**By Ebenezer M. Ashley (PhD.).**One of the essential “tools” identified as constituting an integral part of financial and economic trade in all countries across the globe is interest rate. The latter influences the extent to which individuals, households, businesses and economies are able to access loans and funding from lending institutions and multilateral institutions. Participants within the financial market interact with different forms of interest rate; and their determining factors. Notable among these include real interest rate, inflation premium, default risk premium, maturity risk premium, reinvestment rate risk, interest rate risk and liquidity premium. Succinct explanation on each of the foregoing variables is provided below. However, it is worth-stressing this article commences with continued discussion on “Determinants of Market Interest Rates” in the previous Article; and concludes with discussion on valuation of bonds.

**Real Interest Rate: **This is also called ** real rate of interest **or

**. It is the interest rate paid by debt issuers to debtholders when no inflation is expected; and the security is devoid of risk. The real interest rate is often the rate offered by governments on short-term Treasury Bills in periods of no inflation. The real interest rate is not fixed; it varies with the level of economic conditions such as changes in the performance of corporate bodies; changes in the earnings expectations on productive assets by corporate borrowers; and changes in the present and future tastes and preferences of consumers. Although**

*real risk-free interest rate**short-term risk-free rate*is estimated based on the rate of government Treasury bill,

*long-term risk-free rate*is determined based on the rate of government Treasury bond.

**Inflation Premium: **The purchasing power of a nation’s currency is negatively impacted when there is inflation. Inflation is said to have occurred when there is constant increase in the prices of goods and services in a given economy over a considerable time period. Inflation has negative implications for the value of investments in the affected economy. To compensate for expected fall in value of investment resulting from inflation, investors factor *inflation premium* into the computation of the nominal interest rate. Generally, the average inflation rate expected over the life of the security is called inflation premium.

To illustrate, assume Investor B invests $100 in securities at 6% per annum when the price of apple per box is $5. At the end of the year, Investor B would receive $106.00 ($100 + ($100 x 0.06) = $100 + $6 = $106) from the investment in securities. At the beginning of the year, Investor B could purchase 20 boxes ($100 ÷ $5 per box = 20 boxes) of apple at $100. Further, assume the average inflation during the year was 12%. The inflation rate had an equal effect on all goods and services in the economy. In this case, the price per box of apple at the end of the year would be $5.60 ($5 + ($5 x 0.12) = $5 + $0.60 = $5.60). The number of boxes of apple Investor B could buy with the $100 at the end of the year would be 18 boxes ($100 ÷ $5.60 = 17.857 = 18 boxes), that is, 2 boxes short of the total number (20 boxes) he could purchase at the beginning of the year.

One observes the total cost of 20 boxes of apple at the end of the year would be $112 ($5.60 x 20 boxes = $112) while total proceeds from the investment would be $106 ($100 + ($100 x 0.06) = $100 + $6 = $106). Therefore, the investor would be worse off in real terms because the interest income of $6 is not enough to settle the additional cost of $12 on the 20 boxes of apple he could purchase with the investment amount at the beginning of the year. The illustration suggests Investor B would be economically better off investing in the apples than purchasing the securities.

To avert this investment shortfall, debtholders demand inflation premium in the determination of the nominal interest rate. Mathematically, the nominal interest rate with inflation premium is expressed as: ** r = RR + IP**………. (i). Where r represents

*nominal interest rate*; RR connotes

*real interest rate*; and IP reflects

*inflation premium*.

**indicates the interest rate to be charged by debtholders under this circumstance.**

*Equation 1***Default Risk Premium: **In some cases, the amount received by debtholders at the end of the investment period or maturity date does not equal the promised amount; the debt issuer is likely to default in payments to debtholders. To compensate for this probable risk, debtholders request for the inclusion of default risk premium in the computation of the nominal interest rate by debt issuers. A debt issuer with a greater default risk is expected to pay higher default risk premium, vice versa. Generally, the yield-to-maturity (YTM) on debts with high default risk is greater than those with low default risk.

