FINS5513 Chap.9 Yield Curves and Forward Rates
Yield Curves and Forward Rates
Week 9 replaces one yield-to-maturity discount rate with a term structure of zero-coupon spot rates. Each spot rate discounts a cash flow at one maturity, so a coupon bond can be priced by applying the appropriate spot discount factor to every promised payment. Bootstrapping reverses that valuation.
Starting with the shortest maturity, it uses observed bond prices to solve sequentially for unknown spot rates after earlier cash flows have been valued with already known rates. Forward rates are rates implied today for borrowing or investing over a future interval. They follow from a no-arbitrage equality between investing for the full horizon and rolling through shorter investments.
The powers in the equation must match the lengths of the intervals; an annualised two-year forward rate is not found by simply subtracting spot rates. The observed yield curve can be upward sloping, flat, downward sloping or humped.
Expectations theory connects forwards to expected future short rates, while liquidity-premium and preferred-habitat explanations allow maturity-related compensation to separate forwards from pure expectations. A realised multi-period return can differ from a quoted YTM because coupons may be reinvested at different rates and the bond may be sold at a different yield before maturity.
Every calculation therefore needs a timeline, a compounding convention and a clear distinction among YTM, spot, forward and realised rates.
What this chapter covers
- 01
Realised compound yield from coupons, reinvestment and sale proceeds
- 02
Spot rates and discount factors for maturity-specific cash flows
- 03
Coupon-bond valuation using a complete spot-rate curve
- 04
Sequential bootstrapping from short to longer maturities
- 05
Forward rates derived from no-arbitrage holding-period equality
- 06
Correct exponents for one-year and multi-year forward intervals
- 07
Upward, flat, inverted and humped yield-curve shapes
- 08
Expectations, liquidity premium and preferred-habitat interpretations
Worked example · free
Value a bond from maturity-specific spot rates
- +1Discount the first coupon at the one-year spot rate: $5 divided by 1.07 equals about $4.67.
- +1Discount the second coupon at the two-year spot rate: $5 divided by 1.075 squared equals about $4.33.
- +1Discount the final $105 at the three-year spot rate: $105 divided by 1.08 cubed equals about $83.35.
- +1Adding the three present values gives a bond price of approximately $92.35.
Key terms
- Spot Rate
- The current discount rate applying to a single default-free cash flow at one maturity.
- Discount Factor
- The present value today of one currency unit paid at a stated future date.
- Bootstrapping
- Sequential recovery of spot rates from prices of bonds with progressively longer maturities.
- Forward Rate
- A rate fixed or implied today for an investment beginning at a future date.
- Yield Curve
- A graph of interest rates or yields against maturity at one point in time.
- Liquidity Premium
- Additional expected compensation associated with holding less liquid or longer-maturity securities.
Yield Curves and Forward Rates FAQ
Why not discount every coupon at YTM?
YTM is one internal rate summarising a bond's price. Spot valuation instead assigns a market discount rate to each maturity and is the no-arbitrage approach when the term structure is available.
Why must bootstrapping be sequential?
A longer coupon bond contains payments at several dates. Earlier spot rates must be known so those earlier payments can be removed from price before solving for the discount rate on the final payment.
Is a forward rate a forecast?
It is first an arbitrage-implied rate under today's curve. It can contain expected future short rates and term premiums, so treating it as an unbiased forecast requires an additional expectations assumption.
How is a multi-year forward annualised?
Equate the compounded full-horizon investment with the product of the initial-period investment and the future-interval investment. Then take a root equal to the number of years in that forward interval.
What can an inverted curve indicate?
Short rates exceed longer rates. This can be consistent with expected rate declines, but interpretation must also consider term premiums, liquidity conditions and maturity preferences rather than relying on expectations alone.
Why can realised yield differ from YTM?
Coupons may be reinvested at rates different from the initial YTM, the investor may sell before maturity at a changed yield, and promised payments may carry credit risk. Those outcomes change terminal wealth.
Exam move
Put every fixed-income problem on a timeline before selecting an equation. For spot valuation, list each cash flow, maturity, spot rate, exponent and present value in a table, then verify that the total price moves inversely with the curve. For bootstrapping, solve from the shortest maturity outward and retain precision in earlier discount factors; rounding too soon contaminates later rates.
For forward rates, write the two alternative investment paths over the same total horizon. Match the compounding powers to the dates, isolate the forward accumulation factor and take the correct root for its interval. Test the answer against the curve's slope without assuming that a simple difference of spot rates is valid.
In explanation questions, separate a mathematical forward rate from an expected future short rate and explain the role of a term or liquidity premium. For realised-yield problems, accumulate coupons at their stated reinvestment rates, include the sale or redemption value, and solve the compound holding return over the investor's actual horizon. Label every annual, semiannual or periodic rate so the units cannot silently drift.
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