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Compound a balance and regular contributions with AI-powered step-by-step solutions

$25,000 plus $500 a month at 7% compounded monthly for 30 years

How much per month reaches $1,000,000 in 25 years at 6% from $50,000?

What is $812,898 in 30 years worth in today's dollars at 3% inflation?

Monthly withdrawal a $800,000 balance supports for 25 years at 5%

A Balance Has Two Halves

A retirement or savings projection is a lump sum and a stream of deposits, compounded separately and added:

$FV = P(1+i)^N + PMT \cdot \frac{(1+i)^N - 1}{i}$

$P$ — the balance you start with
$PMT$ — the deposit made every period, assumed at the end of each period
$i$ — the rate per period: annual rate $\div$ periods per year
$N$ — total number of periods: years $\times$ periods per year

The first term is plain compound growth. The second is the future value of an ordinary annuity: each deposit compounds for a different length of time, and the fraction sums that geometric series. If deposits land at the start of each period (an annuity-due), multiply the second term by $(1+i)$ .

An employer match is simply a larger $PMT$ . Reinvested dividends are already inside $i$ if you use a total-return figure. The formula does not know or care what account holds the money; it compounds whatever you enter.

Solving Backwards, and Adjusting for Inflation

The deposit needed for a target

$PMT = \bigl(FV - P(1+i)^N\bigr)\cdot\frac{i}{(1+i)^N - 1}$

The time needed

$N = \frac{\ln\!\left(\dfrac{FV \cdot i + PMT}{P \cdot i + PMT}\right)}{\ln(1+i)}$

Real versus nominal

A balance decades out is quoted in future dollars. Convert with the exact Fisher relation, not by subtracting:

$i_{\text{real}} = \frac{1 + i_{\text{nom}}}{1 + \pi} - 1$

At $7\%$ against $3\%$ inflation that is $3.8835\%$ , not $4\%$ .

Drawing the balance down

The level withdrawal a balance $B$ supports for $N$ periods reverses the annuity:

$PMT = B \cdot \frac{i}{1 - (1+i)^{-N}}$

Tax treatment differs sharply between account types and jurisdictions, and it changes the rate that actually belongs in $i$ . This page computes the arithmetic on the return, contribution and horizon you supply — it is not financial advice.

Common Mistakes to Avoid

Annual rate with monthly periods: if $PMT$ is monthly then $i = r/12$ and $N = 12 \times$ years. Mixing the two scales is the single most common error.
Using the lump-sum formula for contributions: $P(1+i)^N$ alone ignores every deposit. The annuity term is not optional.
Subtracting inflation from the return: $7\% - 3\% = 4\%$ is an approximation that drifts over 30 years. Use the ratio form.
Treating an average return as a guaranteed one: the formula compounds a constant $i$ . Real returns vary year to year, and the same average with more volatility ends somewhere else.
Ignoring fees: an annual fee reduces $i$ directly. A $0.75\%$ fee on a $7\%$ return compounds as $6.25\%$ .
Forgetting that the answer is in future dollars: a seven-figure projection is not seven figures of today's purchasing power.

示例题目

Step 1:

i = 0.07/12 \approx 0.0058333

N = 12 \times 30 = 360

Step 2:

(1+i)^{360} \approx 8.1164975

Step 3: Lump sum:

25{,}000 \times 8.1164975 \approx \

202{,}912.44$

Step 4: Deposits:

500 \times \dfrac{8.1164975 - 1}{0.0058333} \approx 500 \times 1219.971 \approx \

609{,}985.50$

Step 5:

FV \approx 202{,}912.44 + 609{,}985.50 \approx \

812{,}897.93$

Step 6: Of that,

25{,}000 + 360 \times 500 = \

205{,}000$ is money you put in; the rest is growth

Answer: About

\

812{,}898

, of which roughly

$607{,}898$ is compound growth

Step 1:

i = 0.005

N = 300

(1.005)^{300} \approx 4.4649698

Step 2: The existing balance grows to

50{,}000 \times 4.4649698 \approx \

223{,}248.49$

Step 3: Deposits must supply

1{,}000{,}000 - 223{,}248.49 = \

776{,}751.51$

Step 4:

