Common mistake 1 with Time value of money & NPV/IRR

Mixing a per-period rate with an annual period count, or the reverse. Monthly payments need a monthly rate and a count in months.

Common mistake 2 with Time value of money & NPV/IRR

Getting the signs wrong. In the accumulation form used here, money you put in and money you receive are both positive; a loan is a positive amount borrowed with a negative payment and a future value of zero.

Common mistake 3 with Time value of money & NPV/IRR

Ranking mutually exclusive projects by IRR. A small project can have a higher IRR and a lower NPV; follow NPV.

Common mistake 4 with Time value of money & NPV/IRR

Trusting IRR when the cash flows change sign more than once. There can be several IRRs, or none.

EconStatsCraft › Time value of money & NPV/IRR

Time value of money & NPV/IRR

A financial calculator that explains itself: solve for the present value, future value, payment, number of periods or rate of an annuity, or value a stream of cash flows with NPV, IRR, MIRR and payback, with the NPV profile drawn.

The intuition

A rupee or dollar today can be invested, so it will be worth more tomorrow. Discounting runs that backwards: a payment due in the future is worth less today, by exactly the amount you would need to invest now to reach it. Every valuation in finance, from a savings plan to a pension or a share, is this one idea applied to a list of cash flows.

When to use it

Saving towards a target, pricing a loan or a mortgage, comparing rates quoted with different compounding, valuing a perpetuity, or deciding whether a project is worth its cost. It is the arithmetic under every other valuation method, from bonds to pensions.

Where the estimator comes from

One unit grows to (1+i) after a period and (1+i)^n after n, so a single sum has FV=PV(1+i)^n. A level payment PMT at the end of each period accumulates to PMT\sum_{k=0}^{n-1}(1+i)^k = PMT\,\frac{(1+i)^n-1}{i}, a geometric series. Payments at the start of each period earn one extra period, multiplying by (1+i). Letting n\to\infty in the present value PMT\frac{1-(1+i)^{-n}}{i} gives the perpetuity PMT/i. For uneven flows, \text{NPV}(r)=\sum_t CF_t(1+r)^{-t}, and the IRR is the root of \text{NPV}(r)=0, found numerically.

The model

FV = PV(1+i)^n + PMT\,\frac{(1+i)^n-1}{i},\qquad \text{EAR} = \left(1+\tfrac{r}{m}\right)^m - 1,\qquad \text{NPV}=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t},\quad \text{NPV}(\text{IRR}) = 0

What it assumes

One rate applies to every period. Real borrowing and lending rates differ, and rates change over time.
Cash flows are known with certainty. Risk is handled by raising the discount rate, not by this formula.
For an annuity, every payment is the same size and the periods are equally spaced.
IRR assumes interim cash flows are reinvested at the IRR itself, which is usually optimistic. MIRR replaces it with a stated reinvestment rate.

Statistical properties

Present value falls as the rate rises and as the payment moves further into the future.
More frequent compounding at the same nominal rate always gives a higher effective rate; the limit is continuous compounding, e^r − 1.
An annuity due is worth (1 + i) times the ordinary annuity with the same payments.
NPV is additive across projects; IRR is not.
A conventional project (one outflow then inflows) has exactly one IRR. Each extra sign change can add another.

How to read the output

Compare rates on their effective annual basis. 12% compounded monthly is 12.68% a year, more than 12% compounded annually.
An annuity due (payments at the start of each period) is worth exactly (1 + i) times the ordinary annuity, because every payment earns one more period of interest.
A positive NPV means the project earns more than the required return and adds that much value today. Accept it.
IRR is the break-even required return. If it exceeds your cost of capital, NPV at that cost is positive.
Payback ignores everything after the payback date and the time value before it. Use it as a liquidity check, never as the decision rule.

A worked reading

Save 500 a month for ten years at 6% a year compounded monthly. The rate per month is 0.5% and there are 120 payments. The balance reaches 81,939.67, although only 60,000 was put in: 21,939.67 is interest, and most of it is interest earned on earlier interest. The effective annual rate is 6.1678%, not 6%, because of monthly compounding. In cash-flow mode, a project costing 1,000 that returns 300, 400 and 500 over three years has an NPV of 17.63 at 8% and an IRR of 8.896%, so it just clears an 8% hurdle.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Future value is negative	Payments entered as negative while saving	In this calculator money put in is positive
No rate solves the problem	Signs inconsistent with the target	A loan: positive PV, negative PMT, FV of 0
Payment looks 12 times too large	Annual rate used with monthly periods	Set compounding to 12 and enter years
Two IRRs, or none	Cash flows change sign more than once	Decide with NPV at your required return
IRR and NPV rank projects differently	Projects differ in size or timing	Follow NPV

Common mistakes

Mixing a per-period rate with an annual period count, or the reverse. Monthly payments need a monthly rate and a count in months.
Getting the signs wrong. In the accumulation form used here, money you put in and money you receive are both positive; a loan is a positive amount borrowed with a negative payment and a future value of zero.
Ranking mutually exclusive projects by IRR. A small project can have a higher IRR and a lower NPV; follow NPV.
Trusting IRR when the cash flows change sign more than once. There can be several IRRs, or none.

Where to read this properly

CFA Institute, Quantitative Methods, "The Time Value of Money in Finance" (Level I curriculum).
Kellison, The Theory of Interest (2009), ch. 1–4.
Brealey, Myers and Allen, Principles of Corporate Finance (2022), ch. 2 and 5, on NPV and IRR.

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · Merton distance to default · EMI / home loan planner · CAPM / beta estimation · Multi-factor model · Portfolio optimization · Value at Risk & Expected Shortfall · Performance & drawdown · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Binomial option pricing · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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Time value of money & NPV/IRR

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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