Week 5 — Drills
Exam format. Show all work. Core items are for everyone; stretch items for whoever finishes early. Solutions at the end, worked the way an examiner wants to see them.
Running case: Arno Bikes holds a small bond portfolio as a cash reserve, and is asked by the Board whether rising rates should worry it.
Unless stated otherwise: annual coupons, face value €100, prices to the cent.
A. Pricing a bond
D1 (core). A 5-year bond, 4% annual coupon, face €100, yield 6%. (a) Write the price as a sum. (b) Compute it. (c) Is it at a premium or a discount, and why — in one sentence, without arithmetic?
D2 (core). The same bond at a yield of 4%. Then at 2%. (a) Both prices. (b) State the rule that connects coupon, yield and price.
D3 (core — resit form). Select the correct alternative and justify in one line. A bond trades above par. Therefore: (i) its coupon is above its yield (ii) its yield is above its coupon (iii) it must be close to maturity
D4 (core). A zero-coupon bond pays €100 in 8 years. The yield is 5%. (a) Its price today. (b) If the yield rises to 6%, the new price and the percentage change.
B. Duration
D5 (core). For the 5-year 4% bond priced at a 6% yield (D1): (a) Compute Macaulay duration, showing the weight of each cash flow. (b) Convert it to modified duration. (c) Estimate the price change if the yield rises by 50 basis points.
D6 (core). A 10-year bond with a 5% coupon trades at par (yield 5%). Its modified duration is 7.72. (a) Estimate the new price if the yield rises to 6%. (b) The actual price at 6% is €92.64. Compare, and say which way the estimate errs.
D7 (stretch). Without computing anything, order these three by duration, longest first, and justify each comparison in one clause: (i) a 10-year zero-coupon bond (ii) a 10-year 8% coupon bond (iii) a 3-year 8% coupon bond
C. Convexity
D8 (core). Using the bond in D6 (P = 100, D_mod = 7.72, convexity C = 75.0): (a) Estimate the price at a yield of 6% using duration and convexity. (b) Compare with the actual €92.64 and with the duration-only estimate from D6.
D9 (core — the Board sentence). In two sentences, no formulas: what does convexity buy the holder of a bond, and why does a risk manager care about it?
D10 (stretch). Two bonds have the same duration; one has higher convexity. Rates move sharply, direction unknown. Which would you rather hold, and why?
The session closes with a mock of the final exam’s calculus question, thirty minutes in exam conditions.
D. Putting it together
D11 (core). Arno Bikes holds €400,000 of the D6 bond (at par) and €600,000 of a 2-year bond with modified duration 1.90. (a) The portfolio’s modified duration. (Weight each by market value.) (b) The estimated loss in euros if yields rise by 1% across the board. (c) One sentence to the Board.
E. Immunisation — beyond the syllabus
The step after duration: using it to make a payment safe. A zero’s Macaulay duration is its maturity; yields are flat and move once, the day after purchase.
D12 (core). Arno Bikes must pay a supplier €500,000 in exactly 3 years. It can buy 1-year and 5-year zero-coupon bonds, both yielding 4%. (a) How much must it set aside today? (b) What share goes into each zero so that the portfolio’s duration is 3 years? (c) The euros in each, and the amount each zero pays at maturity.
D13 (core). The day after the purchase in D12, yields jump to 5% and stay there. (a) The 1-year zero’s payment, reinvested at 5% until year 3: what is it worth then? (b) The 5-year zero, still two years from maturity at year 3: what is it worth then? (c) The total against the €500,000 owed. Then repeat (a)–(c) with yields falling to 3% instead, and say what you notice.
D14 (stretch). Go back to the slide: €1M due in 4 years, flat yield 5%, money set aside today €822,702.47. Replace the 6-year zero with a 6-year 5% coupon bond at par. (a) Its Macaulay duration, with the table of weights. (b) The share in the 2-year zero that makes the portfolio’s duration 4 years, and the euros in each. (c) One year later, yields unchanged: the payment is 3 years away, and the coupon bond is now a 5-year 5% bond at par, with Macaulay duration 4.55. By how much did each duration fall? What must the treasurer do, and how often?
Solutions
D1. (a) P = Σ(t=1..5) 4/(1.06)^t + 100/(1.06)^5. (b) Annuity part: 4 × [1 − (1.06)⁻⁵]/0.06 = 4 × 4.212364 = 16.85. Face: 100/(1.06)⁵ = 74.73. P = **€91.58**. (c) A discount: the bond pays 4% on its face while the market demands 6%, so it must be cheaper than face to make up the difference.
D2. (a) At 4%: coupon equals yield, so P = **€100.00** (par). At 2%: 4 × [1 − (1.02)⁻⁵]/0.02 + 100/(1.02)⁵ = 4 × 4.713460 + 90.573 = 18.854 + 90.573 = **€109.43**. (b) Coupon > yield → premium; coupon = yield → par; coupon < yield → discount. Price and yield move in opposite directions, always.
D3. (i) its coupon is above its yield. Investors pay more than face only to obtain a coupon better than the going rate.
D4. (a) P = 100/(1.05)⁸ = 100/1.477455 = **€67.68**. (b) P = 100/(1.06)⁸ = 100/1.593848 = **€62.74**; change = (62.74 − 67.68)/67.68 = **−7.30%**.
