FI 6 Fixed-Income Bond Valuation: Prices and Yields

Valuing a fixed-rate bond is an exercise in discounted cash flow analysis. The price is the present value of the promised stream of coupons plus the return of face value at maturity, each cash flow discounted back to today at a single rate per period. That rate is the market discount rate: the return investors require given the risk they take on. It goes by several names that all mean the same thing, including the required yield and the required rate of return.

The value today of one coupon arriving in t periods, when the rate per period is r, follows the standard time-value-of-money relation.

Summing the present values of every coupon and the final principal gives the general pricing formula on a coupon date, where FV is the face value and N counts the periods left until maturity.

Discount, par, and premium

When a bond is first issued at a coupon rate exactly equal to the market discount rate, its price is 100 percent of par: the coupon stream is worth precisely the face value. Once the two rates diverge, the price moves away from par. Consider the Bright Wheels Automotive (BRWA) bond, a five-year note paying a fixed 3.2 percent annual coupon, which is 1.6 percent every six months on a semiannual basis. At issuance the market wanted 1.6 percent per period, so the bond priced at par.

Now let the market discount rate climb to 2.0 percent per period. Freshly issued bonds would have to offer a 2.0 percent coupon to sell at par, so the older BRWA note, stuck at 1.6 percent, looks underpaid relative to the market. Its coupon is deficient, and buyers will only take it at a price below par. Discounting the BRWA cash flows at 2.0 percent gives 96.41, and any bond priced below 100 is said to trade at a discount. If instead the market discount rate falls to 1.2 percent per period, the BRWA coupon of 1.6 percent is now generous, or excessive, relative to the market, so the price is bid above par to 103.75. A bond priced above 100 trades at a premium.

Cash flow is 1.6 per period for the first nine periods and 101.6 at period 10.

The word discount does double duty here. A discount bond is one priced below par; the market discount rate is the rate used to discount cash flows. The two uses are unrelated, so it is worth keeping them apart. The three relationships are summarized below.

Rates per period are turned into stated annual rates by multiplying by how many periods occur within a year. So an equivalent way to describe the BRWA note is that it commands a premium whenever its stated annual coupon of 3.2 percent tops the stated annual market discount rate. Unless a source says otherwise, quoted interest rates are annual.

Pricing runs the other way when the market price is already known. Given a price, the same discounting equation can be solved backward for the rate that makes it hold. That rate is the yield-to-maturity (YTM), the internal rate of return on the bond cash flows: the single uniform rate at which the present values of all future coupons and principal add up to the current price. It is an implied or observed market discount rate, and market participants usually shorten it to simply the yield.

For instance, if the five-year, 3.2 percent BRWA bond is quoted at 108.15, its yield-to-maturity is the value of r that makes the pricing equation hold at that price, which works out to an annual 1.50 percent. A spreadsheet YIELD or IRR function returns the same figure.

When the realized return equals the YTM

The yield-to-maturity is a promised yield, not a guaranteed one. An investor actually earns the stated YTM only if three conditions all hold.

1. The bond is held all the way to maturity.
2. The issuer pays every coupon and the principal in full and on schedule, so there is no default.
3. Each coupon received is reinvested at a rate equal to the YTM.

If any one of these fails, for example a coupon reinvested at a lower rate or an issuer that misses a payment, the return the investor books over the life of the bond will differ from the YTM computed at purchase.

Negative yields-to-maturity

A yield-to-maturity can be positive, zero, or even negative. Negative yields spread across many sovereign bond markets after central banks eased policy aggressively from 2012 onward to counter inflation running below target. A bond ends up with a negative yield in two typical ways: it was issued long ago at a higher yield and has since appreciated sharply in price, or it is a newly issued zero-coupon government bond sold at a premium to par. Coupons themselves are almost never set below zero.

In 2020 the People’s Republic of China sold its first negative-yielding sovereign bond denominated in euros: a five-year, zero-coupon note on an actual/actual day count, issued at a price of 100.763 for 750 million euros of principal. With no coupons, its yield-to-maturity comes from discounting the single principal payment back over five years.

Bonds rarely change hands exactly on a coupon date. When a trade settles partway through a coupon period, the buyer will collect the entire next coupon even though the seller held the bond for part of that period. To make this fair, the price splits into two pieces: the flat price and the accrued interest owed to the seller. Their sum is the full price, the amount actually paid.

The flat price, also called the quoted or clean price, is the full price stripped of accrued interest, and it is what dealers quote. The full price, also called the invoice or dirty price, is what the buyer hands over and the seller receives on the settlement date.

