Bond Investment Strategies

Using Math to Understand Bonds

Mathematics is one of the keys to understanding the bond market. But it is important not to get too hung up on this. Grasping the concepts that underlie the bond market is the crucial task. Most of the data needed to make an investing decision either comes ready calculated and published on web sites or in the press, or can be worked out by punching a few keys on a financial calculator.

Bond market maths can be reduced to a few key concepts: compounding and discounting; accrued interest; running yields; redemption yields; and spreads. Armed with knowledge of these concept and hefty amount of common sense, finding an undervalued bond in which to invest becomes less of a lottery.

Even simple mathematical concepts can help.

Compound interest is one of those neat concepts much underrated by the average investor. For example, find an investment with a 7% annual return and the power of compounding is such that it would double an investor’s money in 10 years. One with a 10% return would double capital in seven years. See why below.

Compounding means earning interest on interest. Lend €1,000 at 10% and at the end of year one you receive €100. You now have €1,100 to lend out. At the end of year two you receive €110. You now have €1,210 to lend out. This earns interest of €121 so at the end of year three you have €1,331. And so on. Instead of receiving a simple €100 a year, you are actually getting more interest because the effect of compounding means that earlier interest payments are themselves earning too.

The maths here is relatively simple. The future value of an investment like this using compound interest is based on the principal (the amount invested or lent out), an interest rate and a term (the number of years the investment runs). This can be expressed algebraically as follows:

Future Value (FV) = P x (1+R)n

In this formula, P = principal, R = the annual rate of interest expressed as a decimal and n = the term in number of years.

Say a bank account pays 5% interest, with the interest paid once a year on the anniversary of the initial deposit. On a €10,000 deposit the amount that would have accrued at the end of five years would be:

Future Value (FV) = 10,000 x (1.05 to the power 5), or 10,000 x 1.276, which equals €12,760.

Understanding this idea is important because interest on interest is a crucial component used when we measure the true returns received from a bond between the time an investors buys it and its maturity date.

Discounting is the mirror image of compounding. The easiest way of understanding this is to look at the effect of inflation. Assume there is 5% inflation and you expect to get €1,000 in a year’s time. In today’s money, at the time you received the €1,000 it would only be worth the equivalent of €950, because inflation in the intervening period cuts its purchasing power by 5%, or €50. In other words, the present (i.e. today’s) value of the amount you expect to receive a year hence has been ‘discounted’ by 5%.

You can express this as follows:

Present Value (PV) = FV x (1-D)n

It is easy to see how similar this formula is to the one used for compounding. In the brackets a discount rate is deducted rather than an interest rate added, the present value (principal) and the future value change places, and that’s it.

Discounting is an important concept in bond investing because investing in a bond involves precisely this type of calculation.

With a bond an investor invests a certain amount at the outset and receives flows of cash, in the form of coupon payments, regularly throughout the bond’s life followed by the return of the principal at maturity.

One way of calculating the true return on a bond (i.e. its redemption yield, or yield to maturity – see below) is as the discount rate that equates the projected flows of cash received through a bond’s life to the price paid and the amount invested today.

If, for instance, an investor pays 90% of face value for a €10,000 bond with a coupon of 4% and ten years to go to maturity, he or she can expect the true return to be higher than the nominal 4% coupon.

This is because it would take a discount rate higher than 4% to equate the flows of coupon payments from the bond, which are based on face value, and the return of the €10,000 face value of the bond in ten years time, to the €9,000 actually paid for the bond today.

Put another way, the true return on the bond is a function of the price paid for the bond (which will almost always differ from its face, or par, value), the coupon rate and the length of time to go to maturity.

Accrued interest

One of the little quirks of the bond market is that when bonds are traded, the market price is adjusted to reflect the accruing of interest since the last coupon payment date.

Let’s say interest on a bond with a 5% coupon is paid at 31st December. A trade in €10,000 nominal of the bond settles on 31st March at a price of 102. The seller of the bond has, however, accrued 90 days of interest since the last payment date.

In this case an amount of €123.29 of interest has accrued, making the price to be paid by the buyer including accrued interest €103.2329. In bond market parlance this is known as the ‘dirty price’. In effect the buyer is paying the seller an extra amount over the market price reflecting the interest that the seller would have been entitled to but will not now receive, since the buyer will receive all of the subsequent coupon payment.

The formula for calculating accrued interest is therefore:

Accrued Interest (AI) = C x (TD – ID)/365

Where C = coupon, TD – ID is the number of days between the trade settlement date and the last interest payment date.

Different conventions for counting dates are used in different countries. Some are based on the actual numbers of days, some assume each month has 30 days and that a year has 360 days, and some ignore the effect of leap years.

