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Explain cash flow additivity and apply it to derive an implied forward rate

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

The idea

Cash flow additivity says the present value of a stream of cash flows equals the sum of the present values of its parts, each discounted at the rate for its own date. It follows that two strategies producing identical cash flows must cost the same, or an arbitrage profit is available. Applied to interest rates, investing for two years at the two-year spot rate must equal investing for one year and rolling over at the implied forward rate: (1 + S2)² = (1 + S1) x (1 + f). Solving gives the forward rate, and the same logic prices a coupon bond as a bundle of zero-coupon bonds.

A practice question

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

The one-year spot rate is 3% and the two-year spot rate is 4%. The implied one-year rate one year from now is closest to:

  1. 5.01% — correct
  2. 3.50%Why not: Averaged the two spot rates; the forward must exceed both because the two-year rate is the average.
  3. 1.00%Why not: Subtracted the one-year spot rate from the two-year spot rate.

(1.04)^2 = 1.03 x (1 + f), so f = 1.0816 / 1.03 - 1 = 5.01%.

A harder one

The two-year spot rate is 2.5% and the three-year spot rate is 3.2%. The implied one-year forward rate two years from now is closest to:

  1. 2.85%Why not: Averaged the two- and three-year spot rates instead of backing out the rate for year three.
  2. 0.70%Why not: Took the difference between the two spot rates as the forward rate.
  3. 4.61% — correct

(1.032)^3 = (1.025)^2 x (1 + f), so f = 1.099105 / 1.050625 - 1 = 4.61%.

The hardest tier

The one-year spot rate is 2.0% and the three-year spot rate is 3.5%. The implied annual rate for a two-year loan starting one year from now is closest to:

  1. 8.70%Why not: Stopped at the two-year cumulative growth and did not take the square root to annualise it.
  2. 4.26% — correct
  3. 1.50%Why not: Subtracted the one-year spot rate from the three-year spot rate.

(1.035)^3 = 1.02 x (1 + f)^2, so (1 + f)^2 = 1.086978 and f = 4.26% a year.

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