Homework 5 — bring it to the next session

No session of mine follows this one, so this homework is not corrected in class. Send a photo or a scan of your working by email by Wednesday 28 October, 23:59, and it comes back with written comments before the midterm deadline.

The last homework of my half. Lighter than usual on purpose: Reading Week is for the midterm, which is due Sunday 1 November on Turnitin.

Part 1 on paper, Part 2 typed. Expect about two hours.

Part 1 — Drill set (paper, show all work)

H1 (pricing). A 6-year bond, 5% annual coupon, face €100. (a) Its price at a yield of 7%. (b) At 5%. (c) At 3%. (d) State in one line the rule connecting coupon, yield and price that your three answers illustrate.

H2 (zero-coupon). A zero pays €100 in 12 years, yield 4%. (a) Its price. (b) The price if the yield rises to 5%, and the percentage change. (c) Why is that percentage change larger than for a coupon bond of the same maturity?

H3 (duration). For the 6-year 5% bond at a 7% yield (H1a): (a) Macaulay duration, with the table of weights. (b) Modified duration. (c) The estimated price change for a 75 basis point rise in yield.

H4 (duration, the rules). Without computing, say what happens to duration when, other things equal: (a) maturity lengthens (b) the coupon rises (c) the yield rises One clause of justification each.

H5 (convexity). A bond has P = 100, D_mod = 6.50, C = 55.0. (a) Estimate the price after a 2% rise in yield, using duration only. (b) Now with the convexity term as well. (c) Which of the two estimates is closer to the truth, and in which direction was the first one wrong?

H6 (portfolio). A reserve holds €300,000 of a bond with D_mod = 8.1 and €700,000 of one with D_mod = 2.4. (a) The portfolio’s modified duration. (b) The estimated change in value if yields fall by 0.5%. (c) The Board wants the portfolio’s duration down to 3.0 without selling either bond entirely. Which way must the weights move? A sentence, no algebra required.

H7 (immunisation — beyond the syllabus). Arno Bikes must pay €2,000,000 in exactly 5 years. It can buy 3-year and 9-year zero-coupon bonds, both yielding 4%. (a) How much must it set aside today? (b) The share in each zero that makes the portfolio’s duration 5 years, the euros in each, and what each zero pays at maturity. (c) The day after the purchase, yields jump to 5%. The value of the portfolio at year 5 — the 3-year zero’s payment reinvested for two years, the 9-year zero with four years still to run — against the €2,000,000 owed. (d) In one sentence: why must the treasurer look at this portfolio again next year, even if rates never move?

Part 2 — One page for the Board (200–300 words, typed)

Arno Bikes keeps €1M in the bond reserve described in H6. The Board has read that rates may rise by a point over the next year and has asked: should we be worried?

Answer with: the loss duration predicts, one sentence on why that figure is an estimate and not a fact, what convexity does to it, and what you would change about the portfolio if the Board’s view is right. End with what you would need to see to change your recommendation.

Self-check before submitting

Every price shows the annuity part and the redemption part separately.
The duration table has a weights column that sums to 1. If it does not, something is wrong before the duration is.
Modified duration is distinguished from Macaulay duration everywhere, including in the units.
The convexity term is ½ C (Δy)² — the half and the square are both easy to drop and both change the answer.
In H7 the duration you match is Macaulay, in years, against the date of the payment — not the modified duration.

And then: the midterm

Due Sunday 1 November, 23:59, on Turnitin; presentations in the week of 2 November. Everything in weeks 1–3 was aimed at its function study, and the briefing in week 2 is its specification. Reading Week has one optional online office hour — bring a draft, not a blank page.