Discount factors from the previous lesson tell us the price of every future cash flow. But a desk does not quote bond trades in discount factors — it quotes them in rates. This note converts the discount function into two rate representations that practitioners actually use: spot rates (the annualised return from a zero-coupon investment of a given term) and forward rates (the rate locked in today for a loan starting at a future date). The two representations are equivalent and no-arbitrage links them by a single identity. Understanding that identity also explains the slope of the yield curve: a term structure that slopes up is simply a statement that forward rates exceed current spot rates.
The forward rate concept carries forward throughout this unit. It reappears in the term structure models of Topics 3 and 4 as the expected future short rate (up to a convexity correction), and again in the HJM framework where the entire term structure is parameterised by the instantaneous forward rate.
Learning objectives. By the end of this note you should be able to:
- convert between discount factors and semiannually compounded spot rates;
- compute forward rates from spot rates using the no-arbitrage identity;
- state and prove the slope proposition: the spot curve slopes up if and only if forward rates exceed spot rates;
- explain the maturity-vs-bond-price relationship using forward rates;
- reproduce the equal-performance argument showing that rolling short loans and buying long bonds are equivalent when forward rates are realised.
In this lesson
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- Semiannual compounding
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- Spot rates
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- Forward rates
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- The slope of the term structure
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- Maturity and bond price: a forward-rate interpretation
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- Rolling short vs buying long: equal performance under realised forwards
Before you start
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