Future Value: Growing a Lump Sum

What FV asks

Future value answers the mirror of PV: you have money today — how much will it be worth at a future date, given a growth rate?

The formula

$FV = PV \cdot (1 + r)^n$

For multiple compounding periods per year ($n$ times/year, $t$ years): $FV = PV \left(1 + \dfrac{r}{n}\right)^{nt}$

Continuous compounding: $FV = PV \cdot e^{rt}$.

Step-by-step solution

Setup: $PV = 5000$, $r = 0.06$, $n = 20$ (annual compounding).

Step 1 — Compute the growth factor. $(1 + 0.06)^{20} = 1.06^{20} \approx 3.20714$

Step 2 — Multiply by principal. $FV = 5000 \cdot 3.20714 \approx \boxed{16{,}035.68}$

Step 3 — Interest earned: $I = FV - PV = 16{,}035.68 - 5000 = 11{,}035.68$

More than double the original principal, all from 6% compounding over 20 years.

Intermediate milestones

To see the curve:

• 5 years: $5000 \cdot 1.06^5 \approx 6691.13$ (+33.8%)
• 10 years: $5000 \cdot 1.06^{10} \approx 8954.24$ (+79.1%)
• 15 years: $5000 \cdot 1.06^{15} \approx 11{,}982.79$ (+139.7%)
• 20 years: $\approx 16{,}035.68$ (+220.7%)

Each 5-year block grows by the same multiplier (1.06⁵ ≈ 1.338), so each block's dollar gain keeps rising — the hallmark of exponential growth.

The role of doubling time

By the Rule of 72, money doubles in about $72/r_{\%} = 72/6 = 12$ years at 6%. Two doublings give $\times 4$; at 20 years that's between 1.67 doublings (since $20/12 \approx 1.67$), i.e. about $2^{1.67} \approx 3.18$. That matches our factor of 3.207 — a one-sigfig Rule-of-72 sanity check.

Additional uses

• Retirement projections: $5K at 22 growing at 8$109K. Front-load the time.
• Comparing offers: project cash flows forward at an assumed growth rate.
• Forward pricing: futures prices are often a "cash-and-carry" FV of spot.

Common mistakes

• Multiplying $PV$ by $r \cdot t$ instead of compounding. That's simple interest and undercounts.
• Using nominal rate with wrong period count. $1.06^{20 \cdot 12}$ would be wildly wrong if the rate is annual.
• Forgetting to reinvest. "6% per year" only compounds into FV if interest is left to grow; spending the interest breaks the model.

Try it in the visualization

Watch the exponential curve climb as you slide rate and term sliders. Stack markers for PV, FV, total interest. Toggle between annual, monthly, and continuous compounding to see them converge.