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Financial strategy: students tend to find discussing debt finance far easier than doing the calculations. Andrew Howarth explains how to deal with yield on debt.

An investor who purchases a corporate bond receives a return from that bond in the form of annual interest (or coupon) payments and, if the bond is redeemable, the final redemption payment. The total effective return is referred to as the yield. The yield of a redeemable bond is also known as the yield to maturity (YTM) or redemption yield.

In F3 exams you will often be asked to calculate the yield or YTM of a bond. In some cases--eg, if the bond is irredeemable--the calculations are simple. But they are more complex if the bond is redeemable or denominated in a foreign currency.

As well as being the effective return to the investor, the yield on a bond is important in many other calculations. For example, an entity's cost of debt, which is used in calculating the weighted-average cost of capital, can be derived by adjusting the yield for the tax relief on debt interest. This post-tax cost of debt, derived from the yield, is also used as the discount rate when evaluating a lease-versus-buy decision.

Irredeemable bonds

Let's start with the simplest of all yield calculations. An irredeemable corporate bond that has just been issued at par (the face, or nominal, value) will have a yield equal to its coupon rate.

Consider a company called Kite, which has just issued some irredeemable 5% coupon-rate bonds at their par value of [pounds sterling]100. It is clear that if an investor were to purchase one of these Kite bonds, they would receive a return of 5% x [pounds sterling]100 = [pounds sterling]5 every year into perpetuity. So their average annual return, as a percentage of the current value of the bond ([pounds sterling]100) is [pounds sterling]5/[pounds sterling]100 = 5%.

In short, for an irredeemable bond, the percentage yield = (annual interest received/current bond price) x 100.


Owing to factors of supply and demand in the marketplace, bond prices fluctuate over time. So it could be that, say, one year later Kite's bonds are trading at [pounds sterling]95. An investor purchasing a bond now has to pay [pounds sterling]95 in order to receive the interest stream of 5% x [pounds sterling]100 = [pounds sterling]5 every year into perpetuity. The interest is always calculated by applying the coupon rate to the bond's par value, irrespective of its current price. So the yield in this case is ([pounds sterling]/[pounds sterling]95) x 100 = 5.26%.
1 Calculating the internal rates of return for Kite's redeemable bond

Time [pounds Discount Present Discount Present
 sterling] factors at value at factors at value at
 a discount 8% a discount 10%
 rate of 8% rate of

Year 0 (100) 1.000 (100.00) 1.000 (100.00)

Years 1-5 1 3.993 3.99 3.791 3.79

Year 5 150 0.681 102.15 0.621 93.15

 6.14 (3.06)

It's notable that bond prices and bond yields are inversely related--ie, the lower the price of a bond, the higher its yield will be. This is because the actual amount of interest (coupon amount) stays constant. So, if the bond's price falls, the interest represents a higher proportion of the price.


Redeemable bonds

The yield on a redeemable bond--ie, one with a specified maturity or redemption date--is a more complex calculation, because the redemption amount and the remaining time before maturity will also influence the yield. To see why, consider the following simple example.

Assume that Kite is about to issue some more [pounds sterling]100 bonds at par, this time with a coupon rate of 1% and a redemption amount of [pounds sterling]150 after five years. An investor buying one of these bonds for [pounds sterling]100 will be entitled to [pounds sterling]1 in interest each year and then [pounds sterling]150 in five years' time. Using the logic from our previous calculations for irredeemable bonds--ie, bonds with an indefinite life--it would be easy to assume that the yield on these new Kite bonds is 1%. After all, an investor buying a bond for [pounds sterling]100 on issue will receive interest of 1% x [pounds sterling]100 = [pounds sterling]1 each year, making a return of 1% of the current bond price. But the return is no longer received in perpetuity. In fact, in five years' time the investor will also receive a lump sum of [pounds sterling]150, which is significantly greater than the original cost of the bond. If we are to calculate the yield properly, it is important to incorporate the impact of this premium on redemption.

In effect, the investor's yield is a 1% return each year and then an extra 50% return in the fifth year. The yield calculation replaces this cumbersome statement with a single effective annual percentage figure, which incorporates both elements of the return. The yield of a redeemable bond equals the internal rate of return (IRR) of the bond price, the annual interest received and the final redemption amount.

Using the figures from the Kite example, we need to calculate the IRR of:

* Year 0: current bond price paid: ([pounds sterling]100).

* Years 1-5: annual interest received: [pounds sterling]1.

* Year 5: redemption amount received: [pounds sterling]150.

The yield can now be calculated by discounting these amounts at any two discount rates and using interpolation, as shown in table 1 above. By interpolation, the annual percentage yield works out as 8% + [2% x (6.14/{6.14 + 3.06})] = 9.33% a year.

So, instead of saying that the investor receives 1% every year in interest and then a 50% return in five years' time, we say that the effective annual return to the investor--ie, the YTM--in this case is 9.33%.

