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Work out, from market prices, the return implied by a bond and the growth rate implied by a stock

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.13.a

The idea

When a price is known, the valuation equation can be run backwards to find the return it implies. For a zero-coupon bond, r is (face value / price) raised to the power 1/n, minus 1. For a coupon bond, the yield to maturity is the single rate that equates the present value of the coupons and principal to the price; it is found by trial or a calculator, and it sits below the coupon rate when the bond trades at a premium. For a stock under constant growth, P0 = D1 / (r - g) rearranges to r = D1 / P0 + g, or to g = r - D1 / P0 for the growth the price implies. Remember that D1 = D0 x (1 + g).

A practice question

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

A four-year zero-coupon bond is priced at 85.00 per 100 of face value. Its implied annual return is closest to:

  1. 4.41%Why not: Divided the total 17.65% gain by four years, ignoring compounding.
  2. 4.15% — correct
  3. 3.75%Why not: Took the 15-point discount as a share of face value and divided by four.

Solve 85 x (1 + r)^4 = 100: r = (100 / 85)^(1/4) - 1 = 4.15%. Simple averaging overstates the compound rate.

A harder one

A share is priced at 60.00, next year's dividend is expected to be 2.40 and dividends are expected to grow at 4% a year indefinitely. The required return implied by the price is closest to:

  1. 8.00% — correct
  2. 4.00%Why not: Stopped at the dividend yield and did not add the 4% growth rate.
  3. 8.16%Why not: Grew the 2.40 by another 4% even though it is already next year's dividend.

r = D1 / P0 + g = 2.40 / 60.00 + 4% = 4.00% + 4.00% = 8.00%.

The hardest tier

A stock trades at 36.00. Its most recent dividend, just paid, was 1.20, and investors require an 8.5% return. The implied constant growth rate is closest to:

  1. 5.17%Why not: Treated the 1.20 just paid as next year's dividend, so the growth in D1 was left out.
  2. 11.83%Why not: Added the dividend yield to the required return rather than subtracting it.
  3. 5.00% — correct

36 = 1.20 x (1 + g) / (0.085 - g) gives 3.06 - 36g = 1.20 + 1.20g, so g = 1.86 / 37.2 = 5.00%.

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