8.2 Yield Measures, Spot Rates, and Forward Rates
Key Takeaways
- Yield to maturity is the single discount rate that equates a bond's promised cash flows to its market price.
- Current yield, yield to call, simple yield, money market yield, and effective annual yield answer different questions.
- Spot rates discount individual cash flows, while forward rates are implied future rates between two dates.
- Bootstrapping derives spot rates from prices of default-free coupon bonds when zero-coupon rates are unavailable.
- Exam questions often test the difference between a quoted yield, the realized return, and the curve used to value cash flows.
Yield is a measure, not a promise
A bond yield summarizes price and cash flows. It is tempting to treat yield as the investor's future return, but that can be wrong. The realized return depends on reinvestment rates, sale price, holding period, default experience, taxes, and embedded option exercise.
CFA Level I questions usually ask which yield measure is most appropriate for the situation. Current yield focuses on coupon income. Yield to maturity solves for one rate across all promised cash flows. Yield to call solves to a call date and call price. Spot rates discount each cash flow at its own maturity rate.
Common yield measures
Current yield equals annual coupon divided by full price. It ignores principal gains or losses and reinvestment. A 5 percent coupon bond priced at 95 has a current yield of 5.26 percent, but that does not capture the pull toward par if held to maturity.
Yield to maturity, or YTM, is the internal rate of return that equates present value of promised cash flows to price. For a conventional fixed-rate bond, if price is below par, YTM exceeds the coupon rate. If price is above par, YTM is below the coupon rate.
Yield to call, or YTC, is calculated like YTM but uses the call date and call price rather than final maturity. For callable bonds trading at a premium, the lower of YTM and YTC is often called yield to worst when the bond has no other relevant embedded option outcomes.
Periodicity and compounding
Bond yields must be compared on the same compounding basis. A semiannual-pay bond has a periodic yield per half-year. The bond-equivalent yield is commonly two times the semiannual yield. The effective annual yield compounds the semiannual yield: EAY = (1 + semiannual yield)^2 - 1.
Money market instruments may use discount yield, add-on yield, or bond-equivalent yield conventions. The exam point is that quoted conventions can differ from economic return. Always identify the denominator, day count, and compounding convention.
Spot rates
A spot rate is the discount rate for a single payment at a specific maturity. A one-year spot rate discounts a one-year cash flow. A two-year spot rate discounts a two-year cash flow. If a bond has several cash flows, no-arbitrage valuation discounts each cash flow using the spot rate for its payment date.
The value of an option-free bond is the sum of each promised cash flow discounted at its matching spot rate. This is more precise than using one YTM because the yield curve is rarely flat. YTM is a summary rate. Spot rates are the valuation inputs.
Bootstrapping intuition
Bootstrapping derives spot rates sequentially. If a one-year zero-coupon bond price is known, the one-year spot rate is known. Then a two-year coupon bond price can be used to solve the two-year spot rate after discounting the first-year coupon at the one-year spot rate.
The process continues maturity by maturity. Each new bond adds one unknown spot rate if earlier spot rates have already been solved. The result is a zero-coupon spot curve consistent with observed bond prices.
Forward rates
A forward rate is a rate agreed today for a loan or investment that starts in the future. In no-arbitrage terms, investing for two years at the two-year spot rate should produce the same ending value as investing for one year and then reinvesting at the one-year forward rate implied today.
For annual rates, the one-year forward rate one year from now is: f(1,1) = [(1 + z2)^2 / (1 + z1)] - 1. Here z1 is the one-year spot rate and z2 is the two-year spot rate. With semiannual or other periods, match the compounding basis.
Structured aid: yield selection table
| Question asks for | Use | Key limitation |
|---|---|---|
| Coupon income relative to price | Current yield | Ignores capital gain or loss |
| Single rate to maturity | YTM | Assumes promised cash flows and reinvestment at YTM |
| Callable bond outcome | YTC or yield to worst | Depends on option exercise assumption |
| No-arbitrage value | Spot rates | Requires a spot curve |
| Future rate implied today | Forward rate | Implied, not a guaranteed forecast |
Exam focus
Do not mix yield concepts. A discount bond has YTM above coupon rate because par repayment adds return. A premium bond has YTM below coupon rate because price declines toward par. Current yield lies between coupon rate and YTM for many conventional fixed-rate bonds, but it is still incomplete.
Forward rates are easy to overinterpret. A forward rate can be viewed as a break-even reinvestment rate implied by the current spot curve. It is not automatically an unbiased forecast. If the question asks for no-arbitrage valuation, use spot and forward relationships rather than a narrative forecast.
A fixed-rate bond sells at a discount to par. Its yield to maturity is most likely:
The rate used to discount a single cash flow due at a specific future date is best described as a:
Given one-year and two-year spot rates, the one-year forward rate one year from now is most directly interpreted as: