Macaulay Duration

The Macaulay duration is defined as the average time it takes to receive all the cash flows of a bond, weighted by the present value of each of the cash flows. It measures the number of years required to recover the true cost of a bond, considering the present value of all coupon and principal payments received in the future. Essentially, it is the payment-weighted point in time at which an investor can expect to recoup his or her original investment.

The Macaulay duration is quoted in “years” and it is the only type of duration with this unit. Interest rates are assumed to be continuously compounded.

Given its relative ability to predict price changes based on changes in interest rates, duration allows for the effective comparison of bonds with different maturities and coupon rates. For example, a 5-year zero coupon bond may be more sensitive to interest rate changes than a 7-year bond with a 6% coupon. By comparing the bonds’ durations, you may be able to anticipate the degree of price change in each bond assuming a given change in interest rates.

Utilizing Duration

Duration can help predict the likely change in the price of a bond given a change in interest rates. As a general rule, for every 1% increase or decrease in interest rates, a bond’s price will change approximately 1% in the opposite direction for every year of duration. For example, if a bond has a duration of 5 years, and interest rates increase by 1%, the bond’s price will decline by approximately 5%. Conversely, if a bond has a duration of 5 years and interest rates fall by 1%, the bond’s price will increase by approximately 5%.

Example

For example, for a two-year bond with a \$1000 face value and one coupon payment every six months of \$50, the duration (calculated in years) is:

Rules of Duration

When thinking about duration, a few general rules apply. With everything else being equal:

• The duration of any bond that pays a coupon will be less than its maturity, because some amount of coupon payments will be received before the maturity date.

• The lower a bond’s coupon, the longer its duration, because proportionately less payment is received before final maturity. The higher a bond’s coupon, the shorter its duration, because proportionately more payment is received before final maturity.

• Because zero coupon bonds make no coupon payments, a zero coupon bond’s duration will be equal to its maturity.

• The longer a bond’s maturity, the longer its duration, because it takes more time to receive full payment. The shorter a bond’s maturity, the shorter its duration, because it takes less time to receive full payment.

Convexity

The duration measure indicates that regardless of whether interest rates increase or decrease, the approximate percentage price change is the same. However, while for small changes in yield the percentage price change will be the same for an increase or decrease in yield, for large changes in yield this is not true. This suggests that duration is only a good approximation of the percentage price change for small changes in yield.

For example, using a 5% 20-year bond selling to yield 4% with a duration of 13.09. For a 10 basis point change in yield, the estimate was accurate for both an increase or a decrease in yield. However, for a 200 basis point change in yield, the approximate percentage price change was off considerably.

The reason for this result is that duration is in fact a first (linear) approximation for a small change in yield. The approximation can be improved by using a second approximation. This approximation is referred to as “convexity.” The use of this term in the industry is unfortunate because the term convexity is also used to describe the shape or curvature of the price/yield relationship. The convexity measure of a security can be used to approximate the change in price that is not explained by duration.

Capped Floater

A capped floater is a floating-rate bond with a cap provision that prevents the coupon rate from increasing above a specified maximum rate. As a consequence, the cap provision protects the issuer against rising interest rates and is thus an issuer option.

Because the investor is long the bond but short the embedded option, the value of the cap decreases the value of the capped floater relative to the value of the straight bond:

Value of Capped Floater = Value of Straight Bond – Value of Embedded Cap

Interest Rate Risk

The interest rate risk is the risk that an investment’s value will change due to changes related to the interest rate. These changes include variations in the absolute level of interest rates, in the spread between two rates, in the shape of the yield curve, or in any other interest rate relationship.

Measure

Some key measures of interest rate risk include duration and convexity.

Duration

The duration of a bond measures the sensitivity of the bond’s full price (including accrued interest) to changes in the bond’s yield to maturity (in the case of yield duration measures) or to changes in benchmark interest rates (in the case of yield-curve or curve duration measures). Duration is among the most important characteristics of a fixed income security. It is often used as a measure of risk in bond investing.