PMT = 776{,}751.51 \times \dfrac{0.005}{4.4649698 - 1} = \dfrac{3{,}883.7576}{3.4649698}

Step 5:

\approx \

1{,}120.86$ per month

Answer: About

\

1{,}120.86$ a month

Step 1: Deflate:

1.03^{30} \approx 2.4272625

Step 2:

812{,}897.93 / 2.4272625 \approx \

334{,}903.18$ in today's purchasing power

Step 3: Withdrawals:

i = 0.05/12 \approx 0.0041667

N = 300

(1+i)^{-300} \approx 0.2871693

Step 4:

PMT = 800{,}000 \times \dfrac{0.0041667}{1 - 0.2871693} = \dfrac{3{,}333.33}{0.7128307}

Step 5:

\approx \

4{,}676.72$ per month before tax

Answer: About

\

334{,}903

in today's dollars;

$800{,}000

supports roughly

$4{,}676.72$ a month for 25 years

常见问题

FV = P(1+i)^N + PMT·((1+i)^N − 1)/i, where i is the rate per period and N the number of periods. The first term compounds what you already have; the second is the future value of the deposit stream.

The standard formula assumes the end of each period (an ordinary annuity). If you deposit at the start, multiply the annuity term by (1 + i). Over 30 years of monthly deposits at 7% that is worth about 0.58% more.

Either divide the final balance by (1 + π)^years, or run the whole projection at the real rate (1 + i)/(1 + π) − 1. Both give the same answer in today's dollars; subtracting inflation from the return does not.

No — the compounding arithmetic is identical for any account. What differs is the tax treatment of contributions and withdrawals, which varies by account type and jurisdiction and changes the effective rate you should enter. Check the rules that apply to you.

Retirement and Savings Calculator

Compound a balance and regular contributions with AI-powered step-by-step solutions

A Balance Has Two Halves

Solving Backwards, and Adjusting for Inflation

The deposit needed for a target

The time needed

Real versus nominal

Drawing the balance down

Common Mistakes to Avoid

示例题目

常见问题

What formula grows a balance with regular contributions?

Should contributions go at the start or end of the period?

How do I adjust a projection for inflation?

Does the account type change the math?

相关求解器

免费试用 AI-Math

Retirement and Savings Calculator

Compound a balance and regular contributions with AI-powered step-by-step solutions

A Balance Has Two Halves

Solving Backwards, and Adjusting for Inflation

The deposit needed for a target

The time needed

Real versus nominal

Drawing the balance down

Common Mistakes to Avoid

示例题目

Problem: $$$25{,}000nowplusnow plusnowplus$500amonthat7 a month at 7% compounded monthly for 30 yearsamonthat7

Problem: WhatmonthlydepositreachesWhat monthly deposit reaches Whatmonthlydepositreaches$1{,}000{,}000in25yearsat6in 25 years at 6% compounded monthly, starting fromin25yearsat6$50{,}000???

Problem: WhatisthatWhat is that Whatisthat$812{,}898worthintoday′sdollarsafter30yearsof3worth in today's dollars after 30 years of 3% inflation, and what monthly withdrawal wouldworthintoday′sdollarsafter30yearsof3$800{,}000supportfor25yearsat5 support for 25 years at 5%?supportfor25yearsat5

常见问题

What formula grows a balance with regular contributions?

What formula grows a balance with regular contributions?

Should contributions go at the start or end of the period?

Should contributions go at the start or end of the period?

How do I adjust a projection for inflation?

How do I adjust a projection for inflation?

Does the account type change the math?

Does the account type change the math?

相关求解器

免费试用 AI-Math