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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. 056Two stocks both have a 10% cost of equity. One grows its dividends at 8% a year, the other at 2%. Using the Gordon growth model, how much does each price fall if the discount rate rises by 50 basis points?Duration and ratesCoreBlackRockNew York · 2026

    Try it first

    Which stock falls more when the discount rate rises half a point?

    Show the worked solution

    The 8% grower falls 20%; the 2% grower falls about 5.9%. Under Gordon growth, price is next year's dividend over r minus g. For the fast grower that gap widens from 2% to 2.5%, so the price falls to 0.02 over 0.025, or 80% of what it was. For the slow grower the gap goes from 8% to 8.5%, and the price keeps 0.08 over 0.085 of its value.

    Why does the fast grower react so much more?

    Think of two ways to be paid Rs 10 lakh: most of it next year, or a trickle that grows for decades. If someone doubles the rate at which you discount the future, the trickle loses far more, because most of its money is far away. A fast-growing dividend is that trickle: its value sits in cash flows many years out. A stock whose value rests on distant cash flows behaves like a long bond, so the same rise in the discount rate cuts its price far more.

    Same 50 basis point rise, very different price falls4060801001201409.5%10.0%10.5%11.0%11.5%12.0%Discount rate (cost of equity)both 100 at 10%g = 2%: 94.1, down 5.9%g = 8%: 80.0, down 20.0%Equity duration = 1 / (r - g)50 years against 12.5 years
    Both stocks are priced at 100 with a 10% cost of equity. A rise to 10.5% takes the 8% grower to 80, a fall of 20%, and the 2% grower to 94.1, a fall of 5.9%, because the fast grower's price rests on a gap of only 2 points between r and g.

    How do you turn this into a duration number?

    Differentiate the price with respect to r and divide by price: the answer is 1 over (r minus g). That gives the fast grower an equity duration of 50 years and the slow grower 12.5 years. Duration times 0.5% predicts falls of 25% and 6.25%; the exact falls are a little smaller, 20% and 5.9%, because the price curve bends, the same convexity a bond has.

    The relationship
    P=D1r−gDeq=−1PdPdr=1r−gP = \frac{D_1}{r-g} \qquad D_{\text{eq}} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g}
    D_1next year's dividend
    rthe cost of equity, 10%
    gthe constant dividend growth rate, 8% or 2%
    What it says in wordsAn equity's sensitivity to the discount rate is one over the gap between the discount rate and growth.

    The limitation is that Gordon growth assumes growth never changes and runs for ever, which exaggerates duration for a fast grower that will slow. The direction survives any sensible model: growth stocks carry more rate risk than stocks priced on today's cash.

    Where candidates lose it

    The trap is answering that both fall by about the same amount because the rate change is the same. The rate change is the same; the base it lands on is not. The fast grower's r minus g is a quarter of the slow grower's, so the same half point is four times as large relative to it.

    The second slip is quoting the duration answer, 25%, as exact. Give 20% and say duration overstates it because the price curve bends.

    What the interviewer asks next

    • What happens to each price if growth expectations for the fast grower fall to 7% at the same time?
    • Why might a portfolio of growth stocks behave like a long-duration bond fund?
    • What does equity duration mean for a pension fund that holds equities against long liabilities?

    Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? Technical and behavioural on VaR, market views and stock valuation.