Risk Management puzzles, solved step by step

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All topicsCapital and leverage6Compounding and drawdowns8Correlation and diversification8Counterparty exposure and collateral7Credit risk arithmetic10Duration and rates7Liquidity and balance sheet7Logic, estimation and brainteasers7Operational loss and fraud7Options and Greeks7Probability and base rates8Statistics and estimation10VaR and expected shortfall8
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  1. 006Write NPV as the product of two vectors. The cash flows are minus 100, 30, 40, 50 and 20 in years 0 to 4, and the discount rate is 10%. What is the NPV?Duration and ratesWarm upMoody'sNew York · 2018

    Try it first

    Which operation turns the two vectors into the NPV?

    Show the worked solution

    NPV is the dot product of the cash flow vector and the discount factor vector, and here it is about 11.56. The discount factors at 10% are 1, 0.909, 0.826, 0.751 and 0.683. Multiplying term by term gives minus 100, 27.27, 33.06, 37.57 and 13.66, which sum to 11.56. A positive NPV means the project earns more than 10%.

    Why is NPV a dot product at all?

    Think of a grocery bill. One list holds the quantity of each item, another holds each item's price, and the bill is quantity times price for each line, added up. NPV has the same shape: one vector holds the cash flows, the other holds what one rupee in each year is worth today, and the NPV is the sum of their products. Writing it this way separates the project, which is the cash flow vector, from the market, which is the discount factor vector. Change the rate and only the second vector changes.

    The relationship
    NPV=c⋅d=∑t=04ct dt,dt=1(1+r)t\text{NPV} = \mathbf{c} \cdot \mathbf{d} = \sum_{t=0}^{4} c_t\, d_t, \qquad d_t = \frac{1}{(1+r)^t}
    cthe cash flow vector, year 0 to year 4
    dthe discount factor vector, one entry per year
    rthe discount rate, 10%
    What it says in wordsMultiply each cash flow by the value today of one rupee in that year, and add the results.
    Multiply the two vectors term by term, then add the productsYear 0Year 1Year 2Year 3Year 4Cash flow-10030405020xDiscount factor1.0000.9090.8260.7510.683=Present value-100.0027.2733.0637.5713.660-100+27.27+33.06+37.57+13.66+11.56Year 0Year 1Year 2Year 3Year 4NPVRunning sum
    Multiplying the cash flows by the discount factors year by year gives present values of minus 100, 27.27, 33.06, 37.57 and 13.66, and adding them from minus 100 upward reaches an NPV of 11.56.

    What does the interviewer want to hear beyond the number?

    The thought process was part of the question, so say it in order. First build the discount factor vector from the rate, then take the dot product, then sanity check the sign and size. The undiscounted inflows are 140 against 100 out, so a positive but much smaller NPV is expected once four years of 10% are taken out. And in a spreadsheet the same idea is one SUMPRODUCT of two ranges, which is why the vector form is how a model is usually built.

    Then give the extension that shows range. With a term structure of rates, only the discount factor vector changes: each entry uses its own year's rate. With several scenarios, stack the cash flow vectors into a matrix and one matrix multiplication gives every scenario's NPV at once. The limitation is that the vector form assumes the cash flows are known; uncertain cash flows need expected values or scenarios first.

    Where candidates lose it

    Candidates reach for the NPV formula and start adding fractions, which gets the number but misses the question. The interviewer asked for two vectors precisely to see whether you can separate what the project pays from what time is worth.

    The other slip is discounting year 0. The first discount factor is 1; the minus 100 is already in today's money.

    What the interviewer asks next

    • How would you write the IRR condition using the same two vectors?
    • The rate for year 1 is 8% and for later years 10%. What changes in the vector form?
    • How would you compute the NPV for 1,000 cash flow scenarios in one operation?

    Asked at Moody's, Analytics, New York, 2018 (Wall Street Oasis): Construct an NPV formula using 2 vectors and show me your thought process.

  2. 031A floating rate note pays a coupon that resets every three months to the market rate, and it matures in 7 years. Roughly what is its interest rate duration?Duration and ratesWarm upTreasury and ALM

    Try it first

    Pick the duration before you reason it out.

    Show the worked solution

    About 0.25 years at most, and on average about an eighth of a year. Every coupon after the next one resets to the market rate, so a change in rates today is passed straight into future coupons and leaves their value unchanged. Only the next coupon is fixed, so the note behaves like a three-month bill: modified duration of about 0.25 just after a reset, falling towards zero before the next one.

    Why does maturity not drive a floater's rate risk?

    Think of renting a flat with the rent reset to the market every quarter. If rents jump tomorrow, your lease is not a bargain or a burden for more than a few months, because the next reset catches up. A bond loses value when rates rise only because its cash flows are fixed below the new market rate; a floater's cash flows are not fixed beyond the next reset. At each reset date the note is worth par again, provided the issuer's credit has not changed, so for rate purposes it is a claim to par plus one known coupon, three months away.

    Only the next coupon is locked in; everything after it resetsprincipalnext coupon: fixed today27 more coupons, each reset to the market rate0y1y2y3y4y5y6y7yrate risk ends here, at the next resetModified durationFloater, just after reset0.25 yearsFloater, quarter averageabout 0.12 years7-year fixed 8% bond5.32 yearsA 1 point rate rise costs the floater about 0.25 per 100 and the fixed bond about 5.32
    A 7-year quarterly floater has only its next coupon fixed, so its rate duration is about 0.25 years just after a reset, while a 7-year fixed 8% bond has a modified duration of about 5.32 years and loses about twenty times as much for a one point rise in rates.

    How do you check the number quickly?

    Right after a reset, the note is worth the present value of par plus one coupon, received in a quarter. That is a single cash flow a quarter away, so Macaulay duration is 0.25 years and modified duration is 0.25 divided by 1.02, about 0.25. A day before the next reset it is almost zero. Compare a 7-year fixed bond paying 8% quarterly at par: its modified duration is 5.32, so a one point rate rise costs it about 5.3 per 100 against about 0.25 for the floater.

    Now say where the 7 years still matter. The coupon is the market rate plus a fixed quoted marginThe fixed spread over the reference rate that a floating rate note pays, set at issue and unchanged for the life of the note.. If the issuer's credit worsens and investors demand a wider spread, that fixed margin is too low for all 28 remaining quarters, so the note's spread duration is close to a fixed bond's, several years. A treasury desk that files floaters under no rate risk and forgets spread risk has the right answer to the wrong question.

    Where candidates lose it

    The instinct is to answer 7 years because the note matures in 7 years, or about 5 because that is what a 7-year bond usually carries. Both confuse maturity with the length of time cash flows are fixed.

    The overcorrection is saying zero. Until the next reset one coupon is locked, and the credit spread is locked for the whole life. Give the rate answer, then name spread duration unprompted.

    What the interviewer asks next

    • What is the floater's spread duration, roughly, and why?
    • An inverse floater pays 16% minus the market rate. What is its duration?
    • How would you hedge a fixed-rate loan book funded with floating-rate deposits?