payments - Purchase Price)/Purchase Price

= (113.19 + 4.5*6 - 100)/100

= 40.19%