Bond pricing, yield and duration

Key takeaways

• A bond's price is the present value of its coupons and face value, discounted at the yield per period; price and yield move in opposite directions.
• Bonds are quoted at a clean price; the buyer pays the dirty price, which adds interest accrued since the last coupon (30/360 for US corporates, actual/actual for Treasuries).
• Yield to maturity is the single discount rate that equates the price with the cash flows, and it assumes coupons are reinvested at that same yield.
• Modified duration is the approximate percentage price change for a one point change in yield; DV01 is the dollar change for one basis point.
• Longer maturity and lower coupons mean higher duration; a zero coupon bond's Macaulay duration equals its maturity.
• Convexity corrects the duration estimate: for an ordinary bond, prices rise more when yields fall than they fall when yields rise.

Pricing a coupon bond

A standard US fixed rate bond pays half its annual coupon every six months and returns its face value at maturity. Prices are quoted per $100 of face value. A 6 percent bond therefore pays $3 every six months on each $100, and a five-year bond has ten coupon payments, the last of them paid with the $100 of principal.

To price it, discount each payment at the yield per period, which for a semiannual bond is the annual yield divided by two, over the number of half-year periods. The coupons form an annuity and the face value is a single sum, so the price is the annuity formula plus the present value of the face. When the yield equals the coupon rate, the bond prices at par ($100). When the yield is below the coupon, it prices above par, at a premium; when the yield is above the coupon, below par, at a discount.

The relationship between price and yield is not a straight line. It is curved, convex toward the origin, and the curvature is stronger for long bonds. The table below shows that a 30-year bond swings much more than a 5-year bond for the same change in yield, which is the first sign of what duration measures.

Clean price, dirty price and accrued interest

Bonds rarely trade on a coupon date. Between coupons, the seller has earned part of the next coupon but will not receive it, because the whole coupon goes to whoever holds the bond on the record date. The buyer therefore compensates the seller for accrued interest, the portion of the coupon earned since the last payment.

Bond prices are quoted clean, without accrued interest, so the quote does not jump every time a coupon is paid. The amount actually paid is the dirty price (also called the full or invoice price): clean price plus accrued interest. The dirty price is the true present value of the remaining cash flows; the clean price is a quoting convention.

How accrued interest is counted depends on the day count convention. US corporate, agency and municipal bonds use 30/360, which treats every month as 30 days and every year as 360. US Treasuries use actual/actual, counting the actual days elapsed over the actual days in the coupon period. On the same dates the two methods can give slightly different accrued interest.

Yield to maturity and current yield

Yield to maturity (YTM) is the single discount rate that makes the present value of a bond's cash flows equal to its dirty price. It is the bond's IRR, and like IRR it is solved by trial and error. In the US it is quoted on a semiannual bond equivalent basis: twice the yield per half year.

YTM is the realized return only if three things hold: the bond is held to maturity, every payment is made in full, and every coupon is reinvested at the YTM itself. The last assumption is the same reinvestment problem as IRR. If rates fall after purchase, coupons are reinvested at lower rates and the realized return falls short of the YTM, which is called reinvestment risk.

Current yield is a much cruder measure: the annual coupon divided by the price. It ignores the gain or loss as the price converges to par at maturity, so for a discount bond it understates the true return and for a premium bond it overstates it. For callable bonds, investors also compute yield to call and quote the lower of the two as yield to worst.

Macaulay and modified duration

Macaulay duration is the weighted average time until a bond's cash flows are received, each time weighted by the share of the price that cash flow represents. A zero coupon bond pays everything at maturity, so its duration equals its maturity. A coupon bond returns some value earlier, so its duration is shorter than its maturity, and the higher the coupon, the shorter the duration.

Modified duration turns that time measure into a price sensitivity. It equals Macaulay duration divided by one plus the yield per period, and it gives the approximate percentage change in price for a one percentage point change in yield. A bond with modified duration of 4.30 loses about 4.30 percent of its value if its yield rises by one point.

Duration grows with maturity and falls with coupon and with the level of yields. This is what makes it so useful: a pension plan can compare its liabilities, an insurer its assets and a credit fund its loan book on one number, and match or hedge them. The approximation holds well for small moves; for large ones convexity matters, as the next section shows.

DV01 and convexity

DV01, the dollar value of a basis point (also called PV01 or PVBP), converts duration into money: it is the change in a bond's price for a one basis point change in yield. Traders and risk managers use it because it adds across positions. A portfolio's DV01 is the sum of the DV01s of its bonds, and a hedge is sized so that the DV01 of the hedge offsets the DV01 of the position.

Duration treats the price yield curve as a straight line. It is really curved, and convexity measures the curvature. For an ordinary bond convexity is positive: as yields fall the price rises faster than duration predicts, and as yields rise it falls more slowly. The duration estimate therefore understates the price after any move, in either direction, and adding the convexity term corrects it.

Convexity matters most for long bonds and large moves. Holders value it, because it means gains outrun losses for equal moves either way, so bonds with more convexity tend to trade at slightly lower yields. Callable bonds and mortgage-backed securities can show negative convexity, where the issuer's right to repay early caps the price gain when yields fall.

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