Calculating Savings Rate

How much do you need to set aside each year? It comes down to three things: how fast your investments grow, how fast inflation erodes the target, and how far away the target sits. The notation below uses gross multipliers rather than rates — $g = 1.08$ means 8% annual growth, $i = 1.032$ means 3.2% inflation:

— base annual contribution, in today’s dollars
— investment growth multiplier (e.g. $1.08$ ), assumed greater than $i$
— inflation multiplier (e.g. $1.032$ )
— number of years
— target amount, in today’s dollars

Suppose you index your contributions to inflation each year — your salary keeps pace, so you save a constant share of a growing paycheck. To reach a goal worth $T$ in today’s dollars, you must accumulate $T \times i^{n}$ in nominal dollars by year $n$ , and the portfolio you build is:

Accumulated value = c \frac{i (g^{n} - i^{n})}{g - i}

(2.1)

Setting that equal to $T \times i^{n}$ and solving for the contribution gives:

c = T \frac{i^{n - 1} (g - i)}{g^{n} - i^{n}}

(2.2)

If contributions are not indexed to inflation — you save the same nominal amount every year — each dollar has to do more work, so the required contribution is higher:

Accumulated value = c \frac{g^{n} - 1}{g - 1}

(2.3)

c = T \frac{i^{n} (g - 1)}{g^{n} - 1}

(2.4)

Take $g = 1.08$ (8% growth), $i = 1.032$ (3.2% inflation), $n = 20$ years, and a target of $T = $ 1, 000, 000$ . With inflation-indexed contributions:

c = $ 1, 000, 000 \times \frac{1.03 2^{19} (1.08 - 1.032)}{1.0 8^{20} - 1.03 2^{20}} = $ 31, 375

You would start at $31,375 a year and raise that figure with inflation. Holding the contribution flat instead costs more — $41,029 a year:

c = $ 1, 000, 000 \times \frac{1.03 2^{20} (1.08 - 1)}{1.0 8^{20} - 1} = $ 41, 029

For the latter case you can also use the PMT function in Excel or Google Sheets to calculate your savings rate:

c = - PMT (0.08, 20, 0, 1000000 \times {(1.032)}^{20}) = $ 41, 028.84

or for same case, but with bi-weekly contributions (assuming 26 payments per year):

c = - PMT ((1.0 8^{\frac{1}{26}} - 1), 20 \times 26, 0, 1000000 \times 1.03 2^{20}) \times 26 = $ 1, 520.34 (per paycheck) \times 26 = $ 39, 520

You may have different goals with different present values, so you’d need to calculate multiple savings rates. For example, you may want to save for a house, a car, a vacation, etc.

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