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Explain the way a bond's YTM is calculated when its structure and price are known.

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

The idea

There is no algebraic formula for the YTM of a bond with several coupons. The equation to solve is Price = Σ CFt ÷ (1 + y)^t. Solving it means finding the root of a polynomial whose degree equals the number of periods. In practice you iterate. Guess a yield, compute the price, and adjust until it matches the market price, or interpolate between two trial yields. A financial calculator does this once you enter N as the number of periods, PMT as the coupon per period, PV as the negative of the price and FV as the face value, and then compute I/Y. A zero-coupon bond is the exception, because y = (Face ÷ Price)^(1÷n) − 1. For a semi-annual bond the answer is a per-period rate, so double it for the quoted bond-equivalent YTM. Common mistakes are mixing annual and semi-annual inputs, and entering PV and FV with the same sign.

Formulas

Yield to maturity of a zero-coupon bond

y = (F ÷ P)^(1 ÷ n) − 1

F
the face value
P
the price
n
the number of periods to maturity

Only zero-coupon bonds have this closed form; coupon bonds need iteration. With semiannual periods, y is per half-year and is doubled to quote.

A practice question

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

Why can the YTM of a coupon bond with several remaining payments generally not be found by rearranging the pricing formula?

  1. The YTM depends on the investor's reinvestment rate, which is unknown in advance.Why not: Mixes the realized-return interpretation with the calculation; the YTM follows from price and promised cash flows alone.
  2. The market price is not known until the YTM has been calculated.Why not: Reverses the dependence: the price is observed, and the YTM is derived from it.
  3. Each cash flow is discounted at a different rate, which cannot be isolated algebraically.Why not: The YTM uses one rate for all cash flows; the difficulty is solving the polynomial, not multiple rates.
  4. The price equation is a polynomial in the discount factor, so the yield must be found by iteration or a calculator's solver. - correct

With n payments the equation price = sum of cash flow / (1 + y)^t has no general closed-form solution once there are more than a few payments; you iterate, or let a calculator do it. A zero-coupon bond is the exception because y = (face/price)^(1/n) - 1.

A harder one

A 3-year bond with face value 100 pays a 6% annual coupon and is priced at 96.50. What is its YTM with annual compounding, closest to?

  1. 6.22%Why not: Reports the current yield (coupon / price) and ignores the pull to par.
  2. 7.34% - correct
  3. 6.00%Why not: Reports the coupon rate, which equals the YTM only when the bond is priced at par.
  4. 9.84%Why not: Adds the whole 3.50 discount to one year's coupon instead of spreading the gain over the life.

N = 3, PMT = 6, PV = -96.50, FV = 100, CPT I/Y = 7.34. The bond is at a discount, so the YTM is above the 6% coupon.

The hardest tier

A 3-year bond pays a 5.5% annual coupon on face value 100 and trades at 100.50. A trader finds that discounting its cash flows at 5.00% gives a price of 101.36 and at 6.00% gives 98.66. Using linear interpolation between these two points, the estimated YTM is closest to:

  1. 5.68%Why not: Measures the interpolated distance from the wrong end of the 5% to 6% range.
  2. 5.50%Why not: Uses the midpoint of the range, which is also the coupon rate, ignoring where 100.50 falls.
  3. 4.68%Why not: Moves the estimate below 5.00% instead of above it, getting the direction wrong.
  4. 5.32% - correct

Price falls 2.70 as the yield rises 1.00%. Reaching 100.50 needs a fall of 0.86 from 101.36, a fraction 0.319 of the way. YTM is about 5.00% + 0.319 x 1.00% = 5.32%.

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