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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.