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Investment Growth, Fee, and Goal Formulas

Review investment formulas for monthly growth, recurring contributions, annual fees, fee drag, inflation, return scenarios, and target contribution solving.

How do the Investment Calculator formulas work?

This formula page defines the variables used by the Investment Calculator, shows how those values are combined, and explains the order of operations behind the result. Use the worked steps to verify an estimate, compare scenarios, and understand which assumptions or rounding rules can affect the displayed answer.

Formula Overview

The calculator uses a monthly recurrence rather than one closed-form future-value equation. That allows it to apply contribution timing, contribution increases, gross return, fees, goal tracking, and annual schedule totals in the same order every month.

Separate projections use the base return, the lower and higher return assumptions, and a fee-free comparison. The target contribution is solved numerically with the same monthly projection.

Variable Definitions

SymbolMeaningUnit or convention
BmBalance after month mCurrency
PInitial investmentCurrency
CStarting monthly contributionCurrency per month
CmContribution used in month m after annual increasesCurrency
raExpected effective annual returnDecimal rate
rmEquivalent monthly gross returnDecimal rate
faAnnual investment feeDecimal rate
fmEquivalent monthly fee deductionDecimal rate
gAnnual contribution increaseDecimal rate
piAnnual inflation assumptionDecimal rate
tInvestment lengthWhole years
GTarget investment balanceFuture nominal currency
vReturn variancePercentage-point range

Effective Annual Return to Monthly Return

The input is treated as an effective annual return. Its equivalent monthly rate is:

rm=(1+ra)1/12−1r_m=(1+r_a)^{1/12}-1

This preserves the annual growth factor because (1 + rm)12 = 1 + ra.

Annual Fee to Monthly Fee

The annual fee is converted into an equivalent monthly deduction:

fm=1−(1−fa)1/12f_m=1-(1-f_a)^{1/12}

The balance retained after 12 fee deductions has the same annual retention factor as 1 - fa, before considering contributions and returns.

Monthly Balance Recurrence

Beginning and end contributions are represented separately:

Bm=((Bm−1+Cb)(1+rm))(1−fm)+CeB_m=((B_{m-1}+C_b)(1+r_m))(1-f_m)+C_e

For beginning timing, Cb = Cm and Ce = 0. For end timing, Cb = 0 and Ce = Cm. Gross return is applied before the monthly fee.

The engine separately accumulates gross return and fee deductions, allowing the annual schedule to reconcile to the ending balance.

Annual Contribution Increase

The monthly contribution increases at the start of each modeled year. For month m:

ym=⌈m12⌉y_m=\left\lceil\frac{m}{12}\right\rceil
Cm=C(1+g)ym−1C_m=C(1+g)^{y_m-1}

A 3% increase keeps the first 12 deposits at C, the next 12 at C(1.03), and so on.

Investment Fees and Fee Drag

Total fees are the sum of monthly deductions. Fee drag compares two full projections:

Fee drag=FVfee free−FVwith feesFee\ drag=FV_{fee\ free}-FV_{with\ fees}

Fee drag is generally greater than the fee total because money removed for expenses no longer earns subsequent returns.

The calculator also reports an effective annual net rate for reference:

rnet=(1+ra)(1−fa)−1r_{net}=(1+r_a)(1-f_a)-1

Inflation-Adjusted Investment Value

Vreal=Bn(1+π)tV_{real}=\frac{B_n}{(1+\pi)^t}

The nominal ending balance remains Bn. The real-value output discounts that balance using the constant inflation assumption entered.

Lower, Base, and Higher Return Scenarios

The lower and higher effective annual return assumptions are:

rlow=max⁡(−0.99,ra−v)r_{low}=\max(-0.99,r_a-v)
rhigh=min⁡(1.00,ra+v)r_{high}=\min(1.00,r_a+v)

Each rate runs through the same contribution, timing, and fee model. The results are sensitivity paths, not probabilities, forecasts, or statistical confidence bounds.

Required Monthly Contribution Solver

The solver seeks the smallest starting monthly contribution C for which the projected ending balance reaches the nominal target:

min⁡Csuch thatBn(C)≥G\min C\quad\text{such that}\quad B_n(C)\ge G

It first checks whether the initial investment alone reaches the goal. Otherwise, it uses 80 bisection iterations between $0 and $1,000,000 per month. The projection inside each iteration retains the entered return, fee, timing, annual contribution increase, and time horizon. It returns no solution when even the upper bound does not reach the target.

Worked Calculation

For the default scenario, P = $10,000, C = $500 monthly, t = 20 years, ra = 7%, fa = 0.25%, end-of-month deposits, no contribution increase, pi = 2.5%, and G = $500,000.

  1. The equivalent monthly return is 0.00565415.
  2. The equivalent monthly fee is 0.00020857.
  3. The initial investment plus 240 deposits equals $130,000.00.
  4. The monthly recurrence produces an ending balance of $282,998.81.
  5. Gross return totals $158,893.27, and fees deducted total $5,894.46.
  6. Net investment growth is $152,998.81.
  7. The fee-free comparison ends at $292,465.03, so fee drag is $9,466.22.
  8. The inflation-adjusted value is $172,705.95.
  9. The $500,000 target has a projected gap of $217,001.19.

Assumptions and Exclusions

  • Each scenario uses a constant effective annual return.
  • The annual fee and inflation assumption remain constant.
  • Contributions occur monthly at the selected timing and increase once per year by the entered percentage.
  • The goal is a future nominal dollar target.
  • The lower and higher paths change only the expected return.
  • Taxes, employer matching, contribution limits, distributions, transaction charges, tiered fees, withdrawals, and account-specific rules are excluded.
  • The engine retains full numeric precision and rounds displayed currency to cents.

Formula FAQ

Why is the expected return converted with a twelfth root?

The input represents effective annual growth. The twelfth root creates a monthly factor that compounds back to the same annual factor. For a useful Investment Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

Why are return and fee not simply subtracted?

The engine applies gross growth and then the fee retention factor. Their interaction is multiplicative. For a useful Investment Calculator formula result, enter realistic values, compare relevant scenarios, check the calculation details, and review the definitions, assumptions, and limitations before using the estimate for an important decision.

Why does contribution timing change the result?

A beginning deposit receives that month's modeled return and fee application, while an end deposit does not enter until afterward. For a useful Investment Calculator formula result, use realistic inputs, compare relevant scenarios, and review the stated assumptions before making a decision.

Why is the target contribution solved numerically?

Annual contribution increases, timing, and monthly fees interact in the recurrence, so bisection uses the complete production model rather than a simplified closed form. Use realistic inputs and review the Investment Calculator formula assumptions before relying on this result for decisions.

References

Use the worked investment examples to reproduce the formulas with complete scenarios.

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