Comparatively, securities issued by corporate bodies have higher default risk rate than securities issued by governments. Due to the foregoing, most government securities do not attract default risk premium. Municipal securities are more comparable to corporate securities in terms of risk than to Treasury securities. The debt issuer’s ability to fulfill promises to debtholders without default may be predicated on a number of factors, including the debt issuer’s financial strength and the contractual provisions, including secured and unsecured securities agreements.

**Maturity Risk Premium: **Generally, it is challenging to make predictions into the future, irrespective of the type of security – Treasury securities or corporate securities. In the long-term, *reinvestment *and* interest rate *stability becomes doubtful. The uncertainties surrounding reinvestment and interest rates result in the inclusion of *maturity risk premium* in the calculation of the yield-to-maturity or nominal interest rate. The inclusion of maturity risk premium in the computation of the nominal interest rate is illustrated in ** Equation 3 **(as outlined in the previous article):

**……………(iii). Explanations on reinvestment risk and interest rate risk are presented in the following section.**

*r = RR + IP +DRP + MRP + LP**Reinvestment Rate Risk***: **At maturity date, debtholders are expected to receive full payments (principal plus interest or coupon payments) from debt issuers. In most cases, debtholders may be interested in reinvesting their earnings in the same firm or another; or others. Suppose the initial investment attracted a coupon rate of 8% per annum. Due to changing economic conditions, the coupon rate reduced to say, 6%. Should the debtholder decide to reinvest his or her funds in the debt issuing firm at 6%, there would be a decrease in income.

Thus, the risk of income decline that debtholders assume as a result of a decrease in coupon rates is known as *reinvestment rate risk*. The effect of reinvestment rate risk is highly felt by investors whose securities are callable prior to the maturity date; the negative impact is highly felt by debtholders with short-term maturity. However, the negative impact of reinvestment rate risk is minimised on debts with long-term maturity and non-callable option.

*Interest Rate Risk***: **The securities market is often saddled with fluctuations in interest rates. These fluctuations affect the value of issued securities in the market. An increase in the interest rate leads to a decrease in value of securities. Investment analysts use the term *interest rate risk *to explain a fall in value of outstanding securities resulting from an increase in interest rates. Due to changing economic conditions, interest rates do change in the securities market; and debtholders are adversely affected; the value of outstanding debts decreases as interest rates increase. Holders of securities with long-term maturity are strongly affected by interest rate increase than holders of debt with short-term maturity.

**Liquidity Premium: **The performance of certain corporate bodies and government agencies such as municipalities makes it difficult for their securities to be traded with relative ease in the financial market. For instance, it may be difficult for a debtholder who wishes to trade or sell debt held in Company A to another investor when Company A is perceived by investors as risky or inactive in the financial market. A company may be described as inactive if its debts are not frequently traded in the financial market. The investor holding debts of Company A may end up selling them at a huge discount.

Due to the foregoing, investors ask for the inclusion of *liquidity premium *in the computation of the nominal interest rate. ** Equation 3** above depicts the inclusion of liquidity premium in determining the nominal interest rate. Generally, there is a relationship between size of a firm and liquidity challenges; debts of large firms are traded faster than debts of small firms. Therefore, liquidity premium of smaller firms tends to be higher than that of larger firms.

**Valuation of Bonds**

The value of bond at time t, or Bt, can be computed as follows:

Bt = __(I____t ____+1)__ + __(I____t ____+2)__ + __(I____t ____+3)__ + (__I____t ____+4)__ + __(I____t ____+5)__ + __(I____t ____+6)__ … + __ F__

(1+r)¹ (1+r)² (1+r)³ (1+r)⁴ (1+r)⁵ (1+r) ⁿ (1+r)ⁿ

Where:

It + n = Coupon payment in period t + n

F = Principal payment or face value

r = Yield-to-maturity or interest rate required by investors

**Illustration of Bond Valuation**

On 8^{th} January, 2017, Fiesta Corporation issued GH¢ 1,000 of 10-year bonds with a year-end interest (coupon or premium) payment of 10% per annum. On 8^{th} January, 2023, yield-to-maturity on these bonds is 8%.

