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

$$PV=\sum _t\frac{C_t}{(1+s_t{)}^t}$$

Zero-coupon price

$$P(t)=(1+s_t{)}^{-t}$$

Par yield

$$\sum _{t=1}^n\frac{c}{(1+s_t{)}^t}+\frac{1}{(1+s_n{)}^n}=1$$

solve for the coupon c that prices at par

Where it comes from

1. A payment at time $t$ can be replicated exactly by a $t$-year zero-coupon bond, so it must be discounted at that bond's rate — the $t$-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

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

• Forward rates
• Net present value

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