Discounting cash flows

In depth

The value of a stream at any date

A stream can be valued at any comparison date $T$, not only today: each cash flow is moved to $T$, compounded if it comes earlier and discounted if it comes later, and the results are added: $V_T = \sum_t C_t (1 + r)^{T - t}$. Values at different dates are linked by a single factor: $V_T = \text{PV} \times (1 + r)^T$. That gives a useful check.

Valued at 1 March 2028 (quarter 4), the note’s payments are worth $150 \times 1.02^3 + 150 \times 1.02^2 + 150 \times 1.02 + 10{,}150 = 10{,}618.24$ dollars: what a holder would have if she reinvested each interest payment at 2% a quarter until the note matured. And $9{,}809.61 \times 1.02^4 = 10{,}618.24$, the same. Choosing the date where the question lives, today for a price and the end for a savings balance, saves work; it never changes the answer.

Matching the rate to the timing

The formulas assume each cash flow arrives at the end of a period, that time 0 is today, and that the rate is the rate per period of the same length as the spacing. With quarterly cash flows you need a quarterly rate. If the rate you are given is quoted differently, convert it first, as the lesson on effective annual rates shows: an APR compounded quarterly is divided by 4; an effective annual rate becomes a quarterly rate of (1 + EAR)^(1/4) − 1.

When cash flows fall at irregular dates, measure each date in years from today, $t_i$, and discount with the effective annual rate: $\text{PV} = \sum_i C_i \div (1 + \text{EAR})^{t_i}$. Worked example 2 does this for payments due in 0, 9, 12 and 24 months. Annual cash flows in a business plan are a simplification: the brewer’s sales arrive week by week, not in one lump at each year end. Treating them as year-end amounts slightly understates their value, and analysts accept that in exchange for a simpler model, or switch to monthly periods when the difference matters.

Signs and whose point of view

Every cash flow is someone’s inflow and someone else’s outflow. From Coastline’s side the note’s payments are inflows worth +$9,809.61; from Lantern’s side the same payments are outflows worth −$9,809.61. Pick one point of view for a whole problem and keep it. Financial calculators and spreadsheets use the same convention, money received positive and money paid negative, which is why a cost today must be entered with a minus sign.

Net present value in more depth

Three choices decide whether an NPV means anything.

1. The rate. The discount rate is the return the money could earn in its next-best use at similar risk: its opportunity cost. A riskier stream is discounted at a higher rate. For a business this is linked to its cost of capital, but a project riskier than the business as a whole deserves a higher rate. The rate on a loan used to pay for the project is not, by itself, the right rate: the loan rate prices the lender’s fairly safe claim, while the project’s cash flows are less certain.
2. The cash flows. Only incremental cash flows count: cash that changes because of the decision. Money already spent and unrecoverable is a sunk cost and is left out. NPV uses cash, not accounting profit: depreciation, for example, is not a cash flow.
3. The horizon. Every cash flow the decision causes belongs in the stream, including those at the end, such as the brewer’s resale value or the cost of removing equipment.

NPVs add up. The NPV of two independent projects together is the sum of their NPVs, and the NPV of a project financed at a fair market rate is not changed by the financing. A financing offer that comes only with the purchase and costs less than the market rate, such as a supplier’s cheap instalment plan, has a value of its own that adds to the project’s NPV; it never changes the rate used to discount the project’s cash flows. When options exclude each other, such as two different brewers for the same counter, choose the one with the highest NPV, provided it is positive. Plotting NPV against the discount rate gives an NPV profile. For a conventional project, one cost followed by inflows, the profile slopes down and crosses zero once, at the IRR.

Estimates are uncertain, so an NPV should be recalculated under other assumptions. If every year’s cash from the brewer were 10% lower than Ana expects, its NPV at 12% would fall from +$1,847.77 to −$578.13. Without the $2,500 resale value it would be only +$258.98. A positive NPV that disappears under a small change in the estimates is a close call, not a clear win.

Internal rate of return: definition and trial

The IRR is the rate that solves $0 = \sum_{t=0}^{n} C_t \div (1 + \text{IRR})^t$. Take a two-year investment of $1,000 that returns $600 at the end of each year. At 10% its NPV is $-1{,}000 + 600 \div 1.1 + 600 \div 1.21 = 41.32$; at 15% it is $-24.57$. The IRR lies between them. A straight line between the two points crosses zero at

With only two periods the equation is a quadratic and can be solved exactly: the IRR is 13.07%. Linear interpolation overshoots slightly, because the NPV profile curves (it is convex), so the straight line lies above it. Checking the NPV at 13.14% gives −$0.96, slightly negative, and a second interpolation closes the gap. With more periods there is no neat formula, and calculators and spreadsheets repeat this kind of step until the NPV is close enough to zero.

For a conventional project, an IRR above the required return and a positive NPV are the same statement, so the two rules agree. The IRR has limits. When cash flows change sign more than once, for example a project with a large clean-up cost at the end, the equation can have more than one IRR, or none. When choosing between options of different sizes or timing, the one with the higher IRR can have the lower NPV: over one year, when money costs 10%, a 50% return on $1,000 adds less value than a 20% return on $100,000. In those cases trust the NPV. F05 Corporate Finance treats these issues fully.

For a loan or a bond bought at a given price and held to maturity, the IRR has another name: its yield to maturity. A buyer who paid $9,809.61 for Lantern’s note on 1 March 2027 and received the four payments would earn exactly 2% a quarter, 8% a year compounded quarterly, because that is the rate used to price it. A buyer who paid $10,000 would earn the note’s own 6%.

Spreadsheets: NPV, IRR and the first cash flow

Spreadsheet functions save time but carry a trap. Excel’s NPV(rate, value1, value2, …) treats the first value as arriving one period from now, at the end of period 1. A cash flow at time 0 must be left out of the range and added separately: =C0 + NPV(r, C1:Cn). Microsoft’s own documentation says the first value must be added to the NPV result rather than included in the values. Put the time-0 cost inside the range and every cash flow is discounted one period too many, so the result is the true NPV divided by 1 + r: for the brewer, $1,649.80 instead of $1,847.77.

IRR(values), by contrast, expects the whole stream including the time-0 flow, and returns the rate per period. IRR() needs at least one negative and one positive value, and it searches from a starting guess (10% unless you give one), so when the cash flows change sign more than once it reports only one of the possible rates. For cash flows at irregular dates, XNPV and XIRR take a list of dates as well as amounts. Whatever the tool, lay out the timeline first, so you know which cash flow sits at which date before you type a formula.