2023 Curriculum CFA Program Level I Fixed Income

## Introduction

Successful analysts must develop a solid understanding of the risk and return characteristics of fixed-income investments. Beyond the vast global market for public and private fixed-rate bonds, many financial assets and liabilities with known future cash flows you will encounter throughout your career are evaluated using similar principles. This analysis starts with the yield-to-maturity, or internal rate of return on future cash flows, introduced in the fixed-income valuation reading. Fixed-rate bond returns are affected by many factors, the most important of which is the full receipt of all interest and principal payments on scheduled dates. Assuming no default, return is also affected by interest rate changes that affect coupon reinvestment and the bond price if it is sold prior to maturity. Price change measures may be derived from the mathematical relationship used to calculate a bond’s price. Specifically, duration estimates the price change for a given change in interest rates, and convexity improves on the duration estimate by considering that the price and yield-to-maturity relationship of a fixed-rate bond is non-linear.

Sources of return on a fixed-rate bond investment include the receipt and reinvestment of coupon payments and either the redemption of principal if the bond is held to maturity or capital gains (or losses) if the bond is sold earlier. Fixed-income investors holding the same bond may have different interest rate risk exposures if their investment horizons differ.

We introduce bond duration and convexity, showing how these statistics are calculated and used as interest rate risk measures. Although procedures and formulas exist to calculate duration and convexity, these statistics can be approximated using basic bond-pricing techniques and a financial calculator. Commonly used versions of the statistics are covered, including Macaulay, modified, effective, and key rate durations, and we distinguish between risk measures based on changes in the bond’s yield-to-maturity (i.e., yield duration and convexity) and on benchmark yield curve changes (i.e., curve duration and convexity).

We then return to the investment time horizon. When an investor has a short-term horizon, duration and convexity are used to estimate the change in the bond price. Note that yield volatility matters, because bonds with varying times-to-maturity have different degrees of yield volatility. When an investor has a long-term horizon, the interaction between coupon reinvestment risk and market price risk matters. The relationship among interest rate risk, bond duration, and the investment horizon is explored.

Finally, we discuss how duration and convexity may be extended to credit and liquidity risks and highlight how these factors can affect a bond’s return and risk. In addition, we highlight the use of statistical methods and historical data to establish empirical as opposed to analytical duration estimates.

## Learning Outcomes

The member should be able to:

• calculate and interpret the sources of return from investing in a fixed-rate bond;<list-type>los</list-type>
• define, calculate, and interpret Macaulay, modified, and effective durations;
• explain why effective duration is the most appropriate measure of interest rate risk for bonds with embedded options;
• define key rate duration and describe the use of key rate durations in measuring the sensitivity of bonds to changes in the shape of the benchmark yield curve;
• explain how a bond’s maturity, coupon, and yield level affect its interest rate risk;
• calculate the duration of a portfolio and explain the limitations of portfolio duration;
• calculate and interpret the money duration of a bond and price value of a basis point (PVBP);
• calculate and interpret approximate convexity and compare approximate and effective convexity;
• calculate the percentage price change of a bond for a specified change in yield, given the bond’s approximate duration and convexity;
• describe how the term structure of yield volatility affects the interest rate risk of a bond;
• describe the relationships among a bond’s holding period return, its duration, and the investment horizon;
• explain how changes in credit spread and liquidity affect yield-to-maturity of a bond and how duration and convexity can be used to estimate the price effect of the changes.
• describe the difference between empirical duration and analytical duration.

## Summary

This reading covers the risk and return characteristics of fixed-rate bonds. The focus is on the widely used measures of interest rate risk—duration and convexity. These statistics are used extensively in fixed-income analysis. The following are the main points made in the reading:

