an introduction

Globally, the fixed income market is a major source of financing for businesses and governments. In fact, the total outstanding market capitalization of corporate and government bonds is much greater than that of equity bonds. Similarly, the fixed income market, which is also called the debt market or the bond market, is a great investment 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 want a relatively stable income often own fixed-income securities. Obviously, understanding how fixed income securities are valued is important to investors, issuers, and financial analysts. We focus on valuing conventional fixed-rate (option-free) bonds, although other debt securities, such as floating-rate bonds and money market instruments, are covered.

First we describe and explain basic bond valuation, which involves pricing the bond using a market discount rate for both future cash flows and pricing the bond using a series of spot rates. Valuation using spot rates allows each future cash flow to be discounted at a rate linked to its timing. This valuation methodology for future cash flows has applications beyond the fixed income market. The relationships between 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 determined and calculated in practice. When bonds are actively trading, investors can monitor the price and calculate various measures of return. However, these yield measures vary by bond type. In practice, different metrics are used for fixed rate bonds, floating rate bonds, and money market instruments.

We then discuss the maturity date or interest rate structure, including an analysis of yield curves, which shows the relationship between yields to maturity and times to maturity on bonds with similar characteristics. Finally, we describe yield spreads, measures of how much additional return on benchmark securities (usually government bonds) investors would expect to take on additional risk.

Learning Outcomes

The member must be able to:

Calculate the price of the bond in view of the market discount rate;

Determining the relationships between the bond price, coupon rate, maturity and market discount rate (yield to maturity);

Determining spot prices and calculating the price of a bond using spot rates;

Description and calculation of the fixed price, accrued interest and full price of the bond;

matrix pricing description;

Calculate the annual yield on a bond for different compounding periods in a year;

Calculation and interpretation of yield measures for fixed rate bonds and floating rate banknotes;

Calculation and interpretation of return measures for money market instruments;

Define and compare the spot rate curve, the coupon bond yield curve, the equity curve and the forward curve;

determining forward rates and calculating spot rates from forward rates, forward rates from spot prices, and the price of a bond using forward rates;

Comparing, calculating and interpreting measures of return spread.

summary

We have covered the principles and techniques used in valuing fixed rate bonds, as well as floating rate bonds and money market instruments. These building blocks are widely used in fixed income analysis. Here are the main points made:

The market discount rate is the rate of return required by the investors given the risk of investing in the bond.

A bond is priced at a premium over face value when the coupon rate is greater than the market discount rate.

A bond is priced at a discount to face value when the coupon rate is lower than the market discount rate.

The amount of any premium or discount is the present value of the “increase” or “decrease” in the coupon payments relative to the yield to maturity.

Yield to maturity, the internal rate of return on cash flows, is the market’s implied discount rate given the price of the bond.

The price of a bond moves inversely with the discount rate in the market.

The relationship between the bond price and the discount rate in the market is a convex one.

The price of a lower coupon bond is more volatile than the price of a higher coupon bond, other things being equal.

In general, the price of a long-term bond is more volatile than the price of a short-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 face value.

Assuming there is no default, premium and discount bond prices are “pulled back” as maturity approaches.

The spot rate is the yield to maturity on a bond without a coupon.

Yield to maturity can be rounded up as a weighted average of the underlying spot rates.

Between coupon dates, the full price (or bill, or “dirty”) of the bond is divided between the fixed (or quoted, or “clean”) price and the interest accrued.

Fixed rates are set so as not to skew the daily increase in the full price as a result of accumulating interest.

The accrued interest is calculated as a proportional share of the next coupon payment using either the actual/actual or 30/360 method of calculating the days.

Matrix pricing is used to value illiquid bonds using prices and returns on similar securities with the same or similar credit risk, coupon rate, and maturity.

The periodicity of the annual interest rate is the number of periods in a year.

The yield on a semi-annual bond basis is an annual rate of two cycles. is the yield per semi-annual period multiplied by two.

The general rule for periodic conversions is that compound frequently at a lower annual rate corresponds to superposition less frequently at a higher annual rate.

Street agreement returns assume that payments are made on scheduled dates, ignoring weekends and holidays. The current yield is the annual coupon payments divided by the fixed rate, and thus is neglected as a measure of the investor’s rate of return, the time value of money, any interest accrued, and the gain from buying at a discount or the loss from buying at a premium.

Simple yield is similar to current yield but includes the amortization of the discount or premium’s straight line.

The pay-to-write on a callable bond is the lowest of the yield on the first call, the yield on the second call, etc., computed using the call price of the future value and the date of the number of periods.

The option-adjusted yield on a redeemable bond is the yield to maturity after adding the theoretical value of the call option to the price.

A floating rate note (Floating Rate or FRN) holds a more stable rate than a fixed rate note because interest payments adjust to changes in market interest rates.

The margin offered on the float is usually the specified return spread on or below the reference price, which we refer to as the market reference price.

Floating Discount Margin is the spread required by investors, which must be set as Offer Margin, in order for the FRN to trade at face value on the price reset date.

Money market instruments, with a term of one year or less to maturity, are priced at the discount rate or on an incremental rate basis.

Money market discount rates lower the investor’s rate of return (and the borrower’s cost of funds) because interest income is divided by the face value or total amount recovered at maturity, not by the investment amount.

Money market instruments must be turned into a common basis for analysis.

The equivalent yield of a money market bond is an additional 365-day rate.

The periodicity of a money market instrument is the number of days of the year divided by the number of maturities. Therefore, money market instruments with different times to maturity have annual rates of different periodicals.

In theory, the maturity structure, or term structure, of interest rates is the relationship between yields to maturity and times to maturity on bonds of the same currency, credit risk, liquidity, tax status, and frequency.

The spot curve is a series of yield-to-maturity bonds without a coupon.

The frequently used yield curve is a series of yield to maturity on coupon bonds.

The parity curve is a series of yield to maturity assuming the bonds are priced at face value.

In the money market, the security and cash payment are delivered on a settlement date within the usual period of time after the trade date – for example, “T + 3.”

In the futures market, the security and cash payment are delivered at a predetermined future date.

The forward rate is the interest rate on a bond or money market instrument traded in the futures market.

The implied forward rate (or forward return) is the equal reinvestment rate that relates the return on investment in a short-term zero-coupon bond to the return on investment in a long-term zero-coupon bond.

An implicit forward curve can be calculated from the concentration 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.

Bond yield to maturity can be separated into benchmark and spread.

Changes in benchmark rates capture the macroeconomic factors that affect all bonds in the market – inflation, economic growth, foreign exchange rates, monetary and fiscal policy.

Changes in spreads usually pick up on microeconomic factors that affect specific bonds – credit risk, liquidity and tax effects.

Standard rates are usually yields to maturity on government bonds or fixed rates on interest rate swaps.

The G spread is the spread above or below the price of a government bond, and I-Spread is the spread above or below the interest rate swap rate.

The G-Spread or I-Spread can be based on a specified standard rate or on a rate interpolated from the standard yield curve.

The Z-shape spread (the spread without volatility) is based on the entire standard point curve. It is the fixed spread that is added to each spot price so that the present value of the cash flows matches the price of the bond.

The option-adjusted spread (OAS) on a callable bond is the Z-spread minus the theoretical value of the underlying call option.