As with irredeemable bonds, the price of redeemable bonds can fluctuate over time. So, if one year later these Kite bonds were priced at [pounds sterling]90 in the market, the yield of these four-year bonds could once again be calculated by interpolation. Using the figures derived from table 2 below, we can use interpolation to work out the percentage yield as 12% + [3% x (8.44/{8.44 + 1.34})] = 14.59% a year.
2 Calculating the internal rates of return for Kite's redeemable bond
taking account of change in market value after year one

Time [pounds Discount Present Discount Present
 sterling] factors at value at factors at value at
 a discount 12% a discount 15%
 rate of rate of
 12% 15%

Year 0 (90) 1.000 (90.00) 1.000 (90.00)

Years 1-4 1 3.037 3.04 2.855 2.86

Year 4 150 0.636 95.40 0.572 85.80

 8.44 (1.34)

Again we can see the inverse relationship between yield and bond price: because the bond price has fallen, the yield has increased.

Irredeemable foreign currency bonds

Yield calculations become more complicated when foreign currency loans are involved. Let's now assume that Kite issues some $100 irredeemable bonds at par, with a coupon rate of 5%. If the exchange rate between the dollar and the pound is expected to stay constant, the yield calculation is the same as it was before, but a problem arises when the rate is expected to change. For example, if the dollar is expected to strengthen by 3% a year against the pound, paying the interest in dollars each year will in effect cost Kite 3% more annually, on top of the 5% coupon rate payable. So the yield on its irredeemable dollar bonds would be(1.05 x 1.03) - 1 =8.15% a year.
3 Calculating cash flows on Kite's redeemable $100 bond taking account
of the strengthening dollar against the pound

Time Item $ [pounds sterling] to $ exchange [pounds
 rate sterling]

Year 0 Current (100) 1.5000 (66.67)
 bond price

Year 1 Interest 5 1.5000 / 1.03 = 1.4563 3.43

Year 2 Interest 5 1.4563 / 1.03 = 1.4139 3.54

Year 3 Interest 5 1.4139 / 1.03 = 1.3727 3.64

Year 4 Interest 5 1.3727 / 1.03 = 1.3327 3.75

Year 5 Interest 115 1.3327 / 1.03 = 1.2939 88.88

4 Calculating the internal rates of return for Kite's redeemable bond
taking account of exchange rate fluctuations

Time [pounds Discount Present Discount Present
 sterling] factors at value at factors at value at
 a discount 8% a discount 10%
 rate of 8% rate of

Year 0 (66.67) 1.000 (66.67) 1.000 (66.67)

Year 1 3.43 0.926 3.18 0.909 3.12

Year 2 3.54 0.857 3.03 0.826 2.92

Year 3 3.64 0.794 2.89 0.751 2.73

Year 4 3.75 0.735 2.76 0.683 2.56

Year 5 88.88 0.681 60.53 0.621 55.19

 5.72 (0.15)

Redeemable foreign currency bonds

Lastly, let's calculate the yield if Kite issues some $100 5% coupon bonds at par, redeemable in five years at $110. Assume that the dollar is expected to strengthen by 3% a year against the pound and that the current (spot) exchange rate is [pounds sterling]1 = $1.50. We know that the yield is the IRR of the current market value, the interest payments and the redemption amount. The complication in this case is that the cash flows will first need to be converted into Kite's home currency, pounds, using exchange rates that are different each year. This is done using table 3. Then we plug those figures into our IRR interpolation calculation in table 4 in order to obtain a percentage yield of 8% + [2% x (5.72 [division] {5.72 + 0.15})] = 9.95% a year (or, by inspection, about 10%).

Bond yields are commonly tested at strategic level and examiners' past comments have indicated that students often find the calculations hard. But I hope that you now understand that the yield is not the same as the coupon rate on the bond. In fact, several other factors also affect the yield. They are as follows:

* The current market value of the bond.

* The capital payment on maturity.

* The coupon rate.

* The time to maturity, or whether the bond is irredeemable.

* Whether the bond is denominated in the company's domestic currency or whether it's in a foreign currency.

The crucial move before calculating a yield is to identify first whether the bond is irredeemable or redeemable and then whether it is denominated in the home currency of the company concerned or a foreign currency.

For an irredeemable bond--ie, one with no maturity date--the percentage yield can be calculated quickly as (annual interest [division] bond price) x 100. If the bond is denominated in a foreign currency, you need to adjust this formula by taking into account the expected annual strengthening or weakening of that currency against the company's domestic currency.

For a redeemable bond--ie, one with a given maturity date--the yield is found by calculating the IRR of the current bond price, the annual interest payments and the redemption payment. If the bond is denominated in a foreign currency, make sure that you convert the cash flows back into the firm's home currency, using the forecast exchange rates, before calculating IRR.

Andrew Howarth is Kaplan Publishing's content specialist for paper F3.

F3 further reading

J Ogilvie, The Official CIMA

Learning System--Financial

Strategy, CIMA Publishing, 2009.
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Title Annotation:study notes
Author:Howarth, Andrew
Publication:Financial Management (UK)
Date:Sep 1, 2010
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