Refresher Reading

Introduction to Fixed-Income Valuation

2023 Curriculum CFA Program Level I Fixed Income

Introduction

Globally, the fixed-income market is a key source of financing for businesses and governments. In fact, the total market value outstanding of corporate and government bonds is significantly larger than that of equity securities. Similarly, the fixed-income market, which is also called the debt market or bond market, represents a significant investing opportunity for institutions as well as individuals. Pension funds, mutual funds, insurance companies, and sovereign wealth funds, among others, are major fixed-income investors. Retirees who desire a relatively stable income stream often hold fixed-income securities. Clearly, understanding how to value fixed-income securities is important to investors, issuers, and financial analysts. We focus on the valuation of traditional (option-free) fixed-rate bonds, although other debt securities, such as floating-rate notes and money market instruments, are also covered.

We first describe and illustrate basic bond valuation, which includes pricing a bond using a market discount rate for each of the future cash flows and pricing a bond using a series of spot rates. Valuation using spot rates allows for each future cash flow to be discounted at a rate associated with its timing. This valuation methodology for future cash flows has applications well beyond the fixed-income market. Relationships among a bond’s price, coupon rate, maturity, and market discount rate (yield-to-maturity) are also described and illustrated.

We then turn our attention to how bond prices and yields are quoted and calculated in practice. When bonds are actively traded, investors can observe the price and calculate various yield measures. However, these yield measures differ by the type of bond. In practice, different measures are used for fixed-rate bonds, floating-rate notes, and money market instruments.

We then discuss the maturity or term structure of interest rates, involving an analysis of yield curves, which illustrates the relationship between yields-to-maturity and times-to-maturity on bonds with otherwise similar characteristics. Lastly, we describe yield spreads, measures of how much additional yield over the benchmark security (usually a government bond) investors expect for bearing additional risk.

Learning Outcomes

The member should be able to:

calculate a bond’s price given a market discount rate;
identify the relationships among a bond’s price, coupon rate, maturity, and market discount rate (yield-to-maturity);
define spot rates and calculate the price of a bond using spot rates;
describe and calculate the flat price, accrued interest, and the full price of a bond;
describe matrix pricing;
calculate annual yield on a bond for varying compounding periods in a year;
calculate and interpret yield measures for fixed-rate bonds and floating-rate notes;
calculate and interpret yield measures for money market instruments;
define and compare the spot curve, yield curve on coupon bonds, par curve, and forward curve;
define forward rates and calculate spot rates from forward rates, forward rates from spot rates, and the price of a bond using forward rates;
compare, calculate, and interpret yield spread measures.

Summary

We have covered the principles and techniques that are used in the valuation of fixed-rate bonds, as well as floating-rate notes and money market instruments. These building blocks are used extensively in fixed-income analysis. The following are the main points made:

