Bond prices and yields

A bond is the simplest traded security whose value comes entirely from discounting: it promises fixed payments on fixed dates, and its price is what those payments are worth today. This lesson prices a bond, defines its yield and explains why prices and yields move in opposite directions. It builds on "Interest, compounding and discounting".

A bond as a list of cash flows

A plain fixed-rate bond has a face value $F$ (also called par or principal), an annual coupon rate $c$ and a maturity of $T$ years. Throughout this track we take $F = 100$ and assume one coupon per year. The holder receives a coupon $cF$ at the end of each year $t = 1, \dots, T$ and, together with the last coupon, the face value $F$.

Example: a 10-year bond with $c = 5\%$ pays 5 at the end of years 1 to 9 and $5 + 100 = 105$ at the end of year 10. Nothing else is uncertain in this description; we assume the issuer does not default.

Price as a present value

Discount every cash flow at one annual rate $y$, using annual compounding as in "Interest, compounding and discounting", and add up the present values. The result is the price $P$:

The coupons form an annuity, so the sum has a closed form:

Each term is a payment times a discount factor $1/(1+y)^t$. The further away a payment is, the more it is shrunk.

Yield to maturity

In the market the direction is reversed: the price $P$ is observed, and $y$ is unknown. The yield to maturity (YTM) is the single rate $y$ that makes the present value of the cash flows equal to the market price. In the language of investment appraisal it is the internal rate of return of buying the bond at $P$ and holding it to maturity.

There is no closed form for $y$ when $T > 1$; it is found numerically (for example by bisection), which is easy because the price is a strictly decreasing function of $y$. The yield is a summary number: it assumes every cash flow is discounted at the same rate.

Yield to maturity: the one discount rate $y$ at which the present value of all promised cash flows equals the bond's price.

The price–yield relation

Every term in the price formula has $y$ in the denominator, so a higher yield gives a lower price and a lower yield a higher price. The relation is not a straight line. Differentiating each term $(1+y)^{-t}$ twice gives a positive result, so the curve $y \mapsto P(y)$ is convex: it falls steeply at low yields and flattens at high yields. A rise of 1% in yield therefore lowers the price by less than a fall of 1% raises it.

Premium, par and discount bonds

Compare the coupon rate with the yield.

• Par: if $c = y$, the price equals $F$. Each year the coupon pays exactly the interest the market demands, so nothing needs to be made up through the price.
• Premium: if $c > y$, the bond pays more than the market rate, and buyers pay more than $F$ for it.
• Discount: if $c < y$, the coupons are too low, and the bond sells below $F$ so that the gain from the price rising to $F$ at maturity makes up the shortfall.

Zero-coupon bonds

A zero-coupon bond has $c = 0$: it pays only $F$ at $T$. Its price is a single discounted amount,

so its yield can be solved directly: $y = (F/P)^{1/T} - 1$. A 10-year zero at $y = 6\%$ costs $100/1.06^{10} \approx 55.84$. Zeros always trade at a discount for positive yields, and they are the building blocks of every other bond: a coupon bond is a bundle of zeros, one for each payment date.

Current yield

A simpler number often quoted is the current yield, the annual coupon divided by the price, $cF/P$. It ignores the gain or loss from the price moving to $F$ at maturity. For a discount bond the YTM is higher than the current yield, which is in turn higher than the coupon rate; for a premium bond the order is reversed. At par all three coincide.

A worked example

Take $c = 5\%$, $T = 10$, $y = 6\%$. The annuity factor is $(1 - 1.06^{-10})/0.06 \approx 7.360$, so

The coupon is below the yield, so the bond trades at a discount. Its current yield is $5/92.64 \approx 5.40\%$, between the coupon rate of 5% and the YTM of 6%. Raising the coupon to 7% at the same yield gives about 107.36, a premium; at $y = 5\%$ the 5% bond is worth exactly 100.

The figure plots the price against yields from 0% to 15%, with a horizontal line at par (100) and a vertical line at the chosen yield. The readouts show the price, whether it is at a premium, at par or at a discount, and the current yield. Move $y$ across the coupon rate and note that the price crosses 100 exactly there. Raise $T$ and note that the curve gets steeper and more curved; set $c$ to 0 to see a zero-coupon bond.

Pull to par

If the yield stays the same, the price of a bond moves towards $F$ as maturity approaches, because fewer coupons are left to be over- or under-priced and the face value is discounted over a shorter time. With $c = 5\%$ and $y = 6\%$, the price is about 92.64 with 10 years left, 95.79 with 5 years left and 99.06 with one year left; at maturity it is 100. This is the pull to par. In the figure, shorten $T$ at a fixed yield and the curve flattens towards the par line.

Summary

• A fixed-rate bond is a list of known cash flows: coupons $cF$ each year and the face value $F$ at maturity.
• Its price is the present value of those cash flows; the yield to maturity is the single rate that reproduces the market price (the bond's IRR).
• Price and yield move in opposite directions along a convex curve.
• The bond trades at par when $c = y$, at a premium when $c > y$ and at a discount when $c < y$; with an unchanged yield its price is pulled to par as maturity approaches.

The next lesson, "Duration and convexity", measures the slope and the curvature of the price–yield curve.