As a quick “first-order approximation” to how much an investment grows over time with regular contributions (from a salary, for example), it can be useful to have a quick closed-form equation ready.

Fixed contribution per period #

Let \(p\) be the fixed payment amount at the end of a period, \(n\) the number of periods, \(c_0\) the initial balance, and \(r > 0\) the interest rate per period. We claim that the balance at the \(n\)-th period is \[ \boxed{c_n = c_0 \left(1 + r\right)^n + \frac{p}{r} \left( \left(1 + r\right)^n - 1\right).} \] We’ll prove by induction. In the base case \(n = 0\) the equality clearly holds. Now suppose the equation holds for \(n = k\). Then we have \[ \begin{aligned} c_{k+1} &= c_0 \left(1 + r\right)^{k+1} + \frac{p}{r} \left( \left(1 + r\right)^{k+1} - 1 \right) \\ &= \left(1 + r\right) \left( c_0 \left(1 + r\right)^k + \frac{p}{r} \left(1 + r\right)^k \right) - \frac{p}{r} \\ &= \left(1 + r\right) \left( c_0 \left(1 + r\right)^k + \frac{p}{r} \left( \left(1 + r\right)^k - 1 \right) + \frac{p}{r} \right) - \frac{p}{r} \\ &= \left(1 + r\right) \left( c_0 \left(1 + r\right)^k + \frac{p}{r} \left( \left(1 + r\right)^k - 1 \right) \right) + p \\ &= \left(1 + r\right) c_k + p \end{aligned} \] which recovers the expected recurrence relation as required. An equivalent proof in Lean 4 is as follows:

def fv_fixed (n : ℕ) (p r c₀ : ℝ) : ℝ :=
  match n with
  | 0 => c₀
  | n + 1 => (1 + r) * fv_fixed n p r c₀ + p

noncomputable def fv_fixed_sol (n : ℕ) (p r c₀ : ℝ) : ℝ :=
  c₀ * (1 + r) ^ n + p / r * ((1 + r) ^ n - 1)

theorem fv_fixed_sol_satisfied (p r c₀ : ℝ) (rpos : 0 < r) :
    ∀ n, fv_fixed n p r c₀ = fv_fixed_sol n p r c₀ := by
  unfold fv_fixed_sol fv_fixed
  intro n
  induction n with
  | zero => simp
  | succ => grind [fv_fixed]

We can “standardise” the equation by dividing it by \(p\), the fixed contribution amount. If we start with an initial ratio of \(c_n / p = 10\), we have the following plot:

We can see that a difference of a few percent in \(r\) can be substantial after a few decades, especially at higher rates. This is why keeping the fees low in any investment is paramount.

Looking at the equation again, we can also immediately see that the ending ratio \(c_n/p\) varies linearly with the initial ratio \(c_0/p\).

Let \(c_0 = 10^5\) and \(p = 10^4\). The following plot shows that if we don’t contribute anything, the balance after many years is substantially lower compared to that with regular contributions, even early on:

This shows the importance of continuing to contribute to the investment on a regular basis rather than simply relying on compounding to do the work.

