Pricing a bond from its yield to maturity

How do you turn a yield into a bond price?

A bond is a promise of cash flows, and its price is the present value of that promise. Discount each coupon and the par value at the market discount rate : the return the market demands on bonds of the same risk and maturity. It also goes by required yield or required rate of return , and a quoted price implies it as the yield-to-maturity .

Market discount ratewhat the market demands for a bond of this risk and maturity

• Required yield Know the idea the investor’s view of it
• Required rate of return Know the idea the same idea, equity-style wording
• Yield-to-maturity Memorize the rate implied by a quoted price
  • Price known, rate unknown solve for the rate (an IRR)
  • Rate known, price unknown discount the cash flows

$$PV = \frac{PMT}{(1 + r)^{1}} + \frac{PMT}{(1 + r)^{2}} + \cdots + \frac{PMT + FV}{(1 + r)^{N}}$$

Here $r$ is the market discount rate per period, $N$ the number of periods left, $PMT$ the coupon per period and $FV$ the future value, here the par value. The last term bundles the final coupon with par.

Why do bonds trade at a premium, at par or at a discount?

Compare the coupon rate with the market discount rate. The price follows from that comparison.

Compare the coupon rate with the market discount ratethe price follows, never the other way round

• Coupon rate above the rate Memorize price above par: a premium bond
• Coupon rate equal to the rate Memorize price equals par: a par bond
• Coupon rate below the rate Memorize price below par: a discount bond

A par bond sells at its par value because its coupon equals the rate. A discount bond sells below par: with a 5% coupon when new bonds pay 6%, the bond is short 1 per year, and its price drops by the present value of that shortfall. For the 6-year bond that is 4.917, and 100 − 4.917 = 95.083. A premium bond pays more than the market asks, so buyers pay above par.

Exam tip Premium or discount is a comparison of coupon with yield. A bond with a 2% yield and a 5% coupon is a premium bond, however low the yield looks.

Go deeper Why “discount” means two different things

A bond trading below par is “at a discount”, and the rate used to value it is the “discount rate”. The two are unrelated words. A bond can be at a discount, at par or at a premium whatever discount rate you use. If a question says “discount”, check whether it means the price or the rate.

How do you price a semiannual bond?

Most bonds pay twice a year, and the yield is quoted as an annual rate. Convert everything to the period between coupons before you enter it.

$$N = m \times \text{years}, \qquad r = \frac{y}{m}, \qquad PMT = \frac{\text{coupon rate} \times FV}{m}$$

Entering N = 4, I/Y = 7 and PMT = 6 would treat the bond as annual and give 96.61, close enough to look right and wrong enough to lose the mark.

What is the yield to maturity?

The yield-to-maturity is the single rate that makes the present value of all the promised cash flows equal the price. It is an internal rate of return, so you solve for it instead of calculating it directly.

$$PV_{\text{market}} = \sum_{k=1}^{N} \frac{PMT}{(1 + r)^{k}} + \frac{FV}{(1 + r)^{N}} \;\Rightarrow\; \text{solve for } r, \qquad y = m \times r$$

For a semiannual bond the answer is a rate per half-year. The quoted YTM is that rate times 2.

The price goes in as a negative number because it is money you pay out. Enter it with a positive sign and the calculator reports an error, because the cash flows then all point the same way.

When does the investor really earn the YTM?

The YTM is a promised return, not a guaranteed one. The investor’s realized return equals it only if all three conditions hold.

Realized return equals the yield-to-maturity

• Held to maturity Memorize no earlier sale
• Paid in full, on time Memorize no default or delay
• Coupons reinvested at the YTM Memorize the hardest to meet

Go deeper Spreadsheet functions for price and yield

In a spreadsheet, PRICE(settlement, maturity, rate, yield, redemption, frequency) returns the flat price and YIELD(settlement, maturity, rate, price, redemption, frequency) the annualized yield, both per 100 of par. The exam gives you a calculator, not a spreadsheet, so use these only to check your work.

Can a bond’s yield be negative?

Yes. If the price is higher than the sum of the remaining cash flows, the investor locks in a loss if held to maturity, and the yield to maturity is a negative yield . Zero-coupon bonds show it most clearly.

$$r = \left(\frac{FV}{PV}\right)^{1/N} - 1$$

Go deeper How negative yields arise

Negative yields appeared when central banks pushed policy rates below zero and investors still wanted safe or eligible assets, such as banks needing high-quality collateral. The price is bid above the total of the remaining payments, and the formulas do not change. The growth factor per period is simply below 1.