Work out forward rates from a set of spot rates
The idea
An investor can lock in a return over two years either by buying a two-year zero-coupon bond or by buying a one-year bond and contracting today to reinvest at a forward rate. No arbitrage requires the two to be equal, which pins the implied forward rate: one plus the two-year spot rate, squared, equals one plus the one-year spot rate multiplied by one plus the one-year forward rate one year forward. The implied forward is not a forecast. It is the breakeven rate at which the two strategies tie, and an investor who expects next year's rate to come in below it would, other things equal, favour the longer bond.
Formulas
Forward rate implied by two spot rates
(1 + z₂)² = (1 + z₁) × (1 + f)
- z₁
- the one-year spot rate today
- z₂
- the two-year spot rate today
- f
- the one-year rate one year forward, implied for year two
Two ways of being invested for two years must return the same, or the difference is arbitrage. The forward rate is what today's spot curve implies, not a rate anybody is forecasting.
A practice question
What does an implied forward rate represent?
- The market's forecast of the future spot rateWhy not: The commonest misreading. It is an arbitrage relationship between prices that already exist; whether it proves accurate is a separate question, and empirically it is a biased forecast.
- The breakeven rate at which rolling short-term bonds ties with holding a longer one — correct
- The coupon on a forward-starting bondWhy not: A forward-starting bond would be priced *from* the forward curve. The rate itself is a zero-coupon rate for a future period.
It is an arbitrage relationship between existing prices, not a prediction. Whether it proves accurate is a separate question.
A harder one
The one-year spot rate is 3% and the two-year spot rate is 4%. What is the implied one-year rate, one year forward?
- Approximately 5.01% — correct
- Approximately 3.50%Why not: The average of the two spot rates. The forward must exceed *both* for the two-year compounded average to sit above the one-year rate.
- Approximately 4.00%Why not: The two-year spot rate itself. The forward has to be higher still, to pull the lower first year up to that average.
1.04 squared is 1.0816; dividing by 1.03 gives 1.0501. The forward must exceed both spot rates for the two-year average to be above the one-year rate.
The hardest tier
An investor with a two-year horizon believes rates will be lower next year than the implied forward suggests. What should they do?
- Roll one-year bonds, since lower rates favour reinvestmentWhy not: Backwards. Lower reinvestment rates *hurt* the rolling strategy — that is the very outcome the investor expects and should avoid.
- Either, since the strategies are equivalent by arbitrageWhy not: They are equivalent only if the forward rate is realised. Holding a view that it will not be is precisely a reason to choose.
- Buy the two-year bond, locking in today's higher implied reinvestment rate — correct
The two-year bond has the forward rate embedded in its price. If actual reinvestment rates come in below it, the investor who locked it in is ahead — which is the whole content of the breakeven interpretation.
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