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Explain the law of one price by way of an arbitrage argument, define it, and show how it is put to work in bond pricing.

FRM® Part I · Valuation and Risk Models · Pricing Conventions, Discounting, and Arbitrage · FRM1.VR.9.b

The idea

The law of one price says that securities or portfolios that deliver exactly the same cash flows in every circumstance must have the same price. The argument is arbitrage. If one were cheaper, a trader would buy it, sell the dearer one, collect the difference today and owe nothing later, and the buying and selling would close the gap. In bond pricing, every default-free cash flow due at date t is valued with the same discount factor d(t), whichever bond carries it. A coupon bond is therefore priced as the following sum. Price = Σ coupon × d(t) + principal × d(T). The sum runs over the coupon dates, T is the maturity date, and each cash flow takes the factor for its own date. Bonds with different coupons rightly have different prices because their cash flows differ. The law covers identical cash flows only, and the usual trap is applying it to bonds that are merely similar, such as bonds with equal maturity.

Formulas

Price of a default-free coupon bond by the law of one price

P = Σ C × d(tᵢ) + F × d(T)

C
the coupon paid on each coupon date
d(tᵢ)
the discount factor for coupon date tᵢ
F
the principal repaid at maturity
d(T)
the discount factor for the maturity date T

Every default-free cash flow on the same date uses the same factor, whichever bond carries it. Bonds of equal maturity but different coupons rightly have different prices.

A practice question

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

Which statement defines the law of one price?

  1. Bonds with the same maturity must have the same yield regardless of couponWhy not: Treats equal maturity as enough; the law needs identical cash flows, not just identical maturity.
  2. Securities or portfolios with identical cash flows in every circumstance must have the same price - correct
  3. Bonds from the same issuer must trade at the same priceWhy not: Ignores that different coupons and maturities from one issuer give different cash flows.
  4. Every security must be priced at the sum of its undiscounted cash flowsWhy not: Ignores the time value of money; cash flows are valued with discount factors.

The law of one price concerns identical cash flows: if two claims pay the same amounts on the same dates in all states, they must cost the same.

A harder one

The one-year and two-year zero-coupon Treasury strips trade at 98.00 and 92.00 per 100 face value. By the law of one price, the price per 100 face value of a two-year Treasury note paying an 8% annual coupon is closest to:

  1. 107.20 - correct
  2. 113.20Why not: Swaps the strips, discounting the 8 at the two-year price and the 108 at the one-year price.
  3. 106.72Why not: Discounts every cash flow, 116 in total, with the two-year strip price.
  4. 116.00Why not: Adds the cash flows without discounting them.

The note equals 8 one-year strips plus 108 two-year strips: 8 x 0.98 + 108 x 0.92 = 7.84 + 99.36 = 107.20.

The hardest tier

Two-year and three-year default-free bonds are priced from one discount function with d(1) = 0.9700 and d(2) = 0.9300. A three-year Treasury note with an 8% annual coupon trades at 108.00. By the law of one price, the price per 100 face value of a three-year zero-coupon bond is closest to:

  1. 86.24Why not: Prices the zero at the note's yield to maturity, which ignores that the note's coupons change its yield.
  2. 85.93 - correct
  3. 92.80Why not: Divides by 100 instead of 108 when backing out d(3).
  4. 85.19Why not: Subtracts the coupons without discounting them, 108 - 24, before dividing by 108.

108.00 = 8 x 0.97 + 8 x 0.93 + 108 x d(3), so d(3) = (108 - 15.2) / 108 = 0.85926. The zero is therefore worth 85.93.

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