Time Value and Discounting

The one operation, written four ways

Present value is future cash divided by one plus the rate, raised to the number of periods: 100 / 1.10 is $90.91 today for $100 in a year. That is the entire idea, and every valuation formula is a rearrangement of it. Cash flows that arrive at different times are discounted separately and added, which is why a business with back-loaded cash flows is worth less than one with the same total arriving sooner. A perpetuity is the same operation applied to a stream that never ends: cash over the discount rate. A $100m cash flow at a 9% rate is worth $1,111m, and it is the reason long-lived assets are so sensitive — this formula has no end date to anchor it, so the whole value depends on a rate you chose. Add growth and the formula becomes cash over the difference between the rate and the growth: $100m growing at 2.5% with a 9% discount rate is 100 / 0.065 = $1,538m. That is the Gordon growth form used for every terminal value in F12, and the difference in the denominator is why an assumption that looks small — half a point of growth — moves the answer by hundreds of millions. The discount rate itself is not one thing. It has a base that compensates for the time value of money, which is what the risk-free rate represents, and a premium for the risk that the cash does not arrive, which is where beta and the equity risk premium enter in F11. When people say rates matter for equities they mean this: the risk-free rate is an input to the denominator of every valuation, so a change in it moves every price at once, which is exactly what was visible in 2022. Finally, a way to think about the whole structure: a valuation is a weighted average of time, and different businesses have very different weights. A utility’s value is mostly the next decade of dividends; a young company’s value is mostly a terminal value thirty years out. Both are discounted with the same arithmetic, and the arithmetic tells you which one a rate move breaks first. The same $100m, four ways — $100m in one year at 9%: $91.7m · $100m a year forever at 9%: $1,111m · $100m growing 2.5% forever at 9%: $1,538m · A five-year growing stream plus that perpetuity, discounted: about $1,992m ← A discount rate at or below the growth rate makes the formula produce a negative or infinite value. It is arithmetic, not a discovery — which is why terminal growth in a DCF is almost always set below long-run nominal GDP growth.

Why duration is the right lens

The sensitivity of a value to the discount rate has a name in fixed income and a use in equities: duration. A bond with a ten-year duration loses about 10% of its price when rates rise one point. An equity can be thought of the same way, with a much longer effective duration, because its cash flows run further into the future and are not contractual. The chart in this lesson moves the discount rate across a whole valuation and the picture is the point: the curve is steep, and the machinery of the answer sits in the perpetuity. Two practical consequences follow. First, the ranking of an equity portfolio by rate sensitivity is a ranking by duration: long-dated growth businesses first, short-cash-flow businesses last. When a tightening cycle begins, the first prices to fall are the ones whose value depends most on the distant future, and that is not sentiment — it is the discount rate working through the same formula. Second, the reverse is true in an easing cycle, and it is the mechanical reason the same set of assets leads the recovery. The discipline this imposes on analysis is to state the discount rate before the forecast. If you expect 12% and the model requires a 9% rate to justify the price, the disagreement is not about growth — it is about risk, and arguing about the growth rate is arguing about the wrong term. That is the subject of F21, where the price is inverted to see what it assumes. Discounting is not the same as adjusting for inflation. A nominal forecast discounted at a nominal rate and a real forecast discounted at a real rate give the same answer; mixing a nominal forecast with a real rate is one of the most common errors in a model.