- What is the value of these bonds if the yield-to-maturity between 2017 and 2022 is 10%?
- What is the value of these bonds if the yield-to-maturity on 8
^{th}January, 2023 is 8%? - What are the total premium payments on these bonds if the coupon rate between 2017 and 2022 is 10%?
- What are the total interest payments on these bonds from 8
^{th}January, 2023 to the maturity date, if the coupon rate is 8%? - What is the total value, if the yield-to-maturity throughout the life of these bonds is 10%?

**Solution:**

- Actual date for redeeming these bonds is 31
^{st}December, 2026 - Remaining term to maturity is 4 years (2023 through 2026)
- Yearly interest or coupon or premium payment on bonds is GH¢100.00 (GH¢ 1,000 x 10%)
- Principal payment on 31
^{st}December, 2026 is GH¢ 1,000.00 - Yield-to-maturity of the bonds between 2017 and 2022 is 10% per annum
- Yield-to-maturity of the bonds on 8
^{th}January, 2023, is 8% per annum

F = Principal

n = Number of years

I = Coupon payment in period t+n

r = Interest rate

F = ¢1000.00

n = 10 years (2017 – 2026)

I = ¢1000.00 × __10%__ = ¢100.00

100%

**What is the value of these bonds if the yield-to-maturity between 2017 and 2022 is 10%?**

** **

**Solution:**

F = GH¢1,000

I = GH¢100.00

n = 6 years (2017 – 2022)

r = 10% = 0.10

The value of the bonds between 2017 and 2022 can be computed as follows:

Bt = __(I____t ____+1)__ + __(I____t ____+2)__ + (__I____t ____+3)__ + (__I____t ____+4)__ + __(I____t ____+5)__ + (__I____t ____+6)__ …. + F__ __

(1+r)¹ (1+r)² (1+r)³ (1+r)⁴ (1+r)⁵ (1+r)⁶ (1+r)⁶

Bt ₌ __100 __ + __100 __ + __100 __ + __100__ + __100 __ + __100 __ + __1000 __

(1+0.10)¹ (1+0.10) ² (1+0.10)³ (1+0.10)⁴ (1+0.10)⁵ (1+0.10)⁶ (1+0.10)⁶

₌ 90.9091 + 82.6446 + 75.1315 + 68.3014 + 62.0921 + 56.4474 + 564.4740

**₌**** **** GH****¢**__1,000.0001__

**What is the value of these bonds if the yield-to-maturity on 8**^{th}January, 2023 is 8%?

** **

**Solution:**

F = GH¢1,000.00

I = GH¢100.00

n = 4 years = (2023 – 2026)

r = 8% = 0.08

The following computations depict the value of the bond on 8^{th} January, 2023:

Bt ₌ __100 __ + __100 __ + __100 __ + __100__ + __1000 __

(1+0.08)¹ (1+0.08) ² (1+0.08)³ (1+0.08)⁴ (1+0.08)⁴

₌ 92.5930 + 85.7340 + 79.3832 + 73.5030 + 735.0300

**₌**** **** GH¢ 1,066.2432**

**What are the total premium payments on these bonds if the coupon rate between 2017 and 2022 is 10%?**

** **

**Solution:**

I = GH¢100.00

n = 6 years (2017 – 2022)

r = 10% = 0.10

Total interest payments over the six-year period (2017 to 2022) are determined as follows:

Bt ₌ __100 __ + __100 __ + __100 __ + __100__ + __100 __ + __100 __ __ __

(1+0.10)¹ (1+0.10) ² (1+0.10)³ (1+0.10)⁴ (1+0.10)⁵ (1+0.10)⁶

₌ 90.9091 + 82.6446 + 75.1315 + 68.3014 + 62.0921 + 56.4474

**₌**** **** GH****¢**__435.5261__

** **

**What are the total interest payments on these bonds from 8**^{th}January, 2023 to the maturity date, if the coupon rate is 8%?

