What are bonds and how to value them

WHAT IS A BOND

A bond is an asset that generates to his owner fixed payoffs called coupons for each period until it finally matures and gives the owner its last coupon and a larger payment called the face value or bond’s principal. In order to evaluate a bond we have to calculate the present value of all its cashflows. The present value of a bond depends on the opportunity cost the investor faces when taking into consideration its purchase and it is computed as the sum of the present values of all coupons and the final payment of the face value. The payment of all coupons can be computed also using the formula for annuities:

When talking about bonds and their returns we need to introduce the concept of yield to maturity (y) which indicates the percentage of return the bond holder expects from his investment if the bond is held until maturity. And if you know the present value of the bond, the coupons and its face value you can calculate it via a process of trial and error on the formula:

If you keep the value of the bond as constant, the more you receive via coupons, the less you will receive in the final face value payment.

WHAT HAPPENS IF BONDS ARE PRICED DIFFERENTLY THAN THEIR FACE VALUE?

A bond sold at a price higher than its face value is said to sell at a premium. A bond sold at a premium will generate a loss if you only look at its price compared to its face value but, when evaluating the full bond you need to consider also the coupons. The ratio between what you receive in a single payment and the price of the bond is denotes as the current yield. A bond sold at premium will have a current yield higher than the yield to maturity. This means that, during the various years of the bonds, you receive “in percentage” more of the total cost of the bond than the total earnings promised by it (ex. A coupon may repay you 2% of the total cost of the bond while the bond itself promised an end return of 1.5%). This happens because, when the bond matures, you are receiving as face value less than you paid. If a bond is priced below its face value, it is said to be sold at a discount. For this kind

of bonds the situation is the opposite: the current yield is lower than the yield to maturity because in the end the bond will generate revenues considering only issuing price and face value.

THE RELATIONSHIP BETWEEN THE PRICE OF THE BOND AND THE YIELD TO MATURITY

As you can see from the formulas above the bond price and its yield to maturity are inversely related meaning that when the bond price increases the yield to maturity decreases and vice versa. This is intuitive: the more the bond costs the less the “profit” you can make.

THE DURATION AND THE VOLATILITY OF A BOND

Obviously a bond that lasts longer will be affected more by a change in the interest rate than a bond lasting less. To understand why, imagine that you own a bond lasting 1 year, if the interest rate changes, the only cashflow that will be influenced is the final one, while, if instead you have a bond lasting 20 years, then all the coupons and the face value from now to year 20 will be influenced by the change in the interest rate. We have to distinguish among the time a bond needs in order to mature and the duration of the bond. A bond’s duration is defined as the average time needed for a bond to pay back its value. The duration of a bond is computed as the weighted average of the cash payments (expressed in present value) multiplied by the period in which they are received:

The more a bond has high coupons but lower face value the shorter is its duration. This is because a larger part of the bond’s value is paid back earlier with coupons. If the bond’s yield to maturity increases you are much happier if the bond lasts less because the discount you have to apply to coupons and the face value influences a shorter number of them. Conversely, if the yield to maturity decreases, the better bond is the one having a longer duration. The volatility of a bond is a measure of how much bonds prices change when yields to return change. It tells us the percentage change in the our bond price when there is a 1% change in the yield and it is computed as follows:

NOT ALWAYS THERE IS ONLY ONE INTEREST RATE

Until now, we only used one interest rate (interest rate is a synonym for yield to maturity) and, in most cases, this can be used as a fair approximation but there are cases in which is important to distinguish between long term and short term interest rates. The relationship between the two is described as term structure of interest rates. Let’s explain this concept with an example: if you own an asset giving you $10 at the end of the year, to calculate its present value you need to compute: $10/(1+r1) where r1 is the interest rate in period 1 also called the one year spot rate. If your asset instead pays you $10 in 2 years, to calculate its present value you will compute: $10/(1+r2) where r2 is the two year spot rate. If your investment pays you $10 both in the first year and in the second year you will simply sum the two present values of the payments. Let’s put some numbers in, imagine that the spot rate for the first year is 3% and that the spot rate of the second year is 4%. The present value of our asset will be:

By knowing that we can now calculate the single interest rate of the asset by calculating the y in the formula:

So, in order to calculate the yield to maturity of the bond you need the spot rates for all the years. After the yield to maturity is set, you can calculate the price of the asset. The law of one price says that, in a well functioning market, the same commodity must be traded at the same price. This law implies that all the cash payments received or paid on the same date which do not present any risk must be discounted using the same spot rate. The yield to maturity of a bond increases as its maturity increases because the spot rates increase with time.

SPECIAL KIND OF BONDS

There exist some particular kind of bonds that do not give coupons but only a single payment when they mature. This kind of bonds are known as strips or stripped bonds. Other particular types of bonds are the consols which give money only via coupons and therefore they do not have a face value.

WHY, AS TIME PASSES, THE DISCOUNT FACTOR DECLINES

Until now we supposed that the longer you wait to receive money, the less is their present value meaning that the discount factor 1/(1+r1) you apply on cash payments you will receive one year from now is lower than the discount factor 1/(1+r2)^2 you apply on cash payments received two years from now. But is that always true? Imagine that it is not; if this is the case, it would mean that asking for a two years loan and investing that money would generate a profit. This would represent for investors on the market an arbitrage: an easy way to make a lot of money. Of course that would never happen in a well functioning market.

EXPECTATION THEORY

The expectation theory about term structure states that a series of short term bonds should produce the same earnings as a single bond that lasts as long as the sum of the short ones. Because of that,

the term structure is upward sloping (increases with time) only when investors think that in the short run interest rates will rise, while if the investors expect interest rates in the short run to fall you will have a downward sloping term structure. In general, when long run interest rates are higher than the short run ones, investors should be aware that the short run interest rates should rise. When instead the long run interest rates are lower than the short run ones investors should expect short run interest rate to decline.

RISK AND INFLATION

If there is no risk involved in your choices, you will simply opt for the investment that gives you more money but, if some of the assets you can purchase are riskier than others, you could choose to buy the one which gives less earning but carries less risk with respect to the one that may be more profitable but also riskier. Inflation plays an important role when evaluating bonds: if you expect inflation to rise in the future it is not convenient for you to invest in long run bonds but you would be happier if investing your money year by year in annual bonds.

DIFFERENCES BETWEEN REAL AND NOMINAL INTEREST RATES

When talking about interest rates, it is important to distinguish whether you are talking about real interest rate or nominal interest rate. The difference between the two is that nominal interest rates does not take into account the presence of inflation while real interest rates does. In order to convert a future nominal cashflow (the money you will receive) into a real cashflow today (present value of the money in the future having same bargaining power as the present) is:

If you invest $1000 at a 10% annual return, in one year you will have $1100 but, if the inflation rate is 6%, you will have 1100/1.06=1037.74 as real money expressed in today’s consumption so in the ned you will have a nominal gain of $100 but a real gain of $37.74 and therefore your investment has a 3.774% real rate of return. In order to calculate the real rate of return you could use the following formula:

As a rule of thumb you can approximate the real interest rate as: r (real) = r(nominal) – inflation rate. Most bonds offer you fixed return that could diminish in real value because of inflation. In order to eliminate this possibility you could buy an indexed bond which promises cash payments corrected for inflation.

HOW BONDS ARE VALUED

There are two scales on which bonds are valued: the Moody’s one and the Standard & Poor’s and Fitch. Those scales differentiate between relatively safe bonds and junk bonds: bonds that may promise high returns but that are highly speculative.