Duration comes in many forms. Yield duration measures, such as modified duration, can be used only for option-free bonds because these measures assume that a bond’s expected cash flows do not change when the yield changes. This assumption is in general false for bonds with embedded options because the values of embedded options are typically contingent on interest rates. Thus, for bonds with embedded options, the only appropriate duration measure is the curve duration measure known as effective (or option-adjusted) duration. Because effective duration works for straight bonds as well as for bonds with embedded options, practitioners tend to use it regardless of the type of bond being analyzed.

Brief History

In 1938, economist Frederick Macaulay suggested duration as a way of determining the price volatility of bonds. ‘Macaulay duration’ is now the most common duration measure.

Until the 1970s, few people paid attention to duration due to the relative stability of interest rates. When interest rates began to rise dramatically, investors became very interested in a tool that would help them assess the price volatility of their fixed income investments. During this period, the concept of ‘modified duration’ was developed, which offered a more precise calculation of the change in bond prices given varying coupon payment schedules.

In the mid-1980s, as interest rates began to drop, several investment banks developed the concept of ‘option-adjusted duration’ (or ‘effective duration’), which allowed for the calculation of price movements given the existence of call features.

Different Duration Measures

While duration comes in many forms, the ones most commonly used by public fund investors include the following:

Macaulay Duration

Developed in 1938 by Frederic Macaulay, this form of duration measures the number of years required to recover the true cost of a bond, considering the present value of all coupon and principal payments received in the future. Thus, it is the only type of duration quoted in “years.” Interest rates are assumed to be continuously compounded.

Modified Duration

This measure expands or modifies Macaulay duration to measure the responsiveness of a bond’s price to interest rate changes. It is defined as the percentage change in price for a 100 basis point change in interest rates. The formula assumes that the cash flows of the bond do not change as interest rates change (which is not the case for most callable bonds).

Effective Duration

Effective duration (sometimes called option-adjusted duration) further refines the modified duration calculation and is particularly useful when a portfolio contains callable securities. Effective duration requires the use of a complex model for pricing bonds that adjusts the price of the bond to reflect changes in the value of the bond’s “embedded options” (e.g., call options or a sinking fund schedule) based on the probability that the option will be exercised. Effective duration incorporates a bond’s yield, coupon, final maturity and call features into one number that indicates how price-sensitive a bond or portfolio is to changes in interest rates.

Callable Bond

A callable bond is a bond that includes an embedded call option. The call option is an issuer option—that is, the right to exercise the option is at the discretion of the bond’s issuer. The call provision allows the issuer to redeem the bond issue prior to maturity. A callable bond is also called a redeemable bond.

Early redemption usually happens when the issuer has the opportunity to replace a high-coupon bond with another bond that has more favorable terms, typically when interest rates have fallen or when the issuer’s credit quality has improved.

Until the 1990s, most long-term corporate bonds in the United States were callable after either five or 10 years. The initial call price (exercise price) was typically at a premium above par, the premium depended on the coupon, and the call price gradually declined to par a few years prior to maturity. Today, most investment-grade corporate bonds are essentially non-refundable. They may have a “make-whole call,” so named because the call price is such that the bondholders are more than “made whole” (compensated) in exchange for surrendering their bonds. The call price is calculated at a narrow spread to a benchmark security, usually an on-the-run sovereign bond such as Treasuries in the United States or gilts in the United Kingdom. Thus, economical refunding is virtually out of question, and investors need have no fear of receiving less than their bonds are worth.

Most callable bonds include a lockout period during which the issuer cannot call the bond. For example, a 10-year callable bond may have a lockout period of three years, meaning that the first potential call date is three years after the bond’s issue date. Lockout periods may be as short as one month or extend to several years. For example, high-yield corporate bonds are often callable a few years after issuance. Holders of such bonds are usually less concerned about early redemption than about possible default. Of course, this perspective can change over the life of the bond—for example, if the issuer’s credit quality improves.

Callable bonds include different types of call features. The issuer of a European-style callable bond can only exercise the call option on a single date at the end of the lockout period. An American-style callable bond is continuously callable from the end of the lockout period until the maturity date. A Bermudan-style call option can be exercised only on a predetermined schedule of dates after the end of the lockout period. These dates are specified in the bond’s indenture or offering circular.

With a few exceptions, bonds issued by government-sponsored enterprises in the United States (e.g., Fannie Mae, Freddie Mac, Federal Home Loan Banks, and Federal Farm Credit Banks) are callable. These bonds tend to have relatively short maturities (5–10 years) and very short lockout periods (three months to one year). The call price is almost always at 100% of par, and the call option is often Bermudan style.