• The three sources of return on a fixed-rate bond purchased at par value are: (1) receipt of the promised coupon and principal payments on the scheduled dates, (2) reinvestment of coupon payments, and (3) potential capital gains, as well as losses, on the sale of the bond prior to maturity.
• For a bond purchased at a discount or premium, the rate of return also includes the effect of the price being “pulled to par” as maturity nears, assuming no default.
• The total return is the future value of reinvested coupon interest payments and the sale price (or redemption of principal if the bond is held to maturity).The horizon yield (or holding period rate of return) is the internal rate of return between the total return and purchase price of the bond.
• Coupon reinvestment risk increases with a higher coupon rate and a longer reinvestment time period.
• Capital gains and losses are measured from the carrying value of the bond and not from the purchase price. The carrying value includes the amortization of the discount or premium if the bond is purchased at a price below or above par value. The carrying value is any point on the constant-yield price trajectory.
• Interest income on a bond is the return associated with the passage of time. Capital gains and losses are the returns associated with a change in the value of a bond as indicated by a change in the yield-to-maturity.
• The two types of interest rate risk on a fixed-rate bond are coupon reinvestment risk and market price risk. These risks offset each other to a certain extent. An investor gains from higher rates on reinvested coupons but loses if the bond is sold at a capital loss because the price is below the constant-yield price trajectory. An investor loses from lower rates on reinvested coupon but gains if the bond is sold at a capital gain because the price is above the constant-yield price trajectory.
• Market price risk dominates coupon reinvestment risk when the investor has a short-term horizon (relative to the time-to-maturity on the bond).
• Coupon reinvestment risk dominates market price risk when the investor has a long-term horizon (relative to the time-to-maturity)—for instance, a buy-and-hold investor.
• Bond duration, in general, measures the sensitivity of the full price (including accrued interest) to a change in interest rates.
• Yield duration statistics measuring the sensitivity of a bond’s full price to the bond’s own yield-to-maturity include the Macaulay duration, modified duration, money duration, and price value of a basis point.
• Curve duration statistics measuring the sensitivity of a bond’s full price to the benchmark yield curve are usually called “effective durations.”
• Macaulay duration is the weighted average of the time to receipt of coupon interest and principal payments, in which the weights are the shares of the full price corresponding to each payment. This statistic is annualized by dividing by the periodicity (number of coupon payments or compounding periods in a year).
• Modified duration provides a linear estimate of the percentage price change for a bond given a change in its yield-to-maturity.
• Approximate modified duration approaches modified duration as the change in the yield-to-maturity approaches zero.
• Effective duration is very similar to approximate modified duration. The difference is that approximate modified duration is a yield duration statistic that measures interest rate risk in terms of a change in the bond’s own yield-to-maturity, whereas effective duration is a curve duration statistic that measures interest rate risk assuming a parallel shift in the benchmark yield curve.
• Key rate duration is a measure of a bond’s sensitivity to a change in the benchmark yield curve at specific maturity segments. Key rate durations can be used to measure a bond’s sensitivity to changes in the shape of the yield curve.
• Bonds with an embedded option do not have a meaningful internal rate of return because future cash flows are contingent on interest rates. Therefore, effective duration is the appropriate interest rate risk measure, not modified duration.
• The effective duration of a traditional (option-free) fixed-rate bond is its sensitivity to the benchmark yield curve, which can differ from its sensitivity to its own yield-to-maturity. Therefore, modified duration and effective duration on a traditional (option-free) fixed-rate bond are not necessarily equal.
• During a coupon period, Macaulay and modified durations decline smoothly in a “saw-tooth” pattern, assuming the yield-to-maturity is constant. When the coupon payment is made, the durations jump upward.
• Macaulay and modified durations are inversely related to the coupon rate and the yield-to-maturity.
• Time-to-maturity and Macaulay and modified durations are usually positively related. They are always positively related on bonds priced at par or at a premium above par value. They are usually positively related on bonds priced at a discount below par value. The exception is on long-term, low-coupon bonds, on which it is possible to have a lower duration than on an otherwise comparable shorter-term bond.
• The presence of an embedded call option reduces a bond’s effective duration compared with that of an otherwise comparable non-callable bond. The reduction in the effective duration is greater when interest rates are low and the issuer is more likely to exercise the call option.
• The presence of an embedded put option reduces a bond’s effective duration compared with that of an otherwise comparable non-putable bond. The reduction in the effective duration is greater when interest rates are high and the investor is more likely to exercise the put option.
• The duration of a bond portfolio can be calculated in two ways: (1) the weighted average of the time to receipt of aggregate cash flows and (2) the weighted average of the durations of individual bonds that compose the portfolio.
• The first method to calculate portfolio duration is based on the cash flow yield, which is the internal rate of return on the aggregate cash flows. It cannot be used for bonds with embedded options or for floating-rate notes.
• The second method is simpler to use and quite accurate when the yield curve is relatively flat. Its main limitation is that it assumes a parallel shift in the yield curve in that the yields on all bonds in the portfolio change by the same amount.
• Money duration is a measure of the price change in terms of units of the currency in which the bond is denominated.
• The price value of a basis point (PVBP) is an estimate of the change in the full price of a bond given a 1 bp change in the yield-to-maturity.
• Modified duration is the primary, or first-order, effect on a bond’s percentage price change given a change in the yield-to-maturity. Convexity is the secondary, or second-order, effect. It indicates the change in the modified duration as the yield-to-maturity changes.
• Money convexity is convexity times the full price of the bond. Combined with money duration, money convexity estimates the change in the full price of a bond in units of currency given a change in the yield-to-maturity.
• Convexity is a positive attribute for a bond. Other things being equal, a more convex bond appreciates in price more than a less convex bond when yields fall and depreciates less when yields rise.
• Effective convexity is the second-order effect on a bond price given a change in the benchmark yield curve. It is similar to approximate convexity. The difference is that approximate convexity is based on a yield-to-maturity change and effective convexity is based on a benchmark yield curve change.
• Callable bonds have negative effective convexity when interest rates are low. The increase in price when the benchmark yield is reduced is less in absolute value than the decrease in price when the benchmark yield is raised.
• The change in a bond price is the product of: (1) the impact per basis-point change in the yield-to-maturity and (2) the number of basis points in the yield change. The first factor is estimated by duration and convexity. The second factor depends on yield volatility.
• The investment horizon is essential in measuring the interest rate risk on a fixed-rate bond.
• For a particular assumption about yield volatility, the Macaulay duration indicates the investment horizon for which coupon reinvestment risk and market price risk offset each other. The assumption is a one-time parallel shift to the yield curve in which the yield-to-maturity and coupon reinvestment rates change by the same amount in the same direction.
• When the investment horizon is greater than the Macaulay duration of the bond, coupon reinvestment risk dominates price risk. The investor’s risk is to lower interest rates. The duration gap is negative.
• When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk. The duration gap is zero.
• When the investment horizon is less than the Macaulay duration of the bond, price risk dominates coupon reinvestment risk. The investor’s risk is to higher interest rates. The duration gap is positive.
• Credit risk involves the probability of default and degree of recovery if default occurs, whereas liquidity risk refers to the transaction costs associated with selling a bond.
• For a traditional (option-free) fixed-rate bond, the same duration and convexity statistics apply if a change occurs in the benchmark yield or a change occurs in the spread. The change in the spread can result from a change in credit risk or liquidity risk.
• In practice, there often is interaction between changes in benchmark yields and in the spread over the benchmark.
• Empirical duration uses statistical methods and historical bond prices to derive the price–yield relationship for specific bonds or bond portfolios.