Compute a bond's value from its coupon and yield.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.c

The idea

A bond's value is the present value of its coupons and principal, discounted at its yield. For a bond with n periods, Price = Σ C ÷ (1 + y)^t + F ÷ (1 + y)^n. Here C is the coupon per period, F is the face value and y is the yield per period, and the sum runs over every coupon date. For semiannual coupons, use half the annual coupon, half the annual yield and twice the number of years, when the yield is quoted with semiannual compounding. The yield and the discount factors must use the same compounding convention. With a continuously compounded yield, Discount factor = e^(−yt). A bond trades at par when its yield equals its coupon rate, at a discount when its yield is higher and at a premium when its yield is lower. The most common errors are using the annual yield for semiannual periods, forgetting to discount the coupons, and mixing compounding conventions.

Formulas

Price of a bond from its coupon and yield

P = Σ C ÷ (1 + y)ᵗ + F ÷ (1 + y)ⁿ

C: the coupon per period
F: the face value
y: the yield per period
t: each coupon period, 1 through n
n: the number of periods to maturity

For semiannual coupons use half the coupon, half the yield and twice the years, provided the yield is quoted semiannually.

Discount factor with a continuously compounded yield

discount factor = e^(−y × t)

y: the continuously compounded yield
t: the years until the cash flow

Use only when the yield is quoted continuously compounded; mixing conventions misprices the bond.

A practice question

From the question bank, with the reason each wrong option is wrong.

A 10-year bond has a 5% annual coupon rate and a yield to maturity of 6.5%. How does the bond's price compare with its face value?

The price is above face value, because the yield exceeds the coupon rateWhy not: Reverses the relationship. A yield above the coupon rate pushes the price below par.
The price equals face value, because coupons are always paid on the face amountWhy not: Price equals par only when the yield equals the coupon rate.
The relationship cannot be known without the bond's durationWhy not: Whether a bond trades at a discount or premium depends only on the yield versus the coupon, not on duration.
The price is below face value, because the coupons are discounted at a rate above the coupon rate - correct

A bond's price is the present value of its coupons and principal at the yield. When the yield exceeds the coupon rate, each cash flow is discounted more heavily than the bond's own coupon rate implies, so the bond trades at a discount.

A harder one

A 10-year zero-coupon bond with face value USD 1,000 has a yield of 6% per year, quoted with semiannual compounding. Its price is closest to:

USD 558.39Why not: Discounts annually at 6% although the yield is quoted semiannually.
USD 553.68 - correct
USD 548.81Why not: Discounts continuously (e^-0.6) although the yield is quoted semiannually.
USD 625.00Why not: Discounts with simple interest, 1,000/(1 + 0.06 x 10).

Price = 1,000 / (1 + 0.06/2)^20 = 1,000 / 1.03^20 = 553.68.

The hardest tier

A 4-year bond with face value USD 1,000 pays an 8% coupon semiannually. Its yield to maturity is quoted as an effective annual yield of 12%. The price is closest to:

USD 875.80Why not: Halves the 12% to 6% per period, which is the nominal semiannual yield, not the 12% effective annual yield.
USD 885.59 - correct
USD 1,135.66Why not: Pays the full annual coupon of USD 80 at every semiannual date.
USD 294.71Why not: Converts to the half-year rate by squaring (1.12² - 1) instead of taking the square root.

Per-period yield = 1.12^0.5 - 1 = 5.830%. Eight coupons of USD 40 plus USD 1,000 discounted at that rate give 885.59.

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