Time value and discounting

9 min read

Everything in valuation is one idea applied repeatedly: a dollar later is worth less than a dollar now, and the says how much less. This lesson is short and it is the one to be genuinely fluent in, because every later page assumes the arithmetic without pausing.

Present value

Present value of one cash flow

PV = \frac{CF_t}{(1+r)^t}

The present value of a cash flow is that cash flow divided by one plus the rate, raised to the number of periods away it is.

$100 in five years at 8% is worth $68.06 today. Check it the other way: $68.06 compounded at 8% for five years is $100.00. Discounting and compounding are the same operation run in opposite directions, which is worth internalizing because it lets you sanity check any discounted number by growing it back.

What $100 five years out is worth at different rates

	4%	8%	12%	16%
Present value	82.19	68.06	56.74	47.61
Present value, 20 years out	45.64	21.45	10.37	5.14

Look at the second row. Twenty years out, the difference between a 4% rate and a 12% rate is the difference between 46 cents on the dollar and 10. The discount rate matters most exactly where a forecast is least reliable, which is an uncomfortable property of the whole method and worth holding onto.

Perpetuities, because forecasts have to end

No one forecasts a business forever, so at some point the model says "and then it continues like this" and values the rest in one number. The Gordon growth formula is that number.

Growing perpetuity

PV = \frac{CF_1}{r - g}

The present value of a cash flow growing forever is next year's cash flow divided by the discount rate less the growth rate. It requires the growth rate to be below the discount rate, and it is at the valuation date one period before the first cash flow.

The mid-year convention

Mid-year discount factor

DF_t = \frac{1}{(1+r)^{t-0.5}}

Discount each year from its midpoint rather than its end, on the reasoning that cash arrives through the year.

ConventionThe mid-year convention applies to the explicit forecast years. The terminal value already sits at the end of the final forecast year, so it is discounted the full number of periods either way. Getting that wrong is a common and quiet error that inflates a valuation by about half a year of the discount rate on the largest component.

Internal rate of return

Turn the question around. Instead of asking what a stream is worth at a given rate, ask what rate makes it worth exactly what you paid. That is the .

Cash flows of negative 100 today and positive 250 in three years give an IRR of 35.72%. There is no closed form for the general case, so a spreadsheet solves it iteratively, and so does the code behind these pages.

IRR has two properties worth remembering. It assumes intermediate cash flows are reinvested at the IRR itself, which is usually generous. And it can have multiple solutions when the cash flows change sign more than once, which is why the implementation on this site returns nothing rather than a confident guess when the flows never change sign at all.

Same math, other direction

The compound interest calculator

Discounting run forwards. Useful for building intuition about how fast the exponent takes over.

Test yourself

01What is $1,000 in ten years worth today at 7%?

$1,000 / 1.07^10 = $508.35. The rule of 72 gets you close without a calculator: 72 / 7 is about 10, so money doubles in ten years at 7%, so a dollar ten years out is worth about half a dollar today. That is $500 against the exact $508, which is close enough to sanity check someone else's model in your head.

02Why does the mid-year convention increase a valuation?

It discounts each year's cash from the middle of the year instead of the end, so every flow is discounted half a period less. The effect is roughly half the discount rate applied once, so a 9% WACC lifts the explicit forecast by about 4.4%. It is more realistic, because cash arrives through the year rather than in a single December payment. It also has to be stated, because a reader who assumes year-end will not reproduce your number.

03A perpetuity grows at 3% and is discounted at 3%. What is it worth?

Infinity, which means the question is wrong rather than the answer being large. If cash grows as fast as it is discounted, every future year contributes the same present value and the sum never converges. This is not a curiosity: it is one edge of a range that a terminal value calculation approaches whenever the growth assumption creeps up toward the discount rate, and the value explodes long before it gets there.

A tutor that knows this lesson. It asks before it explains, and it will not tell you what to do with your own money.

Every figure in this lesson that names a company comes from Target Corporation's Form 10-K for Fiscal 2025, the year ended 2026-01-31, filed 2026-03-11. Educational material, not investment advice.