Compound Growth and Fees

Compound Growth and Fees

Compound Growth and Fees: How It Works

What it does

The page answers one question: if you start with some money, add to it regularly and leave it invested, what is it worth at the end, and how much of that does a yearly fee take? It works the answer out twice, once with no fee and once with the fee, and reports the difference in money, as a share of the final balance and in today's buying power. Every step is printed as formula, numbers put in, answer.

The method

Symbols. P₀ is the starting amount; C_p the contribution paid in at the end of each period (month or year); m_y the number of contributions a year (12 or 1); N_y the number of years; n_p = m_y N_y the number of periods; r the expected yearly return; f the yearly fee as a fraction of the balance; h the yearly inflation. Rates are used as decimals: 6 % is 0.06.

Compound growth. Money that grows at a rate i per period is multiplied by (1 + i) each period, so after n periods it has been multiplied by (1 + i)ⁿ. The growth of earlier periods earns growth itself; that is what "compound" means.

Rate per period. The return entered is the growth over a whole year. With monthly contributions the page uses the monthly rate that compounds to exactly that yearly figure, i = (1 + r)^1/12 − 1, which is slightly less than r / 12. With yearly contributions i = r.

Future value. The balance at the end has two parts:

FV = P₀ (1 + i)ⁿ + C_p [(1 + i)ⁿ − 1] / i

The first term is the starting amount compounded over all n periods. The second is the sum of the contributions, each compounded from the day it is paid in: C_p [(1 + i)^{n − 1} + … + (1 + i) + 1], a geometric series whose sum is the bracket divided by i. When i = 0 nothing grows and FV = P₀ + C_p n.

The fee. A fee of f a year is taken from the whole balance, so each year the balance is multiplied by G_f = (1 + r)(1 − f). The return kept is G_f − 1, close to r − f. The same future-value formula is then used with the rate per period after the fee, i_f = G_f^1/m − 1. With subscript 0 for no fee and f for after the fee:

Cost = FV₀ − FV_f,   Share = Cost / FV₀

Today's money. Prices rise by h a year, so an amount N years ahead buys what that amount divided by (1 + h)^N buys today. This is the real value: FV_real = FV_f / (1 + h)^N.

Paid in and growth. Paid = P₀ + C_p n is the money you put in; Growth = FV_f − Paid is what the investment added after the fee.

Advanced options. Rising contributions: the contribution of year k is C_p (1 + g_c)^{k − 1}. One year of contributions of 1 is worth s = [(1 + i)^m − 1] / i at that year's end, and with G = (1 + i)^m the yearly growth factor, FV = P₀ G^N + C_p s [G^N − (1 + g_c)^N] / [G − (1 + g_c)]. Tax drag: a share t_x of each year's return is paid in tax, so r is replaced by r (1 − t_x). Lump sum or spreading in: the starting amount is invested at once, or in M equal monthly parts beginning today while the rest waits and earns nothing; with j the monthly rate after the fee and T the months of the plan, FV_lump = P₀ (1 + j)^T and FV_spread = (P₀ / M) (1 + j)^T [1 − (1 + j)^−M] / [1 − (1 + j)⁻¹]. Rule of 72: years to double ≈ 72 / (return in percent), beside the exact ln 2 / ln(1 + r_n) at the return after the fee r_n. Target: the future-value formula solved for the contribution, C = (Target − P₀ (1 + i_f)ⁿ) i_f / [(1 + i_f)ⁿ − 1].

What it assumes

The same return, fee and inflation every year for the whole plan. Contributions at the end of each month or year, never missed, for whole years. The fee as one percentage of the balance, taken continuously through the year; sales charges, trading costs and fixed account fees are not modelled unless you fold them into the percentage. No taxes in Simple mode. The tax drag of Advanced mode taxes all of each year's return at one rate, which overstates the tax on gains that are not realised each year.

When it misleads

Returns are not constant. A 6 % average return does not arrive as 6 % each year. Good and bad years come in an unknown order, and with contributions or withdrawals the order changes the final balance. The page shows one smooth path; the real outcome is a wide range around it. The table of final balance against return gives a first feel for that range.

The return is a guess. Small changes in the assumed return compound into large changes in the result over decades. The fee, by contrast, is known in advance, which is why the page separates the two.

The no-fee balance is a yardstick, not an offer. Almost every way of investing has some cost. The comparison that matters in practice is usually between two fees, which the fee table shows.

Lump sum against spreading in. At a constant positive return the lump sum always ends ahead, simply because the money is invested sooner. That says nothing about a market that may fall next month; with real returns either choice can end ahead.

Inflation applies to contributions too. A level contribution shrinks in buying power each year. Use the yearly raise in Advanced mode to let it keep pace.

Worked example

The default case: 10,000 to start, 500 at the end of every month for 30 years, a 6 % return, a 1 % fee and 2.5 % inflation.

Periods: n = 12 × 30 = 360. No fee: i = 1.06^1/12 − 1 = 0.004868, and (1 + i)³⁶⁰ = 1.06³⁰ = 5.7435. FV₀ = 10,000 × 5.7435 + 500 × (5.7435 − 1) / 0.004868 = 57,435 + 487,256 = 544,691.

With the fee: G_f = 1.06 × 0.99 = 1.0494, so the return kept is 4.94 %; i_f = 1.0494^1/12 − 1 = 0.004026 and (1 + i_f)³⁶⁰ = 4.2485. FV_f = 10,000 × 4.2485 + 500 × (4.2485 − 1) / 0.004026 = 42,485 + 403,406 = 445,890.

Cost of the fee: 544,691 − 445,890 = 98,801, which is 98,801 / 544,691 = 18.1 % of the no-fee balance. A fee of 1 % a year takes about 18 % of the result over 30 years.

Today's money: 1.025³⁰ = 2.0976, so the balance is worth 445,890 / 2.0976 = 212,575 and the cost of the fee 47,103 in today's buying power.

Paid in: 10,000 + 500 × 360 = 190,000. Growth after the fee: 445,890 − 190,000 = 255,890.

The page's results carry full precision; the hand figures above are rounded, so a last digit can differ.

A calculator for education and analysis, not investment advice.