Tax-exempt municipal bonds (often called “munis”), a type of non-sovereign (local) government bond issued in the United States, are almost always callable at 100% of par any time after the end of the 10th year. They may also be eligible for advance refunding—a highly specialized topic that is not discussed here.

Although the bonds of US government-sponsored enterprises and municipal issuers account for most of the callable bonds issued and traded globally, bonds that include call provisions are also found in other countries in Asia Pacific, Europe, Canada, and Central and South America. The vast majority of callable bonds are denominated in US dollars or euros because of investors’ demand for securities issued in these currencies. Australia, the United Kingdom, Japan, and Norway are examples of countries where there is a market for callable bonds denominated in local currency.

References

2018 CFA Program Level II Volume 5 Fixed Income and Derivatives.

Endogenous Term Structure Models

In endogenous term structure models, the current term structure of rates is an output rather than an input of the model. Examples of endogenous term structure models include the Vasicek model, the Dothan model, and the Cox-Ingersoll-ross model.

Yield to Maturity

Yield to maturity (YTM) is a way to measure the expected rate of return of a fixed-income security.

There are two ways to understand this concept:

• YTM is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, that all coupon and principal payments will be made in full when due, and that coupons are reinvested at the original YTM.
• YTM is the discount rate at which the sum of all future cash flows from the bond (coupons and principal) is equal to the current price of the bond. It is the single interest rate that equates the present value of a bonds cash flows to its price.

YTM is perhaps the most familiar pricing concept in bond markets. To understand this concept, it is important to clarify how YTM is related to spot rates and a bond’s expected and realized returns.

How is the yield to maturity related to spot rates?

In bond markets, most bonds outstanding have coupon payments and many have various options, such as a call provision. The YTM of these bonds with maturity T would not be the same as the spot rate at T. But, the YTM should be mathematically related to the spot curve. Because the principle of no arbitrage shows that a bond’s value is the sum of the present values of payments discounted by their corresponding spot rates, the YTM of the bond should be some weighted average of spot rates used in the valuation of the bond.

The following example addresses the relationship between spot rates and yield to maturity.

Is the yield to maturity the expected return on a bond?

In general, it is not, except under extremely restrictive assumptions. The expected rate of return is the return one anticipates earning on an investment. The YTM is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM. However, the assumption regarding reinvestment of coupons at the original yield to maturity typically does not hold. The YTM can provide a poor estimate of expected return if (1) interest rates are volatile; (2) the yield curve is steeply sloped, either upward or downward; (3) there is significant risk of default; or (4) the bond has one or more embedded options (e.g., put, call, or conversion).

If either (1) or (2) is the case, reinvestment of coupons would not be expected to be at the assumed rate (YTM). Case (3) implies that actual cash flows may differ from those assumed in the YTM calculation, and in case (4), the exercise of an embedded option would, in general, result in a holding period that is shorter than the bond’s original maturity.

The realized return is the actual return on the bond during the time an investor holds the bond. It is based on actual reinvestment rates and the yield curve at the end of the holding period. With perfect foresight, the expected bond return would equal the realized bond return.

Yield Curve

The yield curve is a graphical representation of the different interest rates at various maturities.

There are different types of yield curves: spot, forward, par etc. No matter what type of yield curve, it represents different interest rates (at different maturities) at a single point of time.

Slope of the Yield Curves

In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity; that is, the yield curve “flattens” at longer maturities. Because nominal yields incorporate a premium for expected inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing or at least level future inflation (associated with relatively strong economic growth). The existence of risk premiums (e.g., for the greater interest rate risk of longer-maturity bonds) also contributes to a positive slope.

An inverted yield curve is somewhat uncommon. Such a term structure may reflect a market expectation of declining future inflation rates (because a nominal yield incorporates a premium for expected inflation) from a relatively high current level. Expectations of declining economic activity may be one reason that inflation might be anticipated to decline, and a downward-sloping yield curve has frequently been observed before recessions.

A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.

A humped yield curve, which is relatively rare, occurs when intermediate-term interest rates are higher than short- and long-term rates.