**Solution:**

I = GH¢100.00

n = 4 years = (2023 – 2026)

r = 8% = 0.08

Total interest payments over the four-year period (2023 to 2026) are determined as follows:

Bt ₌ __100 __ + __100 __ + __100 __ + __100__ __ __

(1+0.08)¹ (1+0.08) ² (1+0.08)³ (1+0.08)⁴

₌ 92.5930 + 85.7340 + 79.3832 + 73.5030

**₌**** **** GH¢ 331.2132**

**What is the value, if the yield-to-maturity throughout the life of these bonds is 10%?**

** ****Solution:**

F = GH¢1,000

I = GH¢100.00

n = 10years (2017 – 2026)

r = 10% = 0.10

Bt ₌ __100 __ + __100 __ + __100 __ + __100__ + __100 __ + __100 __ + __100 __+ __100 __ + __100__ +

(1+0.10)¹ (1+0.10) ² (1+0.10)³ (1+0.10)⁴ (1+0.10)⁵ (1+0.10)⁶ (1+0.10)⁷ (1+0.10)⁸ (1+0.10)⁹

__100 __ + __1000__

(1+0.10)¹º (1+0.10)¹º

₌ 90.9091 + 82.6446 + 75.1315 + 68.3014 + 62.0921 + 56.4474 + 51.3158 +46.6507 +

42.4098 + 38.5543 + 385.5433

**₌**** **** GH****¢**__1,000.0000__

From the foregoing computations, we affirm the indispensable role of the yield in the measurement of bonds value. It is worth emphasising factors such as default risk, current interest rates, and expected inflation play a major role in the calculation of expected or desired yield, when determining the value of bonds.

**Yields in Bond Transactions**

Determination of yields in bond transactions is “indexed” to prevailing market conditions. Unlike the premium rate which often remains unchanged to the bond’s maturity date, a yield may vary in relation to conditions existing in the market from time-to-time. Generally, activities in the bond market could result in the derivation of three distinct yields. These include *current yield, yield-to-maturity, *and* yield-to-call*.

**Current Yield: **This is also called ** interest yield**. It is determined by measuring the annual coupon payment as a fraction of the bond’s prevailing market price. For instance, in our earlier illustration using Fiesta Corporation, the annual coupon payment was determined as GH¢100.00; the value of the bond was given as GH¢1,000.00. Suppose Fiesta Corporation’s bond is currently trading at GH¢990.00. The current yield would be 10.10% ((GH¢100 ÷ GH¢990) x 100% = 0.1010 x 100% = 10.10%). One observes an increase in yield (10.10%) as the price or value of the bond falls from GH¢1,000.00 to GH¢990.00. Conversely, an increase in price of the bond would result in a decrease in the current yield. In our example, assume the prevailing market price of Fiesta Corporation’s bond is GH¢1,020.00. The current yield would be 9.80% ((GH¢100 ÷ GH¢1,020) x 100% = 0.0980 x 100% = 9.80%).

It is worth emphasising the current yield is not the rate investors envisage as a return on their bond investment. Rather, it provides bond investors with ample information on the amount of cash income their bonds would generate in a given year. However, capital gains or losses to be realised should the bond be called or held to the maturity date are not factored into the computation of the current yield. As a result, it is argued the use of current yield as determinant of the total expected rate of return on bond investment is inaccurate.

**Yield-to-Maturity (YTM): **This refers to the rate of return a bondholder earns, should he or she decide to hold on to the bond to the maturity date. If the bond issuer does not call the bonds prior to the maturity date, and does not default in payment, the bondholder would earn the ** promised rate of return**, which is the yield-to-maturity. Assume on 8

^{th}January, 2017, Fiesta Corporation issued GH¢ 1,000 of 10-year bonds with a year-end coupon payment of 10% at a price of GH¢1,450.00. The maturity date for the bonds issued is 7

^{th}January, 2027. Should the bond investor hold on to the bond to the maturity date, he or she would earn

**as a return on the investment as depicted in the following equation.**

*4.35%*Bt = GH¢1,450 = __100__ + __100__ + __100 __+ … __100__ + __1000__

(1+r)¹ (1+r)² (1+r)³ (1+r)¹º (1+r)¹º

To determine the yield-to-maturity in the above equation, we would have to substitute presumed values for “r” until we arrive at the value that would allow the GH¢1,450.00 on the left-hand side to equal the figures to the right of the equation. Indeed, this may be mathematically laborious. However, with the aid of an *Excel* or a *financial calculator*, we could determine the yield-to-maturity with relative ease. The calculations revealed the above equation would “balance,” if we substitute 4.35% for *r*.

