FNCE30001 Investments
Term Structure of Interest Rates
There is not one interest rate but a whole schedule of them — one spot (zero) rate for each horizon. Plotting those rates against maturity gives the yield curve, whose level prices every risk-free cash flow and whose shape (normal, flat or inverted) encodes the market's view of where rates are heading. From the spot curve you can lock in a forward rate today for a future year, pinned by no-arbitrage: rolling a short bond then reinvesting must give the same wealth as buying the long bond. Two hypotheses — pure expectations versus a liquidity premium — explain why the curve has the shape it does. The chapter then turns to managing interest-rate risk: duration measures how far a bond's price moves when yields shift, modified duration turns it into a percentage sensitivity, and convexity is the second-order correction that always helps the holder. Finally, immunisation sets a portfolio's duration equal to its horizon so price and reinvestment risk offset. This is Week 3 — heavily weighted and full of avoidable errors, so slow down.
What this chapter covers
- 013.1 Spot (zero) rates and the shape of the yield curve
- 02Forward rates and the no-arbitrage link (1+z₂)² = (1+z₁)(1+f)
- 03Forward-rate arbitrage: trade toward the implied forward
- 043.2 Expectations vs liquidity-premium hypotheses
- 05Duration, modified duration and what duration ranks
- 06Convexity — the second-order correction that always helps
- 07Immunisation by duration matching
Worked example: the implied one-year forward rate
- +1Identify. Term-structure / forward-rate type — the spot curve fixes the fair forward by no-arbitrage; the forward is not a forecast you choose.
- +1(a) Set up the no-arbitrage link: (1 + z₂)² = (1 + z₁)(1 + f), so 1 + f = (1.05)² / 1.04 = 1.1025 / 1.04 = 1.0601.
- +1Compute: f = 6.01%. Sanity check — on an upward curve the forward sits above both spots (6.01% > 5% > 4%), which it does.
- +1(b) Compare: the market quote 6.50% exceeds the fair 6.01%, so the forward loan is over-priced.
- +1Build the trade: lend at the rich market forward (receive 6.50%) and replicate the funding synthetically from the spot curve at only 6.01% (borrow 1-year, lend 2-year). Net 6.50% − 6.01% = 0.49% locked in, riskless.
Key terms
- Spot (zero) rate
- The single annually-compounded yield on a risk-free zero-coupon bond maturing in T years: z = (Par/P)1/T − 1. It is the average rate per year over the whole horizon [0,T]; the set of all spot rates is the yield curve.
- Forward rate
- The interest rate, agreed now, that will apply between two future dates. It is pinned by the spot curve through no-arbitrage, e.g. (1+z₂)² = (1+z₁)(1+f). A quoted forward that differs from the spot-implied forward is an arbitrage.
- Expectations vs liquidity-premium hypothesis
- Under the pure expectations hypothesis the forward equals the expected future spot (no premium), so curve slope is pure expectations. Under the liquidity-premium hypothesis investors demand a positive, maturity-increasing premium, so a gently upward curve is the normal resting shape even with flat expected rates.
- Modified duration
- D* = D/(1+y), where D is Macaulay duration. It gives the first-order percentage price change for a yield move: %ΔP ≈ −D*Δy. Longer maturity and lower coupon raise duration; a zero's duration equals its maturity, and the longest-duration bond has the biggest percentage price move for a given parallel shift.
- Convexity
- The second-order curvature of the price-yield relationship. The true price-yield curve lies above the straight duration tangent on both sides, so the convexity term +½X(Δy)² is always positive: it cushions the loss when yields rise and adds to the gain when they fall. Of two bonds with the same duration, the more convex one is always worth more to the holder.
- Immunisation
- Setting a portfolio's duration equal to the investment horizon, so price risk and reinvestment risk offset for small parallel yield shifts. It is not set-and-forget: as time passes and yields move, durations drift and the portfolio must be rebalanced to keep D_p equal to the remaining horizon.
Term Structure of Interest Rates FAQ
What is the difference between a spot rate and a forward rate?
The spot rate z is the average rate over the whole horizon [0,T]; the forward rate f is the marginal rate for a single future window [t,T]. They are linked by the no-arbitrage chain (1+z_T)T = the product of (1 + each forward). Discounting a year-2 cash flow at z₁ or at f alone is wrong — use z₂, or compound z₁ with f. A correct forward always sits between consecutive spots and beyond: on an upward curve, f > z₂ > z₁.
How do I use duration to predict a price change?
First-order: %ΔP ≈ −D*Δy, where D* = D/(1+y). This is the tangent line and is accurate only for small parallel shifts. For larger moves add the convexity term +½X(Δy)², which is always positive. So duration alone is too pessimistic on the upside (it overstates the loss when yields rise) and too optimistic on the downside (it understates the gain when yields fall) — convexity corrects both.
Why does convexity always help the holder?
For an option-free bond the price-yield curve is convex (bowed toward the origin), so it sits above the straight duration tangent on both sides. That means when yields fall the price rises more than duration predicts, and when yields rise it falls less. Of two bonds with the same duration, the more convex one gains more and loses less — so it trades at a higher price / lower yield. More convexity is never bad for a long holder.
How does immunisation protect a fund against rate moves?
Set the portfolio's duration equal to the liability horizon by blending bonds of different durations (solve D_p = Σw_i D_i = horizon for the weights). With duration matched, a small parallel yield rise that cuts the bond price is offset by being able to reinvest coupons at the higher rate, and vice versa. It only holds for small, parallel shifts, and durations drift over time, so the portfolio must be rebalanced as the horizon shrinks.
Exam move
This is the heaviest-weighted single week, so bank the mechanics. Build forwards from spots with the no-arbitrage chain and always sanity-check that the forward sits beyond both spots on an upward curve; if a market forward differs from the implied one, state the arbitrage trade explicitly (lend dear, replicate cheap). For interest-rate risk, run the duration-then-convexity sequence: compute D* = D/(1+y), apply −D*Δy, then add +½X(Δy)² and note that the convexity term is always positive. For immunisation, set portfolio duration to the horizon and remember it needs rebalancing. The recurring traps — spot vs forward, duration as only a linear approximation, convexity always helping — are exactly what the exam-trap callouts flag, so rehearse the one-line answer to each.