Where the formulas stop being safe

The perpetuity and growing-perpetuity formulas are the most used equations in valuation and they have conditions attached that are easy to violate without noticing. The growing version divides the next cash flow by the discount rate minus growth, which means two things: growth must be **below** the discount rate for the expression to mean anything, and as the two approach each other the denominator shrinks toward zero and the value goes to infinity. Run a model with ten percent growth and a ten and a half percent discount rate and you do not get a large value, you get a nonsense one — a number that is a statement about the arithmetic rather than about the business. The same fragility shows up through the duration lens this lesson uses. The further out the cash flows, the more a change in the discount rate moves the total, and a perpetuity has the longest duration of all. That is why a company whose value is mostly terminal value is a company whose price is mostly a rate bet: bonds and growth stocks are the same trade at that point, and the 2022 decline in long-duration equities alongside falling bond prices was that identity showing up in public. Two consistency rules keep the formulas honest once you are inside a real model. Cash flows and discount rates must be in the same units and the same tax basis — nominal with nominal, after-tax with after-tax, and the same currency on both sides — and a terminal growth rate has to be something an economy can sustain indefinitely, which in a developed market means roughly the long-run growth of the economy rather than the growth the company has been enjoying. The formula does not know the difference; the analyst has to. The denominator, and why it decides the answer ($100m next-year cash flow) — Discount rate 10%, growth 3%: $100m ÷ 0.07 = $1,429m · Discount rate 10%, growth 7%: $100m ÷ 0.03 = $3,333m — more than double for four points of growth · Discount rate 10%, growth 9.5%: $100m ÷ 0.005 = $20,000m, which is not a valuation ← If a valuation looks implausibly large, check the denominator before you check the business. Growth within a point or two of the discount rate is the single most common way a spreadsheet produces a number nobody should believe.

Inverting the operation: what a price says about the rate

Discounting turns a forecast into a value. The inverse turns a price into a rate, and the inverse is used more often than the forward version without being noticed. Given a stream of cash flows and the price someone will pay for them, the rate that makes the two equal is the internal rate of return; on a bond it is the yield to maturity, and on a share it is the implied cost of capital that makes your own forecast equal the market price. Each of those is the same equation solved for a different unknown, and the arithmetic is trial and error rather than a formula, because the rate appears in every denominator. The measure is useful precisely because it removes the forecast. Two streams can be ranked without agreeing on a discount rate at all: if one offers a higher internal rate over the same duration and risk, it is the better purchase on those terms. That is the logic behind comparing a bond yield with an equity’s implied return, and behind the reverse-valuation habit of asking what a share price assumes. It is also why the forward and inverse operations belong in one lesson: the same machinery answers what a forecast is worth today, and what the market thinks the future looks like. The measure misleads in three specific ways, and each has a tell. First, it assumes the cash flows are reinvested at the same rate, so a project with a very high internal rate over a short period can outrank one with a lower rate over a long period even when the second creates more total wealth. Second, it can return more than one answer when the cash flows change sign more than once — an outlay, a return, then a further outlay — because the equation has as many roots as sign changes. Third, and most relevant to valuation, it ignores scale: a small project with a spectacular rate is not comparable to a large one with a modest rate until the capital each requires is also compared. What to carry forward is that both directions are one discipline pointed at different inputs. The forward version states its assumptions and delivers a number; the inverse takes a number and recovers the assumptions. The habit worth keeping is to move between them: value the business from your own forecast, then invert the market price and see what forecast it implies, and treat the gap between the two as the thing the thesis has to explain. That is what makes a valuation an argument rather than a calculation, and it is the bridge from this lesson to the expectations work later in the subject. • The inverse of discounting is a rate: the internal rate of return, or a bond’s yield to maturity. • It is solved by trial and error, because the rate appears in every denominator. • It removes the forecast, so two streams can be ranked without agreeing on a discount rate. • Three failure modes: the reinvestment assumption, multiple roots when cash flows change sign, and blindness to scale. • Move between directions deliberately: value from your forecast, then invert the price into the forecast it implies. A share has no internal rate of return in the strict sense — a perpetuity with growth has one only while growth stays below the rate, and a company paying nothing has none at all. The useful move is to invert the price into a discount rate given your own forecast and compare that with your required return, rather than to quote an equation that does not hold.

Sources

• Present value, perpetuities and the Gordon growth modelStandard corporate-finance texts; Gordon (1959), "Dividends, Earnings and Stock Prices"
• The discount rate as the opportunity cost of capitalFisher (1930), "The Theory of Interest"; Modigliani & Miller (1958)
• Duration as the sensitivity of value to the discount rateStandard fixed-income arithmetic; Macaulay (1938)