The term structure: spot rates, forward rates and par yields

CM2 · Financial Engineering & Loss Reserving · 11 min read

1Three names for one curve

A government borrows for one year, two years and three years, and the market decides a price for each. That set of prices is the term structure of interest rates, and it can be reported in three different ways. All three contain exactly the same information. Being asked to convert between them is one of the most reliable question types in the asset-valuation half of CM2, and the conversions are short once you know which is which.

Spot rates (also zero-coupon or zero rates). The annual effective rate earned on money lent today for exactly n years with no payments in between. Write it s_n. This is the curve to think in, because every cash flow has exactly one spot rate attached to it.

Forward rates. The rate agreed today for borrowing over a future period. Write f(m, n) for the rate from time m to time n. Nobody has to lend anything today to fix this: it is implied by the spot curve, and arbitrage forces it to be.

Par yields. The coupon rate at which an n-year coupon bond would be priced at exactly its face value. Write it y_n. This is the rate a borrower would actually have to pay on a conventional bond, which is why it is the one quoted in markets.

2Discount factors come first

Every conversion is easier through discount factors than directly.

In words: what one rupee payable in n years is worth today.

Write the discount factors down as the first line of any term-structure answer. They are the quantity every other formula is built from, and once you have them nothing further needs a power.

3Spot rates to forward rates

There are two ways to have money for n years: lend for n years at the spot rate, or lend for m years and agree today to roll the proceeds over at the forward rate. Both are riskless and both are fixed today, so they must give the same answer or there is free money available:

In words: accumulating at the long spot rate must equal accumulating at the short one and then at the forward rate.

Rearranged for the one-year forward rate starting at time m:

The discount-factor form on the right is the one to use. It is a single division, and it makes the free check obvious: multiply all the one-year forward factors together and you must get back the long accumulation factor.

One consequence is worth stating because questions turn on it. When the spot curve is rising, forward rates lie above spot rates. The spot rate is an average of the forwards over its whole term, so for the average to be climbing each new forward must be above the average so far. An upward-sloping curve therefore always implies forward rates above the long rate, and that is arithmetic rather than an economic forecast.

4Par yields

A bond paying coupon c per 100 nominal for n years, redeemed at 100, is worth

The par yield is the coupon that makes that equal 100. Setting P = 100 and solving:

In words: the par yield is one minus the final discount factor, over the sum of all the discount factors.

The one-year par yield equals the one-year spot rate, because a one-year bond has only one payment. After that they part company, and the direction is systematic: with a rising spot curve the par yield sits below the spot rate of the same term. The reason is that a coupon bond pays most of its money early, at the lower short rates, so the single coupon rate that prices it at par is pulled down towards them. The worked example shows a three-year par yield of 5.96% against a three-year spot rate of 6.00%.

There is a third way of reading the par yield that makes this obvious, and it is a useful check. It is a weighted average of the one-year forward rates, with the discount factors as weights. On the example below the weights are 0.3542, 0.3335 and 0.3123 and the weighted average of the three forwards comes to exactly the par yield.

5Bootstrapping: getting the curve out of real prices

Markets quote coupon bonds, not zero-coupon ones, so the spot curve usually has to be extracted. The method is to work outwards one maturity at a time, which is what "bootstrapping" means:

• the one-year bond has a single payment, so its price gives v₁ directly;
• the two-year bond's price involves v₁, which is now known, and v₂, which is not — one equation, one unknown;
• and so on outwards.

Each step uses every discount factor already found, so an error at the short end propagates all the way along the curve. That is the practical warning: check each discount factor by repricing the bond it came from before using it for the next one.

6Why the curve has the shape it does

Three explanations are offered, and a question asking you to "discuss the shape" wants all three with a judgement rather than a winner.

Expectations. The forward rates are the market's expectation of future short rates, so the curve slopes up when rates are expected to rise. This is the cleanest theory and the one most obviously incomplete: curves slope upwards most of the time, which would mean rates are almost always expected to rise, and they are not.

Liquidity preference. Lenders dislike tying money up and want paying for it, so long rates carry a premium over the expected path of short rates. This explains the persistent upward tilt that expectations alone cannot, and it means a forward rate is an expectation plus a premium, so reading it as a pure forecast overstates where rates are going.

Market segmentation. Different investors are confined to different parts of the curve — a pension fund needs thirty-year assets, a bank needs short ones — so supply and demand are set separately at each maturity. This explains kinks and humps that neither of the other theories can, and it is why a curve can be unsmooth in a way no single formula fits.

The honest summary is that the three are complementary. An inverted curve usually does say something about expectations; a persistent upward slope usually includes a premium; a bump at one maturity is usually somebody's regulation.

A worked example

The annual effective spot rates are 5.0% for one year, 5.6% for two years and 6.0% for three years. These rates are invented for this example and are not a real yield curve.

Find the one-year forward rates, the par yields, and the price of a three-year 8% bond. Work to seven decimal places.

