Time Value of Money

Every time value calculation rests on a single relationship between a present amount, a future amount, a periodic rate and a number of periods. Moving forward, the present value is multiplied by one plus the periodic rate raised to the number of periods, which produces the future value. Moving backward, the future value is divided by that same factor, which produces the present value. The divisor is the discount factor, and it always lies between zero and one for positive rates.

Two properties follow directly. First, the relationship is multiplicative, not additive: doubling the number of periods squares the growth factor rather than doubling the accumulated interest. Second, present value falls as either the rate or the horizon rises, and the sensitivity to the rate grows with the horizon. This is why long-dated bonds and distant equity cash flows react far more violently to a shift in yields than near-term claims do.

The rate used to move cash across time is an opportunity cost, not a fee. At Level I it is decomposed into a real risk-free rate, which compensates purely for deferring consumption, plus an inflation premium reflecting expected erosion of purchasing power, plus premia for default risk, liquidity risk and maturity risk where the cash flow is uncertain or the horizon is long. The sum is the nominal required return applied to nominal cash flows; real cash flows must instead be discounted at a real rate for internal consistency.

The real component is strictly positive in normal conditions because capital deployed today can be put to productive use. That is why the deferral penalty survives even when expected inflation is zero, and why a genuinely risk-free instrument still trades below its maturity value.

Most valuation problems involve a sequence of cash flows rather than a single amount. When payments are level and evenly spaced, they form an annuity, and the present value equals the payment multiplied by an annuity factor: one minus the discount factor for the final period, all divided by the periodic rate. An ordinary annuity pays at the end of each period; an annuity due pays at the beginning, and its value equals the ordinary annuity value multiplied by one plus the periodic rate.

A level payment continuing indefinitely is a perpetuity, valued as the payment divided by the periodic rate. If the payment grows at a constant rate below the discount rate, the denominator becomes the discount rate minus the growth rate, the form later reused in dividend discount models. Irregular streams have no shortcut: each cash flow is discounted separately at its own horizon and the results summed.

Precision failures in this topic are almost always convention failures. A stated annual rate must be divided by the number of compounding periods per year to give the periodic rate, and the number of years multiplied by that same frequency to give the number of periods. The effective annual rate is one plus the periodic rate raised to the compounding frequency, minus one; under continuous compounding it is the exponential of the stated rate, minus one. Effective rates, not stated rates, are what make instruments with different compounding schedules comparable.

A candidate should be able to draw a timeline, place each cash flow at its correct index, choose consistent sign conventions for inflows and outflows, solve for any one of present value, future value, payment, rate or number of periods, and convert freely between stated, periodic and effective rates before comparing alternatives.