Time value of money, NPV and IRR

Key takeaways

• Future value compounds a sum forward at (1 + r)^n; present value discounts it back by dividing by the same factor.
• A quoted annual rate means little until you know the compounding frequency; compare investments on the effective annual rate.
• A growing perpetuity is worth next year's cash flow divided by (r - g), and it only works when the discount rate exceeds the growth rate.
• NPV is the decision rule: accept projects with a positive NPV at the cost of capital, and rank mutually exclusive projects by NPV, not by IRR.
• IRR can mislead: non-conventional cash flows can produce several IRRs, and IRR implicitly assumes interim cash is reinvested at the IRR itself.
• MIRR reinvests interim cash flows at an explicit rate, which removes the multiple IRR problem and gives a more realistic return figure.

Present value and future value

If you invest $10,000 at 8 percent a year, after one year you hold $10,800. In year two the 8 percent is earned on $10,800, not on the original $10,000, so interest earns interest. That is compounding, and it is why money grows exponentially rather than in a straight line. After n years, the sum has been multiplied by 1.08 n times.

Discounting runs the same calculation backwards. If $50,000 is due in seven years and the rate you could earn elsewhere at similar risk is 6 percent, the most you should pay today is the amount that would grow to $50,000 at 6 percent over seven years. That amount is the present value, and the rate used is called the discount rate.

The discount rate is the whole argument. It is the return you give up by putting money here instead of in the best alternative of similar risk, which is why it is also called the opportunity cost of capital. A riskier cash flow is discounted at a higher rate, so it is worth less today. Every valuation dispute in an investment committee is, in the end, a dispute about cash flows or about this rate.

Compounding frequency and the effective annual rate

Rates are usually quoted as an annual percentage, but interest may be credited semiannually, quarterly, monthly or daily. A quoted 8 percent compounded quarterly means 2 percent is credited every quarter, and each credit then earns interest itself. The more often interest compounds, the more you end up with by year end, so the same quoted rate can mean different amounts of money.

The fix is to convert every quote into an effective annual rate (EAR), the rate that would give the same result with one compounding a year. Compare loans, deposits and bond yields only on the same basis. US Treasury and corporate bond yields are quoted on a semiannual bond equivalent basis, so a 5.00 percent bond yield is an effective 5.0625 percent a year, a detail that matters when you set it against an annually compounded return.

In the limit, compounding continuously gives a factor of e raised to the rate times time. Continuous compounding is used in derivatives pricing and in some quant work because it makes the maths tidy, but for most deal and portfolio work annual or semiannual compounding is the convention.

Annuities and perpetuities

An annuity is a fixed payment every period for a set number of periods: a bond's coupons, a mortgage, a lease, an equipment loan. Discounting each payment one at a time works, but the sum has a closed form. An ordinary annuity pays at the end of each period; an annuity due pays at the start, so every payment arrives one period earlier and is worth (1 + r) times as much.

A perpetuity is an annuity that never ends. Its value is simply the payment divided by the rate, because the far future payments are discounted to almost nothing. A preferred share paying a fixed $5 a year forever, valued at 8 percent, is worth $62.50. The formula looks too simple to matter, yet it is the backbone of terminal value in every DCF.

The growing perpetuity is the version that matters most. If the payment next year is C and grows at g forever, the value is C / (r - g). Two conditions are essential: C is next year's payment, not this year's, and r must exceed g. As g approaches r the value explodes toward infinity, which is the arithmetic reason a DCF's terminal growth rate must sit well below the discount rate and below the long-run growth of the economy.

Net present value: the decision rule

Net present value (NPV) is the present value of every cash flow a project or investment produces, minus what it costs, all discounted at the cost of capital. A positive NPV means the investment returns more than the cost of capital and adds that many dollars of value today. A negative NPV means investors would be better off with the alternative of similar risk.

NPV has three properties that make it the default rule in corporate finance and in deal work. It is measured in dollars, so it says how much value is created, not just whether some hurdle is cleared. It is additive: the NPV of two projects together is the sum of their NPVs. And it uses the cost of capital explicitly, so the answer reflects what the money could earn elsewhere.

Plotting NPV against the discount rate gives the NPV profile. For a conventional project, one outflow followed by inflows, the curve slopes down: the higher the rate, the lower the value. The point where it crosses zero is the internal rate of return, which is where the next section starts.

Internal rate of return and its pitfalls

The internal rate of return (IRR) is the discount rate that makes NPV exactly zero. It has no closed form beyond the simplest cases, so it is found by trial and error, which is what a spreadsheet's IRR function does. For Halden the IRR is 13.04 percent: above the 10 percent cost of capital, so the project passes, consistent with its positive NPV. IRR is popular because a percentage is easy to compare with a hurdle rate and needs no discount rate to compute.

The first pitfall is ranking. IRR measures a rate, not an amount of value, and it favours projects that return cash early. When two projects are mutually exclusive, the one with the higher IRR can have the lower NPV. The NPV profiles cross at a rate called the crossover rate; below it the ranking by NPV and by IRR disagree, and NPV gives the right answer because it measures dollars created at the actual cost of capital.

The second pitfall is multiple IRRs. When cash flows change sign more than once, for example a mine that requires a large cleanup payment at the end, the NPV equation can have more than one solution. A project with flows of -$100m, +$230m and -$132m has IRRs of both 10 percent and 20 percent. Neither number means anything on its own; only the NPV at the cost of capital does.

The third pitfall is the reinvestment assumption. Discounting at the IRR is mathematically equivalent to assuming every interim cash flow is reinvested at the IRR itself until the end. For a project with a 40 percent IRR that returns cash early, that assumes the firm keeps finding 40 percent opportunities, which is rarely true. The headline IRR then overstates the return the investor actually earns over the period. This is why private equity managers who return capital quickly can report high IRRs with modest multiples of money.

Modified IRR

The modified internal rate of return (MIRR) replaces the hidden reinvestment assumption with an explicit one. Interim inflows are compounded forward to the final year at a stated reinvestment rate, usually the cost of capital. Outflows are discounted back to today at a finance rate, also usually the cost of capital. MIRR is then the single rate that turns the present value of outflows into the future value of inflows.

Because the calculation always collapses to one outflow today and one inflow at the end, MIRR has exactly one answer, which removes the multiple IRR problem. It also brings the return closer to what an investor would actually earn when interim cash can only be redeployed at ordinary rates. For a project whose IRR exceeds the reinvestment rate, MIRR sits below IRR, and the gap widens the more cash comes back early.

MIRR also tends to rank projects closer to NPV. In the example above, reinvesting at 10 percent gives Project A an MIRR of 11.82 percent and Project B 13.19 percent (B has no interim cash, so its MIRR equals its IRR). The ranking now agrees with NPV. It is still a rate, not a dollar figure, so NPV remains the final arbiter when the two rules differ.

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