1. 1Discount factors, always first

   v₁ = 1 / 1.05 = 0.9523810
   1.056² = 1.1151360, so v₂ = 0.8967516
   1.06³ = 1.1910160, so v₃ = 0.8396193

2. 2The one-year forward rates

   1 + f(1,2) = 1.1151360 / 1.05 = 1.0620343, so f(1,2) = 6.20343%
   1 + f(2,3) = 1.1910160 / 1.1151360 = 1.0680455, so f(2,3) = 6.80455%

   The curve rises, so each forward rate sits above the spot rate of the same term — 6.20% against the two-year 5.60%, and 6.80% against the three-year 6.00%. That ordering is a free check on the arithmetic.

3. 3The check: the forwards must rebuild the long rate

   1.05 × 1.0620343 × 1.0680455 = 1.1910160

   which is 1.063 to seven decimal places. Three factors, none of which was used to produce the others, multiplying back to the figure they came from.

4. 4The par yields

   v₁ + v₂ + v₃ = 0.9523810 + 0.8967516 + 0.8396193 = 2.6887518
   y₃ = 100(1 − 0.8396193) / 2.6887518 = 16.0380700 / 2.6887518 = 5.9648761%

   And for two years, using only the first two discount factors:

   y₂ = 100(1 − 0.8967516) / (0.9523810 + 0.8967516) = 10.3248400 / 1.8491326 = 5.5836123%

   The one-year par yield is 100(1 − 0.9523810)/0.9523810 = 5.0000000%, which is the one-year spot rate exactly, as it must be.

5. 5The second check: price the par bond

   A three-year bond paying a coupon of 5.9648761 per 100 nominal, discounted on the spot curve, must come to exactly 100.

   5.9648761 × 2.6887518 + 100 × 0.8396193 = 16.0380700 + 83.9619300 = 100.0000000

   Exactly par. The same calculation on the two-year coupon of 5.5836123 also returns 100.0000000.

6. 6The third check: par yield as a weighted average of forwards

   The weights are each discount factor over their total:

   0.9523810 / 2.6887518 = 0.3542093
   0.8967516 / 2.6887518 = 0.3335197
   0.8396193 / 2.6887518 = 0.3122710
   (they sum to 1.0000000)
   0.3542093 × 5.00000 + 0.3335197 × 6.20343 + 0.3122710 × 6.80455 = 5.9648761%

   The same 5.9648761% to seven decimal places, from a completely different calculation. And the reason the par yield lies below the three-year spot rate is now visible in the weights: a third of the answer is the first year's 5.00%, and that drags the average down.

7. 7Putting the three curves side by side

   Term	Spot rate	One-year forward starting then	Par yield
   1 year	5.0000%	5.0000%	5.0000%
   2 years	5.6000%	6.2034%	5.5836%
   3 years	6.0000%	6.8046%	5.9649%

   Three columns, one curve. On a rising curve the forwards are always the highest, the par yields the lowest, and the spot rates in between — and they all agree at one year, where there is nothing to average. At three years the forward rate and the par yield are 84 basis points apart. Quoting the wrong one of the three is an error of that size here, which on a thirty-year liability is a great deal of money.

8. 8Using it: a three-year 8% bond

   P = 8 × 2.6887518 + 100 × 0.8396193 = 21.5100144 + 83.9619300 = 105.4719

   It trades above par because its 8% coupon beats the 5.9649% the curve demands. The premium of 5.47 is the present value of the excess coupon: (8 − 5.9648761) × 2.6887518 = 5.4719, the same figure, which is the fastest way to price any bond once the par yield is known.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

• Utility theory and risk aversion: non-satiation, A(w) and R(w)
• Measures of investment risk: variance, semi-variance, VaR and Tail VaR
• CAPM and the security market line: beta and the market price of risk
• Ito's lemma explained term by term, and what it does to a share price
• The chain ladder method: estimating outstanding claims from a run-off triangle
• The Bornhuetter-Ferguson method: the chain ladder with a second opinion
• Binomial option pricing: where the risk-neutral probability comes from
• The Black-Scholes formula and put-call parity, read term by term
• Ruin theory: the adjustment coefficient and Lundberg bound explained
• Mean-variance portfolio theory: the efficient frontier and why diversification works
• Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
• The Merton model: company equity as an option, and where a credit spread comes from
• The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
• The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
• The average cost per claim method: developing numbers and severity separately
• Valuing a maturity guarantee by simulation, and checking it against the formula
• The Efficient Markets Hypothesis: the three forms, and how each one is tested
• Single-index and multifactor models: getting a covariance matrix out of a few betas
• Excess-of-loss reinsurance: the retained claim, and choosing the retention
• State price deflators: pricing with real-world probabilities
• What makes a risk insurable, and what to do when it almost is
• Two-state and intensity credit models: backing a default rate out of a bond price
• The martingale representation theorem, and why hedging is possible at all

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.