**Excel Function: **The spreadsheet in ** Excel **could be used to compute the yield-to-maturity. In Excel, the

**could be used to compute the yield-to-maturity after entering the following variables:**

*RATE function***N = 10; PMT = 100; PV = -1450; FV = 1000; Type = 0.**The

**would look like:**

*final input***= RATE(10,100,-1450,1000,0)**. Click “ok.” The

**would be**

*result or output***4.35%**, which is the bond investment’s yield-to-maturity.

The ** RATE function **was used instead of the YIELD function because the former works effectively if the objective is to determine the YTM for a bond transaction with a current date that comes immediately after either a premium payment date or an issue date. In determining yields on bonds on dates other than the foregoing, one could employ the

**in Excel (Brigham and Ehrhardt, 2008, p. 168). The present value (PV) amount entered into the calculator shows a**

*YIELD function***value because it is an**

*negative***from the bond issuer to the bondholder.**

*outflow***Financial Calculator: **In the financial calculator, one would have to input (enter) the following variables to derive the YTM: **N = 10; PV = -1450; PMT = 100; FV = 1000; **and press the key,** I/YR. **

**Yield-to-Call (YTC): **The tendency for a bondholder to hold on to his or her bonds to the maturity date may not be guaranteed if the bond agreement has a call option. A bond with a callable option may be called by the bond issuer if the present market premium rate is less than the rate on the date of signing the bond contract. At the call date, the expected rate of return would be computed and paid to bondholders. Here, the expected rate of return is known as the yield-to-call. In our earlier illustration, we noted Fiesta Corporation issued its 10-year bonds at 10%.

Suppose the bonds issued had a callable option and the current market rate is 6%. Fiesta Corporation may be interested in calling the bonds issued earlier at 10% to save 4% (10% – 6% = 4%) each year on each bond. In monetary terms, Fiesta Corporation would save GH¢40 (GH¢100 – GH¢60 = GH¢40) per annum on each bond issued or sold. This callable option inures to the financial benefits of the bond issuer; bondholders may be compelled to surrender their bond investments to the bond issuer. Mathematically, YTC could be determined as:

Where:

N = Number of years until the issuer can call the bond

Call price = Amount the issuer must pay to the holder to call the bond

r = Yield-to-call

** **The call price is often determined as an equivalent of the bond’s face value and one-year coupon payment (Brigham and Ehrhardt, 2008). Assume the 10-year bond issued by Fiesta Corporation had a clause that allowed the firm to call the bonds 5 years after the issue date. The agreed price at the call date is GH¢1,100 (par or face value (GH¢1000) plus one-year coupon payment (GH¢100)). Further, assume the current price of the bond price after 3 years is GH¢1,250.

The increase in bond price could be attributed to a fall in coupon rates in the bonds market. Since the fall in coupon rates was observed 3 years after the issue date, and the bonds had a life of 10 years, Fiesta Corporation had 2 years more until it can first call the bonds. The YTC for Fiesta Corporation is computed below. N (represented by 1 and 2) in the equation equals years. That is, the years until Fiesta Corporation can first call the bonds. See ** Figure 5**.

As noted in the computation of YTM, presumed values would have to be substituted for “r” or YTC until the accurate rate is obtained. This may be laborious. However, with the aid of an *Excel* or a *financial calculator*, we could determine the yield-to-call with relative ease.

