Future Value
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変数
| 記号 | 名称 | 単位 | 説明 |
|---|---|---|---|
| $FV$ | Future Value | currency | The value of an investment at a future date. |
| $PV$ | Present Value | currency | Current value or principal invested today. |
| $r$ | Interest rate per period | decimal | Return rate per compounding period. |
| $n$ | Number of periods | periods | Total number of compounding periods. |
Future Value Formula
Future Value (FV) calculates what a sum of money invested today will be worth after n compounding periods:
$$FV = PV \times (1 + r)^n$$
This is the compound interest formula rewritten to emphasise the future perspective. It answers: "If I invest $X today at r% per period, how much will I have after n periods?"
FV vs PV
| Formula | Question answered | Direction |
|---|---|---|
| FV = PV(1+r)^n | How much will this grow to? | Forward (investing) |
| PV = FV/(1+r)^n | What is it worth today? | Backward (discounting) |
The two formulas are mathematical inverses. Together they form the foundation of all time-value-of-money analysis.
FV of Regular Contributions (Annuity)
When you invest a fixed amount each period (like monthly pension contributions), use the Future Value of an Annuity:
$$FV_{annuity} = PMT \times \frac{(1+r)^n - 1}{r}$$
Example: $500/month for 30 years at 7% annual (0.583%/month): FV = 500 × [(1.00583)^360 − 1] / 0.00583 ≈ $606,438
Total contributed: $180,000. Compound growth adds $426,438.
Retirement Planning Application
FV is the core formula in retirement projections. The key insight is the enormous sensitivity to time:
| $10,000 invested at 7% | FV |
|---|---|
| 10 years | $19,672 |
| 20 years | $38,697 |
| 30 years | $76,123 |
| 40 years | $149,745 |
Starting 10 years earlier nearly doubles the final value at 30 vs 20 years.
導出と歴史
The Future Value formula is a direct expression of the compound interest principle: each period, the existing balance grows by factor (1+r). After n periods, the original principal PV has been multiplied by (1+r) exactly n times, yielding PV × (1+r)^n.
This geometric growth model was formalised in European commercial mathematics of the 15th–16th centuries. The exponential function e^(rt) arises as the continuous-compounding limit as the compounding frequency n → ∞, discovered by Jacob Bernoulli in 1683 as a byproduct of investigating what happens when compound interest is applied infinitely often.
計算例
Retirement savings
- FV = 5,000 × (1.08)^25
- FV = 5,000 × 6.8485
- FV = $34,242
結果: FV = $34,242 — $5,000 grows to $34,242 in 25 years at 8%
College savings goal
- Use PV = FV/(1+r)^n = 80,000/(1.06)^18
- PV = 80,000 / 2.8543 = $28,027
- Invest $28,027 today at 6% to have $80,000 in 18 years
結果: Need to invest $28,027 today
エッジケースと制限
Inflation: The FV formula gives nominal future value. Real (inflation- adjusted) future value = FV / (1 + inflation)^n.
Variable rates: If the rate changes each period, use: FV = PV × (1+r₁)(1+r₂)...(1+rₙ) — the product of all period factors.
Negative returns: Substitute r < 0 for asset depreciation or deflation scenarios; FV < PV.
実世界での応用
Every retirement calculator, pension projection, and investment planning tool is built on the FV formula. Fund managers present FV projections to illustrate the long-term value of early investing. Insurance endowment products disclose FV of premiums paid. Government savings bond pricing is based on FV calculations. National pension systems (like Korea's NPS or the US Social Security system) use FV modelling to assess long-term funding adequacy.