Compound and discount single sums

One reversible relationship

For a single sum, compounding and discounting use the same growth factor:

FV_n = PV(1+i)^n
PV   = FV_n/(1+i)^n

The discount factor is the reciprocal of the growth factor. Do not learn these as unrelated keyword formulas. If the inputs remain aligned, compounding and then discounting should return the starting amount.

Read the factor before the currency

At i = 0.004/month for n = 36 months:

growth factor   = (1.004)^36 = 1.1545525…
discount factor = 1/(1.004)^36 = 0.8661365…
factor product  = 1 within numerical precision

The growth factor answers how one current dollar maps to month 36. The discount factor answers how one month-36 dollar maps to month 0. Multiplying a cash amount by a factor should not obscure the date attached to the result.

Direction checks

For the bounded model with a positive amount, nonnegative rate, and positive period count:

• future value is at least present value;
• present value is no greater than the later future amount;
• at zero rate, the two are equal; and
• at zero periods, the two are equal.

A violated direction check signals a likely sign, exponent, rate, or operation error. Passing it proves only plausibility.

Precision policy

Keep the full factor and amount through the inverse check. Round only displayed currency. If a rounded $51,968.19 deposit compounds to a few cents above or below $60,000, label the difference as display rounding. Do not overwrite the controlled full-precision value to force equality.

Accounting measurement boundary

Present value appears in accounting, but the formula is not an accounting standard. Before using the number as a recognized or disclosed amount, identify the applicable authoritative guidance, measurement objective, cash-flow basis, timing, risk treatment, rate selection, and presentation requirements. A result is not fair value merely because it was discounted.

Finance decision boundary

Likewise, the formula cannot choose between a borrowing rate, required return, opportunity cost, risk-free rate, or expected return. It does not guarantee a future balance. Name who supplied the rate, the decision it serves, and which risk, fees, taxes, and optional cash flows are outside the model.

Exit check

An amount of $20,000 at t = 0 compounds at 3% annually for 2 years. Compute the $21,218 future value, discount it back to $20,000, and write one sentence that prevents the result from being misread as a forecast.