Bonds: advanced topics

Yield to Maturity
The yield to maturity of a bond is the interest rate that would be required on an investment of the current price of the bond in order to receive all future scheduled payment. This is the yield which is quoted in bond pricing reports, and the average yield of all a bond fund's holdings, minus expenses, is the quoted yield of the bond fund. When a bond price falls, its yield rises, since you can get the same future payments by investing a lower amount at a higher interest rate.

For example, consider a bond which has a $10,000 face value and pays $500 annually until maturity, then pays back the $10,000 principal. If the bond price remains $10,000, the yield to maturity remains 5%. But if the bond price drops to $9,653 four years before maturity, the yield to maturity rises to 6%, as you could get $500, $500, $500, and $10,500 over the next four years by investing $9,653 at 6% interest.

When a bond is downgraded, its price falls because the future payments are less likely to be made, and its yield rises to compensate. This is why high-yield bonds have their high yields.

Yield Curves
The yield curve is a plot of bond yields (usually Treasury yields) against years to maturity. Usually, the curve has an upward slope; long-term bonds have higher yields than short-term bonds as compensation for the increased interest-rate and inflation risk.

When the curve is steep, investors receive a large premium for taking the greater interest-rate and inflation risks associated with long-term bonds. When the curve is flat, long-term bonds and short-term bonds have the same current yields. Sometimes, the curve may even be inverted, with higher yields on short-term bonds than on long-term bonds; this means that the market expects short-term rates to fall, so that a two-year bond reinvested five times is still expected to earn less in ten years than a ten-year bond.

Duration
The duration of a bond, or a bond fund, is a measure of its sensitivity to interest rates. The duration is approximately the amount by which the price of the bond will fall for every 1% rise in interest rates. This is only approximately correct because of convexity (see below). A bond fund with a ten-year duration is twice as sensitive to interest-rate risk as a bond fund with a five-year duration. (In practice, the relative risk is somewhat reduced because long-term rates are usually less volatile than short-term rates.)

The formal definition of duration is the average value of the time to each future payment, weighted by the proportion of the present value of the bond made up by the present value of each future payment. A zero-coupon bond is a promise to make a single payment at a future time, so its duration is equal to its maturity--the time until that payment is made. A conventional bond, or bond portfolio, or bond fund, makes multiple payments; each payment has a present value and a duration as if it were a zero-coupon bond, and the average of all durations is weighted by the present value of the payments divided by the present value of the whole bond. A bond which pays coupons will always have duration less than its maturity. High interest rates make the duration much shorter than the maturity, since a greater proportion of the bond's total value comes sooner. In low interest rate environments, coupon-paying bonds more closely resemble zero-coupon bonds (duration is closer to the maturity), since the present value of the principal returned at maturity is generally much larger than the coupon payments.

Duration has another useful summary property, which is that if the yield curve shifts in parallel, then duration is the point of indifference to interest rate changes. For example, if a bond/portfolio/fund with a duration of 5 years experiences a market interest rate increase of 1%, its value will drop by approximately 5%; however, since the same coupon payment now represents a higher percentage of the bond's value, its yield is higher (it will match the market rate), and the higher yield plus higher market interest on coupon payments will exactly offset the lower NAV after 5 years. To be absolutely assured of receiving a given sum on a future date (assuming parallel shifts of the yield curve), therefore, you should gradually reduce the duration as the date approaches. A zero-coupon bond reduces duration by exactly the amount of time that passes, and is therefore the risk-less choice for meeting a future obligation. However, few investors have such exact demands on their capital. For most purposes, shifting from intermediate- or longer-term bonds to shorter-term bonds as the need for capital approaches will cause little risk; also, in practice non-parallel shifts of the yield curve are not likely to cause very large changes in returns.

Bonds and bond funds also compute an effective duration which takes into account the possibility that the bond will be called (paid back early).

Investopedia's article on bond duration has the mathematical formulas.

Please also see the Duration section of Individual Bonds vs a Bond Fund.

Convexity
Duration is just one number, and cannot completely describe the relationship between bond prices and interest rates, as they are not a straight line; a bond which rises in price by 0.5% for a 0.1% fall in interest rates may not rise by 5% for a 1% fall in interest rates. Convexity is a description of how the price moves away from the linear relation; a bond with positive convexity will be worth more than given by the linear relation if interest rates change significantly (in either direction), while a bond with negative convexity will be worth less. Effectively, a bond with positive convexity increases in duration when interest rates fall (and thus the price rises more from the rate fall than would be indicated by the duration) and decreases in duration when interest rates rise (and thus the price falls less from the rate rise). A bond with negative convexity increases in duration when interest rates rise and decreases in duration when interest rates fall.

Simple bonds, with no call or prepayment features, have slightly positive convexity because of the way duration works out. If a bond is likely to be called, it has negative convexity, because the issuer will redeem it at the call price if interest rates have fallen enough to make it worth more than that price. Callable high-yield bonds often have considerable negative convexity; if the bonds improve to investment grade, the interest rates will fall considerably but the issuer will call them and investors will not get the full benefit of the rate decrease.

Mortgage-backed securities have considerable negative convexity because homeowners can refinance their mortgages. If interest rates fall, more homeowners will refinance, and the mortgage-backed securities will pay back the principal on the refinanced loans, which did not rise with the fall in interest rates. Bondholders can reinvest the returned principal, but they reinvest it at the current (lower) rate and do not benefit from the fall in interest rates.

Given two bonds with the same duration, credit risk, and yield, the bond with higher convexity will always be worth at least as much and may be worth more. Therefore, investors demand higher yields for bonds with negative convexity; this is why yields on GNMA mortgage-backed securities are higher than on Treasury bonds even though GNMAs are also backed by the full faith and credit of the US government. Vanguard even warns about this risk when it quotes the yield on the Vanguard GNMA Fund.

Investopedia's article on bond convexity gives a more complete illustration.

Links
Bond Basics
 * Bond Price, Duration and Convexity Calculator

Tutorial
Easy to understand, fundamental information about bonds. From Investopedia
 * Bond Basics: Introduction
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 * Advanced Bond Concepts: Introduction