**Excel Function: **In our illustration, the date is 3 years after the bonds were issued, not immediately or a year after they were issued. To this end, the ** YIELDMAT function **in Excel could be employed to compute the yield-to-call; the spreadsheet in

**could be used to compute the yield-to-call. The**

*Excel***in Excel would be applied to the calculation of the YTC. The Date function in Excel is entered as DATE(YearMonthDay) without space and commas. Suppose we have a transaction recorded on 23**

*DATE function*^{rd}August, 2015. This would be entered as

**DATE(2015823)**. In Excel, the

**could be used to compute the yield-to-call after entering the following variables:**

*YIELDMAT function***Settlement = 202217 **

**Maturity = 202717**

**Issue = 201718 **

**Rate = 0.10 **

**Pr = 1000 **

Where ** Settlement** refers to the bond’s first call date, which is 7

^{th}January, 2022;

**represents the originally agreed date on which the bond is expected to mature, which is 7**

*Maturity*^{th}January, 2027;

**relates to the date, on which the bond was issued, that is, 8**

*Issue*^{th}January, 2017.

** Rate** refers to the coupon rate at the issue date, that is, 10%; and

**represents the principal or face value of the bond, that is, GH¢1000.**

*Pr*The ** final input** would look like:

**=YIELDMAT(202217,202717,201718,0.1,1000)**. Click “ok.” The

**would be**

*result or output***6.40%,**which is the bond investment’s yield-to-call.

**Financial Calculator: **In the financial calculator, one would have to input the following variables to derive the YTC: **N = 2; PV = -1250; PMT = 100; FV = 1000; **and press the key,** I/YR. **

The output, 6.40%, represents the YTC; it is the rate of return to be earned should a decision be made to purchase the bond at GH¢1,250 and hold until it was called 2 years from today. It is worth emphasising, whether Fiesta Corporation would call the bonds at the first callable date or not would depend essentially on prevailing market conditions; it would depend on the market premium rate at the callable date. Should the premium rate increase to say, 12% at the first callable date and beyond, it would not be economically beneficial for Fiesta Corporation to call the bonds.

However, calling the bonds would be a wise economic decision if the coupon rate at the first callable date is 6%. The cost savings of GH¢40 each year per bond would be an incentive for Fiesta Corporation to call the bonds, in spite of the fact that these cost savings may be reduced by the cost of retiring or calling the bonds. The yield-to-call is 6.40%. This is more than the yield-to-maturity value of 4.35%.

**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 face value of GH¢1000 and coupon rate of 10% would be GH¢1000. The ** Excel input** for the computation is

**=PV(0.1,10,100,1000,0)**. The

**Excel output**is 1000.00. However, computation of semi-annual, quarterly and monthly coupon payments requires different valuation formulae and inputs.

**Semi-Annual Coupon Payments: **Successful determination of the value of a bond with semi-annual coupon payments would hinge on the following steps. First, 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. Second, 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. Third, the annual coupon rate must be divided by 2, that is, r ÷ 2. For Fiesta Corporation, the annual coupon rate was 10%. For semi-annual payments, r = 10% ÷ 2 = 5.

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

Suppose 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

**key to determine the bond’s value.**

*PV*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. First, 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. Second, 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. Third, 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:

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

**, 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**

*financial calculator***key to compute the bond’s value.**

*PV*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. First, 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. Second, 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. Third, the annual coupon rate must be divided by 12, that is, r ÷ 12. The annual coupon rate in Fiesta Corporation’s illustration was 10%. For monthly payments, r = 10% ÷ 12 = 0.83.

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:

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

**would look like:**

*final input***=PV(0.0083,120,8.33,1000,0)**. Click “ok.” The

**output**=

**-1002.27.**Using a

**, the value of the bond (compounded) can be computed as follows: N = 120; I/YR = 0.0083 (0.83%); PMT = 8.33; FV = 1000; and press the**

*financial calculator***key to compute the bond’s value.**

*PV*In using the financial calculator, the value of I/YR is entered as 0.0083, not 0.83%; the percentage (0.83%) is converted into decimal (0.0083) before the application. The output of 1002.27 represents the value of the bond with monthly coupon payments. It is observed the bond’s value (1002.27), when determined using the monthly valuation method is high, relative to the value (1000.00) 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 (1000.00) derived from the annual, semi-annual and quarterly valuation methods.

**Author’s Note**

The above write-up was extracted from a Chapter in “Principles of Corporate Finance: Theory with a Practical Dimension (Second Edition)” by Ashley (In Press).