Yields are the staple currency of bond investors. They are the basis of making judgements about whether a bond is cheap or dear.

Think of them as a standardised measurement that can be used to compare bonds from the same issuer but with different maturity dates, or bonds from different issuers with similar maturities. Redemption yields (yields to maturity) in particular allow this comparison to be made. They also allow us to compare bonds with different coupons on the same standardised basis.

At its simplest the yield on a bond is simply the coupon on a bond expressed as a percentage of the market price. A 20-year bond with a coupon of 6% selling at 120 has a simple yield of 5% (6 x 100/120). This is known as a ‘simple yield’, ‘income yield’, or ‘running yield’. This does not, however, tell the whole story. It ignores, for example, the ‘interest on interest’ element in a bond’s return - the fact that investors can reinvest what they receive in coupons to earn additional interest.

More important, however, it ignores the fact that the price paid for the bond and its redemption value, usually its face value or ‘par’ value, may be quite different. Most bonds are repaid at par (face value), or 100. So if an investor pays 120 for a bond and holds it to maturity, an accurate calculation of the true return has to include the fact that the investor will suffer an inevitable capital loss when the bond is repaid at par.

Ignoring the interest-on-interest element for the moment, in the example above an investor should deduct an amount of approximately 1% a year to allow for the annual loss in capital value as the bond moves to maturity. This is actually quite realistic because, other things being equal, this is precisely what the market will do. Holders of a bond bought above par will generally see a drift in the price of the bond over time, other things being equal, as it moves towards maturity and repayment at no more or no less than 100% of its face value.

The same applies to a bond bought below par, where holders will see their income yield supplemented by a capital gain as the bond moves up from the price they paid towards par as maturity approaches. An approximation of the redemption yield can be worked out by adding or subtracting the difference between par and the price paid divided by the number of years to maturity. This is known as the ‘adjusted current yield’.

The formula for this is:

Adjusted Current Yield (ACY) = SY + (100 – P)/M

Where SY = the simple or ‘running’ yield, P = the ‘clean’ price and M = the number of years to maturity.

In the example above this translates to:

Adjusted Current Yield (ACY) = 5 + (100-120)/20, which in turn equals 5% + (-20)/20, or 5% - 1%, or 4%.

The redemption yield, or yield to maturity, is just a more sophisticated way of getting to the same result. It also includes the interest-on-interest element in the return.

So as not to overload the algebra, the redemption yield formula can be stated in words as follows:

Yield to Maturity (YTM) = running yield + ‘interest-on-interest’ + annualized gain or loss on maturity.

…..or it can be expressed in the following way, as alluded to earlier in the idea of discounting:

Present Value (PV) of bond cash flows discounted at r% = market price + accrued interest

Where r% = redemption yield.

In strict terms, therefore, the redemption yield is the uniform discount rate at which the present value of a bond’s future cash flows (that is to say its coupon payments, interest on coupon payments, and repayment of the bond’s face value) equates to its current price including accrued interest.

Fortunately it is rare that investors need to calculate a redemption yield from scratch. Bond yields are listed in the financial pages, on financial web sites like and, or can be worked out with relative ease using the function in a spreadsheet or a pocket financial calculator.

What is needed to work out a redemption yield is the coupon, the frequency of coupon payments, the price of the bond, the trade settlement date, the maturity date of the bond, and the repayment value (usually 100).

The really important aspect of the redemption yield is that it is the single number that encapsulates all aspects of a bond – where the price stands relative to par, whether the bond is high coupon or low coupon (or indeed zero coupon), and its number of years to maturity. It can therefore be used to compare any bond from any issuer with any other bond from any other issuer.

Redemption yields provide the crucial information you need to assess true worth of the bond. Because redemption yields encapsulate all the relevant information on a bond in a standard format, the differences between yields on different bonds of a similar maturity (known as spreads) tell you a lot about how the market perceives the risk attached to them. A common technique is to calculate the spread against the equivalent government bond, such as a US Treasury bond or a German bund or a UK gilt edge bond of similar maturity.

The spread, or difference in redemption yield, is expressed in ‘basis points’. These are one hundredth of one per cent. So for example a Vodafone 5.9% sterling denominated bond maturing November 2032 currently yields 6.69% versus the gilt edge stock with same maturity yielding 4.4%, a spread of 2.29 percentage points (6.69 – 4.40), or 229 basis points.

What investors have to judge is whether or not the extra return implied by the spread over a government bond that is in practice free of the risk of default, is adequate compensation for the additional risk involved. This is the crucial question, and the one where maths stops and judgement begins.

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