Increasing contributions per period #

We can also write a similar equation but with increasing contributions. Let \(p_0\) be the initial payment amount at the end of the first period, \(n\) the number of periods, \(c_0\) the initial balance, \(r\) the investment returns, and \(r_p\) the growth rate of contributions. Then we have \[ \boxed{ c_n = c_0 \left(1 + r\right)^n + \begin{dcases} p_0 \frac{\left(1 + r\right)^n - \left(1 + r_p\right)^n}{r - r_p} & r \ne r_p \\ p_0 n \left(1 + r\right)^{n - 1} & r = r_p \end{dcases} } \] When \(n = 0\) the equality is clear. We first consider the case where \(r = r_p\). At \(n = k + 1\) we have \[ \begin{aligned} c_{k+1} &= c_0 \left(1 + r\right)^{k+1} + p_0 \left(k + 1\right) \left(1 + r\right)^k \\ &= \left( c_0 \left(1 + r\right)^k + p_0 k \left(1 + r\right)^{k-1} + p_0 \left(1 + r\right)^{k-1} \right) \left(1 + r\right) \\ &= \left( c_k + p_0 \left(1 + r\right)^{k-1} \right) \left(1 + r\right) \\ &= c_k \left(1 + r\right) + p_0 \left(1 + r\right)^k \end{aligned} \] which recovers the recurrence relation. Now consider the case where \(r \ne r_p\). At \(n = k + 1\) we have \[ \begin{aligned} c_{k+1} &= c_0 \left(1 + r\right)^{k+1} + p_0 \frac{\left(1 + r\right)^{k+1} - \left(1 + r_p\right)^{k+1}}{r - r_p} \\ &= \left( c_0 \left(1 + r\right)^k + p_0 \frac{\left(1 + r\right)^{k+1} - \left(1 + r_p\right)^{k+1}}{\left(r - r_p\right) \left(1 + r\right)} \right) \left(1 + r\right) \\ &= \left( c_0 \left(1 + r\right)^k + p_0 \left( \frac{\left(1 + r\right)^k}{r - r_p} - \frac{\left(1 + r_p\right)^{k+1}}{\left(1 + r\right) \left(r - r_p\right)} \right) \right) \left(1 + r\right) \\ &= \left( c_k + p_0 \frac{\left(1 + r_p\right)^k}{r - r_p} - p_0 \frac{\left(1 + r_p\right)^{k+1}}{\left(1 + r\right) \left(r - r_p\right)} \right) \left(1 + r\right) \\ &= c_k \left(1 + r\right) + p_0 \frac{\left(1 + r\right) \left(1 + r_p\right)^k}{r - r_p} - p_0 \frac{\left(1 + r_p\right)^{k+1}}{r - r_p} \\ &= c_k \left(1 + r\right) + \left( p_0 \frac{1 + r}{r - r_p} - p_0 \frac{1 + r_p}{r - r_p} \right) \left(1 + r_p\right)^k \\ &= c_k \left(1 + r\right) + p_0 \left(1 + r_p\right)^k \end{aligned} \] as required. Equivalently in Lean 4, which interestingly shows that we don’t need the hypotheses \(r > 0\) and \(r_p > 0\) mathematically:

def fv_inc (n : ℕ) (p₀ r rₚ c₀ : ℝ) : ℝ :=
  match n with
  | 0 => c₀
  | n + 1 => (1 + r) * fv_inc n p₀ r rₚ c₀ + p₀ * (1 + rₚ) ^ n

noncomputable def fv_inc_sol (n : ℕ) (p₀ r rₚ c₀ : ℝ) : ℝ :=
  c₀ * (1 + r) ^ n +
  if r = rₚ then
    p₀ * n * (1 + r) ^ (n - 1)
  else
    p₀ * ((1 + r) ^ n - (1 + rₚ) ^ n) / (r - rₚ)

theorem fv_inc_sol_satisfied (p₀ r rₚ c₀ : ℝ) :
    ∀ n, fv_inc n p₀ r rₚ c₀ = fv_inc_sol n p₀ r rₚ c₀ := by
  intro n
  induction n with
  | zero => simp [fv_inc, fv_inc_sol]
  | succ n ih =>
    simp [fv_inc, fv_inc_sol, ih]
    by_cases h_r_eq : r = rₚ
    · cases n <;> grind
    · grind

In the following plot, we have \(c_0 = 5 \times 10^5\) and \(p_0 = 10^5\). Observe that a modest annual contribution increase of 3% doesn’t do a lot to the balance in the first few decades compared to having no contribution increase at all. The difference does start to grow substantially after about 25 years, though.

I want to caveat that the analyses above are not completely realistic, primarily because investments that return more than 5% in the real world tend to have substantial volatility on a year to year basis. Nevertheless, they can help improve our intuitions on financial planning.

Impact of pay raise on wealth #

Let’s examine the phenomenon of savings leverage. We can model an individual with a regular per-period income \(I\) and a fixed cost \(F\), which doesn’t depend on \(I\). Hence, the contribution per period is \(p = I - F\).

A pay raise percentage \(m\) is, however, applied on the gross income \(I\) rather than the contribution \(p\). As long as the fraction of income spent on fixed cost is less than one, that is \(F/I < 1\), the effective increase in contribution, which is what actually matters to wealth grow, is multiplied by a greater number, as a result of “savings leverage”. Specifically, \[ \boxed{\text{\% increase in } p = \frac{\left(1 + m\right) I - F}{I - F} - 1 = \frac{m}{1 - \frac{F}{I}},} \] where \(1 - F/I\) is the saving rate. This is mathematically clean, and we can understand it as the ratio of pay raise percentage and saving rate. The higher the saving rate, the lower the percentage increase in \(p\).

In other words, if your saving rate is very high to begin with, then a pay raise doesn’t matter quite as much as if you have a lower saving rate.

