# Appreciating Investment Valuation Models (v)

Coupon payments in bond transactions

Coupon payments to bondholders could be made by bond issuers in different ways, depending on the contractual agreement. For instance, coupons could be paid annually, semi-annually, quarterly or monthly. The last two options are rare in bond transactions, but they are possibilities. Each of these payment options is illustrated below.

Annual Coupon Payments: For annual coupon payments, valuations in the preceding sections, including illustration of bond valuation, YTM, YTC provide ample explanation on its application. To illustrate, the value of a 10-year bond with a face value of GH¢1,000 and coupon rate of 10 per cent would be GH¢1,000. The Excel input for the computation is =PV(0.1,10,100,1000,0). The Excel output is 1,000.00. However, computation of semi-annual, quarterly and monthly coupon payments requires different valuation formulae and inputs.

Semi-Annual Coupon Payments: The successful determination of the value of a bond with semi-annual coupon payments would hinge on the following steps.

1. The number of years, denoted by N, must be multiplied by 2, that is, N x 2. In our Fiesta Corporation’s example, the number of years was 10. For semi-annual payments, N = 10 years x 2 = 20.

2. The annual coupon payments must be divided by 2, that is, I ÷ 2. For Fiesta Corporation, the annual coupon payment amount was GH¢100. For semi-annual payments, I = GH¢100 ÷ 2 = 50.

3. The annual coupon rate must be divided by 2, that is, r ÷ 2. For Fiesta Corporation, the annual coupon rate was 10 per cent. For semi-annual payments, r = 10% ÷ 2 = 5.

Mathematically, the value of bonds with semi-annual coupon payments can be derived as follows:

Bt=∑_(t=1)^2N▒〖(It÷2)/(1+r÷2)t+F/(1+r÷2)2N〗

Supposing Fiesta Corporation’s contractual agreement included semi-annual coupon payments to bondholders, the PV function in Excel could be used to compute the value of the bond.

The final input would look like: =PV(0.05,20,50,1000,0). Click “ok.” The output = -1000.00.

Using a financial calculator, the compounded value of the bond can be determined as follows:

N = 20; I/YR = 0.05 (5%); PMT = 50; FV = 1000; and press the PV key to determine the bond’s value.

The output of 1000.00 is the bond’s value with semi-annual coupon payments. The

frequency of payments and compounding under the semi-annual valuation method

(twice) is more than those under the annual valuation method (once). However, no difference is observed in the value (GH¢1000.00) of bonds between both methods.

Quarterly Coupon Payments: To successfully determine the value of a bond with quarterly coupon payments, the following steps would be helpful.

1. The number of years, denoted by N, must be multiplied by 4, that is, N x 4. In our Fiesta Corporation’s illustration, the number of years was 10. For quarterly payments, N = 10 years x 4 = 40.

2. The annual coupon payments must be divided by 4, that is, I ÷ 4. In our Fiesta Corporation’s example, the annual coupon payment amount was GH¢100. For quarterly payments, I = GH¢100 ÷ 4 = 25.

3. The annual coupon rate must be divided by 4, that is, r ÷ 4. The annual coupon rate in Fiesta Corporation’s illustration was 10%. For quarterly payments, r = 10% ÷ 4 = 2.5.

As noted earlier, quarterly bond payments are rare. However, it is essential to acquaint ourselves with the computation to ease its application, should the need arise in future. The value of bonds with quarterly payments can be derived, mathematically, as follows:

Bt=∑_(t=1)^4N▒〖(It÷4)/(1+r÷4)”t” +F/(1+r÷4)4N〗

Assume Fiesta Corporation’s contractual agreement with bondholders included a provision for quarterly coupon payments. The PV function in Excel could be used to compute the value of the bond.

The final input would look like: =PV(0.025,40,25,1000,0). Click “ok.” The output = -1000.00.

With the aid of a financial calculator, the value of the bond (compounded) can be computed as follows:

N = 40; I/YR = 0.025 (2.5%); PMT = 25; FV = 1000; and press the PV key to compute the bond’s value.

In the financial calculator, the value of I/YR is entered as 0.025, not 2.5%; the percentage (2.5%) is converted into decimal (0.025) before the application. The output, 1000.00, depicts the bond’s value with quarterly coupon payments. Using the quarterly valuation method, the value (1000.00) of bond derived is not different from the values obtained from the annual (1000.00) and semi-annual (1000.00) valuation methods. We observe frequency of payments and compounding did not add more significant value to the quarterly payments than to the annual and semi-annual payments. It is worth emphasising the frequency of payments and compounding under the quarterly valuation method (four times) is more than those under the annual (once) and semi-annual (twice) valuation methods.

Monthly Coupon Payments: The following steps are essential to successful determination of the value of a given bond with monthly coupon payments.

1. The number of years, denoted by N, must be multiplied by 12, that is, N x 12. In Fiesta Corporation’s illustration, the number of years was 10. For monthly payments, N = 10 years x 12 = 120.

2. The annual coupon payments must be divided by 12, that is, I ÷ 12. For Fiesta Corporation, the annual coupon payment amount was GH¢100. For monthly payments, I = GH¢100 ÷ 12 = 8.33.

3. The annual coupon rate must be divided by 12, that is, r ÷ 12. The annual coupon rate in Fiesta Corporation’s illustration was 10 per cent. For monthly payments, r = 10% ÷ 12 = 0.83.

It is observed that the bond’s value (1,002.27), when determined using the monthly valuation method is high, relative to the value (1,000.00) derived from the annual, semi-annual and quarterly valuation methods.

Ebenezer M. Ashley (PhD)

Eben Consultancy

Fellow Chartered Economist &

Council Member, ICEG

Email: [email protected]

Website: www.ebenezerashley.com

As stated earlier, monthly bond payments are rare. However, it is essential to acquaint ourselves with the computation to ease its application in future bond transactions. The value of bonds with monthly coupon payments can be derived, mathematically, as follows:

Bt=∑_(t=1)^12N▒〖(It÷12)/(1+r÷12)”t” +F/(1+r÷12)12N〗

Suppose Fiesta Corporation’s contractual agreement with bondholders included a provision for monthly coupon payments. The PV function in Excel could be used to compute the value of the bond.

The final input would look like: =PV(0.0083,120,8.33,1000,0). Click “ok.” The output = -1002.27.

Using a financial calculator, the value of the bond (compounded) can be computed as follows:

N = 120; I/YR = 0.0083 (0.83 per cent); PMT = 8.33; FV = 1000; and press the PV key to compute the bond’s value.

In using the financial calculator, the value of I/YR is entered as 0.0083, not 0.83 per cent; the percentage (0.83 per cent) is converted into decimal (0.0083) before the application. The output of 1,002.27 represents the value of the bond with monthly coupon payments.

It is observed that the bond’s value (1,002.27), when determined using the monthly valuation method is high, relative to the value (1,000.00) derived from the annual, semi-annual and quarterly valuation methods. It is worth noting the frequency of payments and compounding under the monthly valuation method (twelve times) is significantly more than those under the annual (once), semi-annual (twice) and quarterly (four times) valuation methods. This significant difference in payment and compounding periods may account for the higher bond value (1002.27) under the monthly valuation method than the bond value (1,000.00) derived from the annual, semi-annual and quarterly valuation methods.