D5. (a) With P = 91.58:
| t | CF | PV = CF/(1.06)^t | weight w = PV/P | t · w |
|---|---|---|---|---|
| 1 | 4 | 3.77 | 0.0412 | 0.0412 |
| 2 | 4 | 3.56 | 0.0389 | 0.0777 |
| 3 | 4 | 3.36 | 0.0367 | 0.1100 |
| 4 | 4 | 3.17 | 0.0346 | 0.1384 |
| 5 | 104 | 77.71 | 0.8486 | 4.2432 |
| 91.58 | 1.0000 | 4.611 |
Macaulay duration = 4.61 years. It is the average time to being repaid, weighted by present value — which is why it is under five even though the bond runs five years. (b) D_mod = D/(1 + y) = 4.611/1.06 = **4.350**. (c) ΔP/P ≈ −D_mod × Δy = −4.350 × 0.005 = −2.18%, i.e. about −€1.99, giving roughly €89.59.
D6. (a) ΔP/P ≈ −7.72 × 0.01 = −7.72% → P ≈ 100 × 0.9228 = **€92.28**. (b) The actual price is €92.64, so the estimate is €0.36 too low. Duration alone overstates the loss. It will also understate the gain when yields fall: the straight line lies below the curve on both sides.
D7. (i) > (ii) > (iii). (i) beats (ii): a zero pays everything at the end, so its weighted average time is the full 10 years, while coupons pull (ii)’s average forward. (ii) beats (iii): same coupon, but money returned over ten years instead of three.
D8. (a) ΔP/P ≈ −D_mod Δy + ½ C (Δy)² = −7.72(0.01) + ½(75.0)(0.0001) = −0.0772 + 0.00375 = −0.07345 → P ≈ 100 × 0.92655 = **€92.66**. (b) Duration alone gave €92.28, out by 36 cents below the truth. Adding convexity gives €92.66, out by 2 cents above it. The second term recovers almost all of the curvature the straight line missed; the 2 cents that remain belong to the next term again, and no one on a trading floor cares about them.
D9. Convexity means the price curve bends away from the straight-line estimate in the holder’s favour: when yields fall the bond gains more than duration predicts, and when they rise it loses less. A risk manager cares because a duration-only report overstates losses and understates gains, and because between two bonds of equal duration the more convex one is worth more when rates move a long way.
D10. The more convex one. With equal duration the two behave the same for a small move, but for a large move in either direction the convex bond does better: more upside, less downside. That asymmetry is worth paying for, and in the market you usually do.
D11. (a) Weights 400/1,000 = 0.4 and 600/1,000 = 0.6. D_portfolio = 0.4(7.72) + 0.6(1.90) = 3.088 + 1.140 = **4.228**. (b) ΔV ≈ −4.228 × 0.01 × €1,000,000 = **−€42,280**. (c) “A one-point rise in yields costs the reserve about €42,000, a little over 4% of it; about three quarters of the exposure sits in the ten-year holding, and shortening that would cut it roughly in half.”
D12. (a) PV = 500,000/(1.04)³ = 500,000/1.124864 = **€444,498.18**. (b) Each zero’s duration is its maturity. With a share w in the 1-year zero: 1·w + 5(1 − w) = 3 → 5 − 4w = 3 → **w = ½**. Half in each. (c) €222,249.09 in each. The 1-year zero pays 222,249.09 × 1.04 = **€231,139.05** in year 1. The 5-year zero pays 222,249.09 × (1.04)⁵ = **€270,400.00** in year 5.
D13. (a) 231,139.05 × (1.05)² = 231,139.05 × 1.1025 = **€254,830.80**. (b) 270,400.00/(1.05)² = 270,400.00/1.1025 = **€245,260.77**. (c) 254,830.80 + 245,260.77 = **€500,091.57**, i.e. €91.57 more than the €500,000 owed. At 3%: 231,139.05 × 1.0609 = 245,215.42 and 270,400.00/1.0609 = 254,877.93, total €500,093.35, €93.35 more. In both directions the reinvestment effect and the price effect cancel, and what is left over is a small gain: that is convexity, on the holder’s side again. Without the match — all in the 1-year zero — a fall to 3% would leave 444,498.18 × 1.04 × 1.0609 = €490,430.84, about €9,570 short.
D14. (a) At par, P = 100:
| t | CF | PV = CF/(1.05)^t | weight w = PV/P | t · w |
|---|---|---|---|---|
| 1 | 5 | 4.76 | 0.0476 | 0.0476 |
| 2 | 5 | 4.54 | 0.0454 | 0.0907 |
| 3 | 5 | 4.32 | 0.0432 | 0.1296 |
| 4 | 5 | 4.11 | 0.0411 | 0.1645 |
| 5 | 5 | 3.92 | 0.0392 | 0.1959 |
| 6 | 105 | 78.35 | 0.7835 | 4.7012 |
| 100.00 | 1.0000 | 5.330 |
Macaulay duration = 5.33 years, under six because the coupons come back earlier. (b) With a share w in the 2-year zero: 2w + 5.33(1 − w) = 4 → w = (5.33 − 4)/(5.33 − 2) = 1.33/3.33 = **0.40**. About 40% in the zero (€328,509) and 60% in the coupon bond (€494,193). Less in the short bond than on the slide, because the long bond is now shorter than six years. (c) The payment’s date fell by exactly 1 year (4 → 3); the 2-year zero’s duration too (2 → 1). The coupon bond’s fell by only 5.33 − 4.55 = **0.78**. So even with rates unchanged, durations drift apart as time passes and the match is lost. The treasurer must re-compute the weights and rebalance, in practice at least once a year and after any large rate move: immunisation is a routine, not a one-off purchase.