For instance, suppose \(I = 5000\) and \(F = 3000\), so \(p = 5000 - 3000 = 2000\) and \(1 - F/I = 2/5\). If this individual obtains a pay raise of \(m = 3\%\), then the increase in percentage terms in the contribution \(p\) is \(3\% / \left(2/5\right) = 7.5\%\), which is a fair bit higher than the pay raise percentage. Now instead consider \(F = 1000\) with a higher saving rate of \(1 - F/I = 4/5\). Then the corresponding percentage increase in \(p\) is \(3\%/\left(4/5\right) = 3.75\%\), which is much lower and much closer to \(m\).

Interestingly, suppose the cost \(F\) is not fixed and instead scales up with the pay raise. Then the percentage increase in \(p = m\). This shows the importance of avoiding lifestyle inflation.

Of course, even with a very high saving rate, pay raises still have an impact on wealth. But how big of a game changer is it?

For simplicity and qualitative understanding, it suffices to consider the fixed contribution case, where the contribution after pay raise \(p_h = \left(1 + \gamma\right) p\) and \(\gamma\) is some percentage. The percentage increase in future value as derived as earlier corresponding to the two contribution amounts is \[ \frac{c_{h,n}}{c_n} - 1 = \frac{c_0 \left(1 + r\right)^n + \frac{p_h}{r} \left( \left(1 + r\right)^n - 1 \right)}{c_0 \left(1 + r\right)^n + \frac{p}{r} \left( \left(1 + r\right)^n - 1 \right)} - 1. \] We may examine the asymptotic behaviour of the wealth ratio as \(n \to \infty\). To this end, we divide the numerator and the denominator by \(\left(1 + r\right)^n\) and substituting \(p_h = \left(1 + \gamma\right) p\) to yield \[ \lim_{n \to \infty} \frac{c_{h,n}}{c_n} - 1 = \lim_{n \to \infty} \frac{c_0 + \frac{p \left(1 + \gamma\right)}{r} \left( 1 - \left(1 + r\right)^{-n} \right)}{c_0 + \frac{p}{r} \left( 1 - \left(1 + r\right)^{-n} \right)} - 1. \] If \(0 < r\), this limit exists and may be written as \[ \boxed{\lim_{n \to \infty} \frac{c_{h,n}}{c_n} - 1 = \frac{\gamma}{1 + \dfrac{c_0 r}{p}}, \quad \left(p \ne 0\right).} \] Interestingly, we see a dependence on the initial wealth \(c_0\) at the time of the pay raise or increase in contribution.

• If the initial wealth is \(c_0 = 0\), then the long term percentage increase in wealth is simply \(\gamma\) itself.
• But if we start off with substantial wealth \(c_0\), then the effect on long term wealth is diminished, as the capital is doing more of the heavy lifting.

Observe also that the higher the rate \(r\), the more diminished the wealth increase, although the effect is minor. Changes in the starting contribution amount \(p\) has the opposite effect: if the pay raise occurs when \(p\) is higher, then the long term wealth increase will be higher.

To illustrate, let \(c_0 = 3 \cdot 10^5\), \(r = 5\%\), \(p = 6 \cdot 10^4\), and \(\gamma = 5\%\). Then the long term wealth increase is \(4\%\), i.e. somewhat lower than \(\gamma\). If we instead set \(c_0 = 10^6\), we obtain \(\approx 2.72\%\) instead, which is substantially lower.

If let \(\gamma\) to be the % increase in \(p\), then we can put the two equations together and obtain \[ \boxed{\frac{m}{1 - \dfrac{F}{I} + \dfrac{c_0 r}{I}}.} \] We see that the higher the saving rate \(1 - F/I\), the smaller the effect of pay raise. Likewise, the higher the capital \(c_0\) at the time of the pay raise, the smaller the effect of pay raise as well. On the other hand, the higher the income \(I\) at the time of the pay raise, the higher the effect.

Interestingly, if \(c_0 r - F \ge 0\), then the % increase is less than \(m\), and vice versa. In other words, if the return on the capital is greater than total fixed cost, then the % increase is less than \(m\), and vice versa.

In practice, the discussion above is only useful for a qualitative understanding of the expected returns. Any small % differences will be dwarfed by the wide dispersion of outcome as a result of the volatility inherent in all but the safest investments. Nevertheless, it serves as a power reminder: The less you rely on your paycheck to cover your costs, the less a raise changes your life.