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Explain what a spot rate is and derive discount factors from spot rates.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.b

The idea

A spot rate, or zero rate, is the annualized yield on a zero-coupon bond that pays a single cash flow at maturity T. It is the return for lending from today until T with nothing in between. The discount factor for T, written DF(T), is the present value of one unit paid at T. With annual compounding, the factor for maturity T is given by Discount factor = 1 ÷ (1 + sT)ᵀ. With continuous compounding, the factor for maturity T is given by Discount factor = e^(−sT × T). A coupon bond is a portfolio of zeros, so its price is the sum of each cash flow times its own discount factor. That is why one yield cannot be used for every date unless the curve is flat. Going the other way with annual compounding, Spot rate = DF(T)^(−1 ÷ T) − 1. A common mistake is to discount every cash flow at the spot rate of the bond's final maturity.

Formulas

Discount factor from an annually compounded spot rate

DF(T) = (1 + sT)^(−T)

sT
the annually compounded spot rate for maturity T
T
the maturity in years

Use the spot rate for each cash flow's own date, not the final maturity's rate for all of them.

Discount factor from a continuously compounded spot rate

DF(T) = e^(−sT × T)

sT
the continuously compounded spot rate for maturity T
T
the maturity in years

Only for a continuously compounded quote; an annual quote put into this formula gives the wrong factor.

Annually compounded spot rate from a discount factor

sT = DF(T)^(−1 ÷ T) − 1

DF(T)
the discount factor for maturity T
T
the maturity in years

Take the root equal to the number of years. Forgetting the root gives a cumulative rate, not an annual one.

A practice question

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

The 2-year spot rate is 4.5% with annual compounding. The 2-year discount factor is closest to:

  1. 0.9157 - correct
  2. 0.9569Why not: Uses one year of discounting, forgetting the exponent of 2.
  3. 0.9100Why not: Uses simple interest, 1 - 0.045 x 2, rather than compounding.
  4. 1.0920Why not: Inverts the factor: this is the growth factor, not the present value of one unit.

DF(2) = 1 / 1.045² = 0.9157. The discount factor is the present value of 1 paid in 2 years.

A harder one

The annually compounded spot rates are 2.0% for 1 year, 3.0% for 2 years and 5.0% for 3 years. A 3-year bond with face value 100 pays a 5% annual coupon. Its price is closest to:

  1. 100.00Why not: Discounts every cash flow at the 3-year rate of 5.0% instead of each at its own spot rate.
  2. 104.68Why not: Discounts all flows at 3.33%, the average of the three spot rates, rather than at each flow's own spot rate.
  3. 109.76Why not: Omits the exponent, dividing by (1 + s) rather than (1 + s)^t for each date.
  4. 100.32 - correct

Price = 5/1.02 + 5/1.03² + 105/1.05³ = 100.32. Each cash flow is discounted at the spot rate for its own date; one yield does not apply unless the curve is flat.

The hardest tier

The 1-year spot rate is 2% (annual compounding). A 2-year bond with a 8% annual coupon is priced at par, 100. The 2-year discount factor implied by these prices is closest to:

  1. 0.8533 - correct
  2. 0.8573Why not: Treats the 8% par coupon as the 2-year spot rate, 1/1.08².
  3. 0.9612Why not: Reuses the 1-year spot rate for the second year.
  4. 0.9216Why not: Divides by 100 instead of 108, leaving out the coupon paid at the same time as the principal.

Bootstrap: 100 = 8 x DF1 + 108 x DF2, DF1 = 1/1.02 = 0.9804, so DF2 = (100 - 8 x 0.9804) / 108 = 0.8533. The 2-year spot rate is 8.26%, above the 8% par rate.

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