Mutual Fund Mastery puzzles, solved step by step

Puzzles
100
Traced to a firm
29
Topics
13
Hard
30
Topic
All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 086A client needs her money in exactly three years. Why does a bond fund with a duration of about three years protect her whether interest rates rise or fall? Show it with Rs 1 crore and yields jumping from 7% to 8% on the first day.Bond maths and durationHardPIMCOLondon · 2022

    Try it first

    Yields jump to 8% on day one and the fund's value falls. Where does she stand at year three?

    Show the worked solution

    Because a fall in price and a rise in reinvestment income cancel at a horizon equal to the duration. If yields jump to 8%, Rs 1 crore at a three-year duration drops about Rs 2.74 lakh on day one, then compounds at 8% instead of 7%. By year three it is worth Rs 122.53 lakh against Rs 122.50 lakh had nothing moved. If yields fall to 6%, the gain today offsets the lower reinvestment rate in the same way.

    Why do rising rates both hurt and help a bond investor?

    Think of a tenant who has locked in a lease. If market rents rise, the lease itself is worth less to sell on, because a new buyer could get a better deal elsewhere, but any new space the tenant takes from now on costs more too. A bond investor faces the mirror image. When yields rise, bonds already held fall in price, but every coupon and every maturing bond is reinvested at the new, higher yield. One effect is immediate and the other builds over time. Which one wins depends only on how long you hold.

    Make it concrete with a fund that has a duration of exactly three years. Picture its Rs 100 lakh as two holdings of Rs 50 lakh each at 7%: paper that matures in one year, and paper that matures in five. The value-weighted average of one and five is three, so the fund's Macaulay durationThe average time until a bond portfolio pays back its cash, with each payment weighted by its share of present value. is 3.0 years. Left alone at 7%, both halves grow to Rs 61.25 lakh by year three, Rs 122.50 lakh in all.

    Rs 1 crore with a three-year duration: the gap to the no-change path-3-2-1+1+2+30012345Years after the rate moveRs lakh above or below Rs 100 lakh growing at 7%her date, year 3day one: -2.74 if yields go to 8%day one: +2.88 if yields go to 6%yields 8%yields 6%At year 3, yields at 8%1-year paper, rolled at 8%62.40 (+1.15)5-year paper, sold early60.12 (-1.13)Fund at year 3122.53Target, no change122.50Rs lakhAlone, each leg fails:all short, yields 6%: 119.10all long, yields 8%: 120.25
    A jump to 8% knocks Rs 2.74 lakh off the fund on day one, but reinvesting at 8% wins it back by about year 3.0, so at her three-year date she holds Rs 122.53 lakh against a plan of Rs 122.50 lakh; a fall to 6% gives the mirror image.

    What happens to each holding when yields jump to 8%?

    The one-year paper matures at Rs 53.50 lakh and is rolled for two more years at 8%, reaching Rs 62.40 lakh, Rs 1.15 lakh more than the plan. The five-year paper still has two years to run at year three and must be sold at an 8% yield, fetching Rs 60.12 lakh, Rs 1.13 lakh less than the plan. One leg carries reinvestment risk and the other carries price risk, and a duration equal to the horizon sets them against each other in equal size. Held alone, each leg would fail: all short paper with yields falling to 6% gives only Rs 119.10 lakh, and all five-year paper with yields at 8% gives only Rs 120.25 lakh.

    The relationship
    V3=P(y) (1+y)3P(8%)=53.501.08+70.131.085=97.26  ⇒  97.26×1.083=122.53V_3 = P(y)\,(1+y)^3 \qquad P(8\%) = \frac{53.50}{1.08} + \frac{70.13}{1.08^5} = 97.26 \;\Rightarrow\; 97.26 \times 1.08^3 = 122.53
    P(y)the fund's value today at yield y, Rs lakh
    53.50, 70.13the amounts the one-year and five-year paper pay at maturity
    V_3the value at year three if all cash is reinvested at y
    What it says in wordsWhatever the new yield, the fund's value at year three is today's repriced value grown at that yield, and at a horizon equal to the duration the two changes offset.

    Where does this protection stop working?

    In three places, and naming them is what the interviewer is after. Her horizon shrinks by exactly a year every year, but the fund's duration does not keep pace on its own: maturing paper has to be reinvested and coupons and rate moves shift the average, so the fund has to be rebalanced to stay matched. The offset assumes all yields move together; if short yields rise while long yields fall, the two legs no longer cancel. And the match only protects someone whose date is fixed: an open-ended fund whose manager keeps duration at three years forever is matched to nobody's date in particular. A product that runs down towards a fixed maturity close to her date does the matching more naturally. The small surplus either way, about Rs 0.02 lakh, comes from convexity and is a bonus, not the point.

    Where candidates lose it

    The common answer is that a shorter fund is safer, so she should use a liquid fund. That removes price risk but leaves her fully exposed to falling rates: if yields drop to 6% and stay there, three years of rolling short paper leaves her at Rs 119.10 lakh, about Rs 3.4 lakh short of plan.

    The other loss is saying duration is a measure of price sensitivity and stopping there. This question is about the second meaning of duration, a time horizon at which price risk and reinvestment risk balance. Say both meanings, then show the cancellation with one number.

    What the interviewer asks next

    • After one year, what has happened to the fund's duration and to her horizon, and what should change?
    • Short yields rise 1% and long yields fall 1% on the same day. Does the protection hold?
    • Why does the matched fund end slightly above plan whichever way yields move?

    Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration