The time value of money
Money now is worth more than the same money later, because it can be put to work and later is not guaranteed: compounding grows it forward, discounting is the same equation solved backwards, and every valuation method in this school is this idea plus bookkeeping.
One idea underneath all of finance: money now is worth more than the same money later, because money now can be put to work, and because later is not guaranteed. Every valuation method in this school is this idea plus bookkeeping, so this guide builds it carefully, with numbers.
Compounding: money moving forward
FV = PV x (1 + r)^n $100 at 8% for 10 years = 100 x 1.08^10 = $215.89 ...for 30 years = 100 x 1.08^30 = $1,006.27 rule of 72: doubling time ≈ 72 / rate. 8% -> 9 yrs. 12% -> 6 yrs.
Growth multiplies rather than adds, which produces the two facts every investor eventually internalizes. First, time beats rate over long horizons: the second decade of compounding earns more dollars than the first at the same rate, because it compounds a bigger base. Second, small rate differences become enormous outcome differences: two points of annual fee or drag feels trivial and compounds into wildly different lives.
Discounting: the same arithmetic, run backwards
PV = FV / (1 + r)^n $215.89 promised in 10 years, discounted at 8% = $100.00 today $1,000 promised in 20 years, at 10% = $148.64
Discounting is not pessimism; it is pricing. It answers: what would I pay today for that future amount, given what my money could earn elsewhere? The rate r is therefore an opportunity cost, the return of the alternative you give up, and that is the deep reason risky cash flows carry high discount rates: they must be compared against risky alternatives, which pay more. Risk becomes price through this one mechanism.
At 10%, a dollar in year 5 is worth 62 cents, year 10 is 39 cents, year 20 is 15 cents. Two consequences run all through markets: terminal assumptions dominate the valuation of durable companies (most of their worth sits far away), and long-duration assets (growth stocks, 30-year bonds) swing hardest when rates move, because the discount machine punishes distance and rates set the punishment. 2022's simultaneous crash in long bonds and profitless tech was this paragraph, enacted.
Streams of cash: annuities and perpetuities
Real assets pay repeatedly, and closed formulas price the standard shapes. An annuity pays a fixed amount for a fixed number of periods; a perpetuity pays forever, which sounds exotic until you notice that a company with no planned end date is one.
annuity: PV = C x (1 - (1+r)^-n) / r perpetuity: PV = C / r growing perpetuity: PV = C1 / (r - g) valid only while g < r worked: a 30-year mortgage of $400,000 at 6% (0.5%/month, 360 months) 400,000 = C x (1 - 1.005^-360) / 0.005 -> C = $2,398/month total paid over 30 years = $863,353: the price of time, itemized
The mortgage example is worth staring at: more than half of what a 30-year borrower pays is time itself. The same formula prices bond coupon streams, pensions, structured settlements, and any "$X per month" claim anyone ever offers you: PV it at a fair r and compare to the asking price.
The (r − g) danger zone
The growing perpetuity PV = C1 / (r − g) is the most consequential formula in valuation: it is the terminal value in every DCF and the heart of the dividend discount model. It is also the most dangerous, because value explodes as g approaches r:
| $100 growing at g, discounted at r = 9% | PV |
|---|---|
| g = 0% | $1,111 |
| g = 2% | $1,429 |
| g = 4% | $2,000 |
| g = 6% | $3,333 |
| g = 8% | $10,000 |
| g = 8.9% | $100,000 |
From g of 6% to 8%, the value triples; approach r and it goes vertical. Nothing about the business changed in that table; only an assumption moved. This is why perpetual growth assumptions deserve more suspicion than any other number in any model, why the school's DCF guide caps terminal growth at the economy's (~2-3%), and why an analyst who shows you a valuation without showing you its sensitivity to g is showing you a preference.
The machine asked backwards: IRR and yields
Given a price and a stream, solve for the rate instead of the value: that rate is the internal rate of return (IRR), the discount rate at which the deal exactly breaks even. Private equity speaks in IRR; a bond's yield to maturity is an IRR; and "what return does this price imply?" is often a more honest question than "what is this worth?", because it removes one estimated input.
a bond pays $50/yr for 3 years + $1,000 at maturity, priced at $973.27 973.27 = 50/(1+y) + 50/(1+y)² + 1,050/(1+y)³ -> y = 6.0% price and yield are two readings of one equation; that is the seesaw: pay LESS than 973.27 and your yield is MORE than 6%, mechanically.
| Question | Solve for | Name |
|---|---|---|
| What is this stream worth at my required return? | PV | Valuation |
| What return does this price imply? | r | IRR / yield to maturity |
| What growth justifies this price? | g | Reverse DCF |
| What payment services this loan? | C | Amortization |
One caution on IRR: it assumes interim cash flows reinvest at the IRR itself, which flatters early-return profiles; private equity marketing leans on exactly this, and sophisticated LPs also demand the multiple on invested capital for that reason.
Nominal and real
A rate can count dollars (nominal) or purchasing power (real); they differ by inflation, approximately real = nominal − inflation. At 7% nominal with 3% inflation, money doubles in 10 years but what it BUYS doubles in 18. The rule that prevents silent errors: discount nominal cash flows at nominal rates and real at real, never mixed; most practical valuation runs fully nominal because reported financials are nominal. (The macro guide picks this thread up: real rates, read from inflation-protected bonds, are the price that moves gold and long-duration assets.)
Where this shows up in every later guide
- DCF modeling is this guide applied to a company: FCF streams, a WACC for r, a growing perpetuity for the tail.
- Fixed income is this guide applied to promised payments: price/yield is PV/IRR, duration is the derivative of this arithmetic.
- Other intrinsic methods: the dividend discount model IS the growing perpetuity; bank and REIT math re-skins it.
- Risk and return: CAGR is compounding measured, volatility drag is compounding's arithmetic penalty.
- Macro: rates are the r in everything, which is why the whole market repriced when they moved.
- Present value
- What a future amount of money is worth today, after shrinking it for the time you must wait and the risk you must bear. A dollar promised in five years is worth less than a dollar in hand; present value says exactly how much less.
- Future value
- What money today grows into at a given rate over time: PV times (1 + r) to the n. Compounding read forward.
- Discount rate
- The annual rate used to shrink future cash into today's money. It is the return an investor could demand elsewhere for taking similar risk, so riskier cash flows get higher rates and smaller present values.
- Time value of money
- The principle that money now beats the same money later, because money now can be invested, and because later is uncertain. All of valuation is this principle applied carefully.
- Compounding
- Growth on top of prior growth: each period's return earns returns in every later period. The reason time in the market matters more than almost any other input.
- Rule of 72
- The mental shortcut for doubling time: 72 divided by the annual rate. At 8%, money doubles in about nine years.
- Annuity
- A fixed payment repeated for a fixed number of periods, like a mortgage or a bond's coupons. Its present value has a closed formula, so streams of payments can be priced in one line.
- Perpetuity
- A stream of cash flows assumed to continue forever. A perpetuity growing at a steady rate g and discounted at rate r has a finite value of next year's cash flow divided by (r minus g).
- Gordon growth model
- Terminal value as a growing perpetuity: final-year cash flow, grown one year, divided by the discount rate minus the perpetual growth rate. The growth rate must not exceed the economy's, or the formula quietly claims the company will outgrow the world.
- IRR
- The discount rate at which an investment's cash flows exactly break even against its price. The standard language of private equity returns, and what a bond's yield to maturity is.
- Yield to maturity
- The single rate that makes a bond's remaining coupons and principal worth exactly its current price: the bond's IRR if held to the end and paid in full.
- Nominal vs real
- Nominal counts dollars; real counts purchasing power, which is nominal with inflation removed. The rule: discount nominal cash flows at nominal rates and real at real, never mixed.
- Opportunity cost
- The return of the best alternative you give up by choosing this one. Discount rates are opportunity costs; that is why they rise with risk, since riskier projects must beat riskier alternatives.
- Volatility drag
- The gap between average and compound returns, roughly half the variance: +50% then -33% averages +8.5% and compounds to nothing. The mathematical reason wild rides underperform their marketing.