ActuaryDeck

Home/Exam FM/Spot rates

Exam FMCash flow analysisFree to read

Spot rates and the yield curve

The t-year spot rate is the single rate applying to one payment t years away; a set of them is the term structure, and each cash flow gets its own.

The formulas

Present value
PV=∑tCt(1+st)tPV = \sum_{t} \dfrac{C_t}{(1+s_t)^{t}}
Zero-coupon price
P(t)=(1+st)−tP(t) = (1+s_t)^{-t}
Par yield
∑t=1nc(1+st)t+1(1+sn)n=1\sum_{t=1}^{n} \dfrac{c}{(1+s_t)^t} + \dfrac{1}{(1+s_n)^n} = 1

solve for the coupon c that prices at par

Where it comes from

  1. A payment at time tt can be replicated exactly by a tt-year zero-coupon bond, so it must be discounted at that bond's rate - the tt-year spot rate.
  2. Using one flat yield instead is only correct when the curve is flat; otherwise it misprices every cash flow except by accident.

Worked example

Annual spot rates are s₁ = 4%, s₂ = 4.6%, s₃ = 5.1%. Find the present value of payments of 1,000 at the end of each of the next three years.

  1. Discount each payment at ITS OWN spot rate.
  2. 1,000/1.04=961.5381{,}000/1.04 = 961.538; 1,000/1.0462=913.9801{,}000/1.046^{2} = 913.980; 1,000/1.0513=861.3741{,}000/1.051^{3} = 861.374.
  3. PV=961.538+913.980+861.374=2,736.89PV = 961.538 + 913.980 + 861.374 = 2{,}736.89.
  4. A flat 4.6% would have given 2,743.79 - close, and wrong.

Answer: 2,736.90

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • One rate per DATE, not one rate per bond.
  • sₜ is compounded t times: the exponent and the subscript always match.

Traps

  • Using the n-year spot rate for every cash flow of an n-year bond.
  • Confusing a spot rate with a forward rate.

Related

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.