Why dealers quote the flat price

Quoting the clean price avoids sending a misleading signal about where a bond is actually trading. If dealers quoted full prices, the number would drift upward every day purely because interest keeps accruing, even when the yield had not budged, and then it would lurch downward the moment a coupon was paid. The flat price removes that sawtooth, leaving a quote that reflects genuine changes in value. Note also that accrued interest is unaffected by the yield-to-maturity, so a change in interest rates moves only the flat price.

Measuring accrued interest

Accrued interest is the seller share of the upcoming coupon, prorated by the fraction of the coupon period that has elapsed. If t days have passed since the last coupon and the period spans T days, then the accrued interest is that fraction of the coupon.

The count of days depends on the convention. A day count is fixed by two choices: how days within a period are tallied, and how many days a year is assumed to hold. Two conventions dominate. The 30/360 convention pretends every month has 30 days and every year 360, neither of which is literally true. The actual/actual convention uses the real number of days in each month and in the year, counting weekends, holidays, and leap days.

The full price between coupon dates

To value a bond partway through a period, each cash flow is discounted for the time actually remaining until it arrives. The next coupon is only 1 minus t/T of a period away, the one after that 2 minus t/T periods away, and so on.

Multiplying top and bottom of this expression by (1 + r) raised to t/T collapses it into a tidy shortcut: value the bond as of the last coupon date, using the ordinary coupon-date formula, then carry that value forward by the elapsed fraction of a period.

The bracketed term, PV, is the present value on the previous coupon date, which a spreadsheet computes easily because it involves N evenly spaced periods. It is not the flat price, though. Once the full price is in hand, subtract accrued interest to reach the flat price.

A bond’s own features decide how sharply its price reacts to a change in yield. Four regularities matter: the inverse link between price and yield, the effect of the coupon rate, the effect of maturity, and the curvature known as convexity. A fifth pattern, the pull to par, describes how prices drift over time even when yields hold still.

The inverse relationship

In any present value calculation, a higher discount rate shrinks the value of a fixed future cash flow and a lower rate lifts it. A bond is just a bundle of such cash flows, so its price and its yield-to-maturity always move in opposite directions. This inverse relationship is the anchor for everything else in this section.

The coupon effect

For two bonds of the same maturity, the one with the smaller coupon reacts more, in percentage terms, to a given yield change. A low coupon pushes a larger share of the total cash flow out to the final principal payment, and that distant payment is discounted by the largest power of (1 + r), so a shift in r bites hardest there. The clearest illustration compares a zero-coupon bond, whose entire payoff arrives at maturity, with an otherwise identical coupon bond.

The maturity effect

Time-to-maturity works in the same direction. All else equal, a longer-dated bond generally shows a larger percentage price change than a shorter-dated one for the same yield move, because the higher exponent N amplifies the effect of the discount rate on the distant cash flows.

Exceptions to the maturity effect do exist, but they are rare. They surface only for long-dated bonds that carry a low (though not zero) coupon and trade at a discount. On zero-coupon bonds, and on any bond priced at par or above, the maturity effect is never overturned.

The constant-yield price trajectory

Even with the market discount rate frozen, a bond’s price still drifts over time. As maturity nears, N falls toward zero and the holder moves ever closer to collecting par, provided the issuer does not default. A bond bought at a discount rises toward par; a bond bought at a premium falls toward par. Both are pulled to par. Take two 10-year bonds each valued at a 5 percent yield: a 2 percent coupon bond starts at 76.835 and climbs, while an 8 percent coupon bond starts at 123.165 and declines, and both land at 100 at maturity.

The convexity effect

The size of a price change also depends on the direction of the yield move. A yield fall raises the price by more than an equal yield rise lowers it, so the price gain is larger in absolute value than the price loss. This asymmetry means the price-yield curve is not a straight line but bends, curving toward the origin: the relationship is convex.

Unlike listed shares, most bonds trade infrequently, so a current market price is often simply unavailable, and the same gap exists for bonds that have not yet been issued. Without a price there is no yield-to-maturity to observe. The workaround is matrix pricing: estimating a bond’s price from the observed yields of comparable but more actively traded bonds, picked because they share a comparable maturity, coupon rate, and credit standing. The technique is widely used in bond price quotations.

The four-step process

1. Identify actively traded comparable bonds of similar maturity, coupon, and credit standing.
2. Compute each comparable bond’s yield-to-maturity, then average the yields at each maturity.
3. Linearly interpolate between those average yields to estimate a yield at the target bond’s maturity.
4. Discount the target bond’s coupons and principal at the interpolated yield to estimate its price.

Estimating a yield spread

Matrix pricing also helps set the yield spread when underwriting a new bond, meaning the difference between the bond’s yield and that of a government benchmark of similar maturity. The comparable here can be the issuer’s own outstanding debt, and the benchmark yield can itself be interpolated from nearby government bonds.