The market discount rate is the rate of return required by investors given the risk of the investment in the bond.
A bond is priced at a premium above par value when the coupon rate is greater than the market discount rate.
A bond is priced at a discount below par value when the coupon rate is less than the market discount rate.
The amount of any premium or discount is the present value of the “excess” or “deficiency” in the coupon payments relative to the yield-to-maturity.
The yield-to-maturity, the internal rate of return on the cash flows, is the implied market discount rate given the price of the bond.
A bond price moves inversely with its market discount rate.
The relationship between a bond price and its market discount rate is convex.
The price of a lower-coupon bond is more volatile than the price of a higher-coupon bond, other things being equal.
Generally, the price of a longer-term bond is more volatile than the price of a shorter-term bond, other things being equal. An exception to this phenomenon can occur on low-coupon (but not zero-coupon) bonds that are priced at a discount to par value.
Assuming no default, premium and discount bond prices are “pulled to par” as maturity nears.
A spot rate is the yield-to-maturity on a zero-coupon bond.
A yield-to-maturity can be approximated as a weighted average of the underlying spot rates.
Between coupon dates, the full (or invoice, or “dirty”) price of a bond is split between the flat (or quoted, or “clean”) price and the accrued interest.
Flat prices are quoted to not misrepresent the daily increase in the full price as a result of interest accruals.
Accrued interest is calculated as a proportional share of the next coupon payment using either the actual/actual or 30/360 methods to count days.
Matrix pricing is used to value illiquid bonds by using prices and yields on comparable securities having the same or similar credit risk, coupon rate, and maturity.
The periodicity of an annual interest rate is the number of periods in the year.
A yield quoted on a semiannual bond basis is an annual rate for a periodicity of two. It is the yield per semiannual period times two.
The general rule for periodicity conversions is that compounding more frequently at a lower annual rate corresponds to compounding less frequently at a higher annual rate.
Street convention yields assume payments are made on scheduled dates, neglecting weekends and holidays.The current yield is the annual coupon payment divided by the flat price, thereby neglecting as a measure of the investor’s rate of return the time value of money, any accrued interest, and the gain from buying at a discount or the loss from buying at a premium.
The simple yield is like the current yield but includes the straight-line amortization of the discount or premium.
The yield-to-worst on a callable bond is the lowest of the yield-to-first-call, yield-to-second-call, and so on, calculated using the call price for the future value and the call date for the number of periods.
The option-adjusted yield on a callable bond is the yield-to-maturity after adding the theoretical value of the call option to the price.
A floating-rate note (floater, or FRN) maintains a more stable price than a fixed-rate note because interest payments adjust for changes in market interest rates.
The quoted margin on a floater is typically the specified yield spread over or under the reference rate, which we refer to as the Market Reference Rate.
The discount margin on a floater is the spread required by investors, and to which the quoted margin must be set, for the FRN to trade at par value on a rate reset date.
Money market instruments, having one year or less time-to-maturity, are quoted on a discount rate or add-on rate basis.
Money market discount rates understate the investor’s rate of return (and the borrower’s cost of funds) because the interest income is divided by the face value or the total amount redeemed at maturity, and not by the amount of the investment.
Money market instruments need to be converted to a common basis for analysis.
A money market bond equivalent yield is an add-on rate for a 365-day year.
The periodicity of a money market instrument is the number of days in the year divided by the number of days to maturity. Therefore, money market instruments with different times-to-maturity have annual rates for different periodicities.
In theory, the maturity structure, or term structure, of interest rates is the relationship between yields-to-maturity and times-to-maturity on bonds having the same currency, credit risk, liquidity, tax status, and periodicity.
A spot curve is a series of yields-to-maturity on zero-coupon bonds.
A frequently used yield curve is a series of yields-to-maturity on coupon bonds.
A par curve is a series of yields-to-maturity assuming the bonds are priced at par value.
In a cash market, the delivery of the security and cash payment is made on a settlement date within a customary time period after the trade date—for example, “T + 3.”
In a forward market, the delivery of the security and cash payment are made on a predetermined future date.
A forward rate is the interest rate on a bond or money market instrument traded in a forward market.
An implied forward rate (or forward yield) is the breakeven reinvestment rate linking the return on an investment in a shorter-term zero-coupon bond to the return on an investment in a longer-term zero-coupon bond.
An implied forward curve can be calculated from the spot curve.
Implied spot rates can be calculated as geometric averages of forward rates.
A fixed-income bond can be valued using a market discount rate, a series of spot rates, or a series of forward rates.
A bond yield-to-maturity can be separated into a benchmark and a spread.
Changes in benchmark rates capture macroeconomic factors that affect all bonds in the market—inflation, economic growth, foreign exchange rates, and monetary and fiscal policy.
Changes in spreads typically capture microeconomic factors that affect the particular bond—credit risk, liquidity, and tax effects.
Benchmark rates are usually yields-to-maturity on government bonds or fixed rates on interest rate swaps.
A G-spread is the spread over or under a government bond rate, and an I-spread is the spread over or under an interest rate swap rate.
A G-spread or an I-spread can be based on a specific benchmark rate or on a rate interpolated from the benchmark yield curve.
A Z-spread (zero-volatility spread) is based on the entire benchmark spot curve. It is the constant spread that is added to each spot rate such that the present value of the cash flows matches the price of the bond.
An option-adjusted spread (OAS) on a callable bond is the Z-spread minus the theoretical value of the embedded call option.