BANK3011 · Bank Financial Management
Interest Rate Risk II: The Duration Model
BANK3011 Bank Financial Management at the University of Sydney teaches the duration model as the second half of interest-rate risk: where the repricing/GAP model measures the hit to a bank's earnings, the duration model measures the hit to value — the market value of assets, liabilities and equity. Macaulay duration is the present-value-weighted average life of a security's cash flows, and modified duration turns it into a price-sensitivity rule ΔP = −MD·ΔR·P. Scaled up to a whole balance sheet it becomes the leverage-adjusted duration gap DGAP = D_A − k·D_L, which drives the change in the market value of equity ΔE = −DGAP·A·[ΔR/(1+R)]. All of these formulas sit on the provided exam Formula Sheet, so the marks come from clean, correctly-signed application rather than recall.
What this chapter covers
- 011. Value vs earnings — why duration exists where the repricing/GAP model is blind, and why every input is a market value
- 022. Macaulay duration D — the present-value-weighted average time to receive a bond's cash flows, in years
- 033. Duration features — zero-coupon → D = maturity; coupon bonds → D < maturity; higher coupon or yield → lower D; consol → D = (1+R)/R
- 044. Modified and dollar duration — MD = D/(1+R), dollar duration = MD × P
- 055. Price sensitivity — ΔP = −D·[ΔR/(1+R)]·P = −MD·ΔR·P for a small parallel yield move
- 066. Portfolio duration — D_p = Σ(MV_i·D_i)/Σ MV_i turns a whole asset or liability side into one number
- 077. Leverage-adjusted duration gap and equity — DGAP = D_A − k·D_L and ΔE = −DGAP·A·[ΔR/(1+R)], with k = L/A
- 088. Immunisation and convexity — set DGAP = 0 to insulate equity, and treat duration as a linear approximation that convexity corrects
Price, Macaulay duration and the price change for a small yield move
- +1List the cash flows: at t=1 you receive the $5,000 coupon; at t=2 you receive the $5,000 coupon plus the $100,000 face = $105,000.
- +1Discount at 7% to get the price. PV₁ = 5,000/1.07 = 4,672.90; PV₂ = 105,000/1.07² = 105,000/1.1449 = 91,711.07; so P = 4,672.90 + 91,711.07 = $96,383.96.
- +1Weight each time by the present value that arrives then and sum: 1×4,672.90 + 2×91,711.07 = 4,672.90 + 183,422.14 = 188,095.03.
- +1Macaulay duration = weighted sum ÷ price = 188,095.03 / 96,383.96 = 1.952 years (note D < 2 years because the coupon pulls the weighted life forward).
- +1Modified duration MD = D/(1+R) = 1.952/1.07 = 1.824.
- +1Price change for a +25 bp rise: ΔP = −MD·ΔR·P = −1.824 × 0.0025 × 96,383.96 ≈ −$439 (a rate rise gives a price fall).
Key terms
- Macaulay duration (D)
- The present-value-weighted average time, in years, to receive a security's cash flows. It bundles the coupon and maturity effects into one number and, once modified, measures how sensitive the price is to a change in yield.
- Modified duration (MD)
- Macaulay duration divided by the yield factor, MD = D/(1+R). It is the approximate percentage price change per unit change in yield, so ΔP = −MD·ΔR·P.
- Dollar duration
- Modified duration multiplied by price, MD × P. It converts the percentage sensitivity into the dollar price change for a given yield move.
- Leverage-adjusted duration gap (DGAP)
- DGAP = D_A − k·D_L, where D_A and D_L are the market-value-weighted durations of assets and liabilities and k = L/A is the leverage ratio. It measures the bank's net interest-rate exposure in duration terms.
- Change in equity (ΔE)
- ΔE = −DGAP·A·[ΔR/(1+R)] gives the change in the market value of equity for a small parallel rate move. A positive gap means equity value falls when rates rise.
- Immunisation
- Neutralising interest-rate risk by matching durations rather than maturities. Setting the leverage-adjusted duration gap to zero (D_A = k·D_L) insulates the market value of equity from a small parallel rate shift.
- Convexity
- The curvature of the price–yield relationship that duration ignores. Because duration is a linear tangent to a convex curve, it overstates the price fall when yields rise and understates the gain when they fall; convexity is the correction for large moves.
- Consol (perpetuity) duration
- A consol pays a coupon forever, yet its duration is finite: D = (1+R)/R. At a 7% yield that is 1.07/0.07 ≈ 15.3 years.
Interest Rate Risk II: The Duration Model FAQ
What is the difference between the repricing model and the duration model?
The repricing (GAP) model measures how a rate move changes a bank's net interest income over a horizon, using book values. The duration model measures how the same rate move changes the market value of assets, liabilities and equity. One is an earnings view; the other is a value view, and BANK3011 tests both.
Why is a coupon bond's duration always less than its maturity?
Because you receive coupons before maturity, some of the bond's value arrives early. The present-value-weighted average time is therefore pulled forward, so D is below the final maturity. Only a zero-coupon bond, whose single cash flow lands at maturity, has D equal to its maturity.
How do I compute the change in the market value of equity from a rate move?
First find the leverage-adjusted duration gap, DGAP = D_A − k·D_L with k = L/A. Then ΔE = −DGAP·A·[ΔR/(1+R)]. A positive gap plus a rate rise gives a negative ΔE — equity value falls — so watch the sign, the k term and the (1+R) denominator.
What does it mean to immunise a bank against interest-rate risk?
Immunisation means driving the leverage-adjusted duration gap to zero, so D_A = k·D_L. You can shorten asset duration, lengthen liability duration or change the leverage mix. Matching maturities is not enough — two portfolios with the same maturities can still have very different durations.
Why isn't the duration price estimate exact?
Duration is a linear approximation of a convex price–yield curve, so it is only accurate for small yield changes. It overstates the price fall on a rate rise and understates the gain on a rate fall; the difference is convexity, which becomes material for large shocks.
Can AI help me with the duration model?
Yes — ask Sia to walk through any duration model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains each move — weighting cash flows into D, scaling to modified duration, or signing ΔE off the duration gap — so you can then work the next problem yourself.
Studying with AI? Sia — free AI financial modeling tutor works through BANK3011 step by step.
Exam move
Build fluency in two moves. First, drill the single-bond routine until it is automatic: cash flows → price → PV-weighted times → D → MD → ΔP, always shown as separate lines. Second, practise the balance-sheet routine: compute k, net the durations into DGAP, sign ΔE, and state how you would immunise. Because every formula is on the provided Formula Sheet, spend your revision time on correct signs, the k leverage term and the (1+R) denominator rather than on memorisation, and keep 'convexity' ready as the one-word answer for why the estimate is not exact.
