Bond Pricing

Bond pricing is the method of determining what a bond is worth by discounting its expected future payments back to today. A conventional bond pays a fixed coupon at regular intervals and returns its face value at maturity; its price is the sum of the present value of each of those payments. Investors, dealers, issuers and accountants all use the same basic calculation.

The central relationship is that price and yield move in opposite directions. When market interest rates rise, the fixed payments of an existing bond are worth less and its price falls; when rates fall, its price rises. Pricing therefore links a bond's contractual terms to current market conditions and to the credit standing of the issuer. It is an application of the time value of money.

For a bond paying an annual coupon, the standard calculation is:

• $P$ is the price of the bond.
• $C$ is the coupon paid each period, equal to the coupon rate multiplied by the face value.
• $F$ is the face value, or par value, repaid at maturity.
• $y$ is the yield per period, used as the discount rate.
• $n$ is the number of periods to maturity.

Take a bond with a face value of 100, a 5% annual coupon and three years to maturity, when investors require a yield of 6%. The first coupon is worth 5 divided by 1.06, or 4.72. The second is worth 5 divided by 1.06 squared, or 4.45. The final payment of 105 is worth 105 divided by 1.06 cubed, or 88.16. The price is the total, 97.33. Because the coupon is below the required yield, the bond trades below face value. For bonds paying twice a year, the coupon and yield are halved and the number of periods doubled.

The yield that makes the discounted payments equal to the market price is the yield to maturity. Price and yield are two ways of stating the same thing: given one, the other can be calculated.

Three cases follow from the formula. If the coupon rate equals the yield, the bond is priced at par. If the coupon is lower than the yield, the bond trades at a discount, and the investor's return comes partly from the rise in price towards face value. If the coupon is higher, the bond trades at a premium. In every case, the price is pulled towards par as maturity approaches, provided the issuer does not default.

Bonds are quoted in the market at a clean price, which excludes interest earned since the last coupon date. The buyer pays the dirty price, which is the clean price plus bond accrued interest. Quoting clean avoids the saw-tooth pattern that would otherwise appear as interest builds up and then drops on each coupon date. The accrued amount depends on the bond's day count convention. Prices are normally stated as a percentage of face value, so a price of 97.33 means 97.33 per 100 of nominal.

Several factors determine the yield that investors demand, and so the price:

• Interest rates. The level of risk-free rates for the bond's maturity, read from the government yield curve, is the base.
• Credit risk. Investors add a credit spread for the possibility of default, which widens if the issuer's position weakens.
• Liquidity. Bonds that are hard to trade carry a higher yield.
• Time to maturity. Longer bonds are more sensitive to a given change in yield.
• Embedded options. A right for the issuer to redeem early, as in a callable bond, lowers the price relative to a comparable bond without it.

The sensitivity of price to yield is measured by duration, and the curvature of that relationship by convexity. Together they describe a bond's interest rate risk. A bond with a duration of five will lose roughly 5% of its value if yields rise by one percentage point.

The single-yield formula is a simplification, since it discounts every payment at the same rate. A more precise approach discounts each payment at the spot rate for its own date, taken from the yields of zero-coupon bonds, and then adds a spread.

Other instruments need adjustments. A zero-coupon bond has only the final term of the formula. A floating-rate note (FRN) resets its coupon to market rates, so its price stays near par and moves mainly with changes in the issuer's credit spread. Bonds with call or put features are valued with option models and quoted on a yield to the worst outcome for the investor. Inflation-linked bonds are priced on real yields. Amortising and mortgage-backed bonds require assumptions about the timing of principal repayments.

Most bonds trade over the counter through dealers and do not have a continuous public price. Liquid government bonds are quoted directly. Corporate bonds are usually quoted as a spread over a government bond or swap rate of similar maturity, from which the price is calculated. For bonds that seldom trade, pricing services estimate values from comparable issues, a method known as matrix pricing. Funds and banks use such evaluated prices for daily valuation, and accounting standards rank them by how far they rely on observable market data.