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Explain what a bond's yield to maturity (YTM) is, interpret it, and use it in pricing the bond.

FRM® Part I · Valuation and Risk Models · Bond Yields and Return Calculations · FRM1.VR.11.c

The idea

Yield to maturity is the single discount rate y that makes the present value of a bond's promised cash flows equal its market price. With m payments a year, Price = Σ CFt ÷ (1 + y÷m)^(m × t). YTM is the bond's internal rate of return. For semi-annual bonds it is quoted as twice the half-year rate, which is the bond-equivalent basis. Given y, you price the bond by discounting every coupon and the principal at the same rate. Price falls as y rises, and the bond sells above par when its coupon is above y. YTM equals the return an investor actually earns only if the bond is held to maturity, pays every promised cash flow, and has its coupons reinvested at y itself. Usual errors are forgetting to halve the rate and coupon and double the periods for a semi-annual bond, and treating YTM as guaranteed.

Formulas

Bond price from its yield to maturity

P = Σ CFt ÷ (1 + y ÷ m)^(m × t)

CFt
the promised cash flow at date t
y
the yield to maturity, quoted annually
m
payments a year
t
the date of the cash flow, in years

One rate y discounts every cash flow. For a semiannual bond m is two, so the rate per period is y ÷ 2.

A practice question

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

Which statement best defines a bond's yield to maturity?

  1. The annual coupon divided by the bond's market price.Why not: This is the current yield, which ignores the principal repayment and the time to maturity.
  2. The total return in the first year, made up of the coupon and the price change.Why not: Describes a one-year holding return, not the internal rate of return on promised cash flows.
  3. The single discount rate at which the present value of the promised cash flows equals the bond's market price. - correct
  4. The spot rate for the bond's final maturity date.Why not: A spot rate discounts a single cash flow; the YTM is one rate for all of a bond's cash flows.

YTM is the internal rate of return of the bond's promised cash flows at the market price: price = sum of cash flow / (1 + y)^t with one rate y for all dates.

A harder one

A bond with face value 100 matures in 5 years and pays a 4% coupon semi-annually. Its YTM is 5% on a bond-equivalent basis (semi-annual compounding). What is its price, closest to?

  1. 95.62 - correct
  2. 113.13Why not: Pays the full 4.00 annual coupon every half-year instead of 2.00.
  3. 97.68Why not: Counts 5 periods at 2.5% instead of 10 half-year periods.
  4. 76.83Why not: Uses the annual 5% as the rate for each half-year period.

N = 10, I/Y = 2.5, PMT = 2.00, FV = 100 gives price = 95.62. The coupon is below the YTM, so the bond sells at a discount.

The hardest tier

A semi-annual-pay bond has a YTM of 9.00% quoted on a bond-equivalent basis. To compare it with an annual-pay bond, an analyst wants the equivalent annually compounded yield. What is it, closest to?

  1. 9.20% - correct
  2. 9.00%Why not: Treats the bond-equivalent yield as already annually compounded.
  3. 8.80%Why not: Converts to a continuously compounded rate, not an annually compounded one.
  4. 18.81%Why not: Squares 1.09, treating 9.00% as the rate for each half-year.

Each half-year earns 4.50%. Annually compounded equivalent = 1.045² - 1 = 9.20%. The bond-equivalent yield understates the effective annual yield.

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