Delayed saving

Suppose I want to retire at time $T > 0$ . I make constant payments to a savings account that earns continuously compounded interest $r > 0$ . I want my retirement fund to be worth $V > 0$ today (time $0$ ). How much bigger do my payments have to be if I delay them?

Let $X_{d}$ be the payments I have to make if I start saving at time $d \in [0, T]$ . These payments form an annuity with value $\frac{X_{d}}{r} (1 - e^{- r (T - d)})$ at time $d$ . I want this value to equal $V e^{r d}$ . So my payments must equal $\begin{aligned} X_{d} & = \frac{r}{1 - e^{- r (T - d)}} \times V e^{r d} \\ = \frac{r V}{e^{- r d} - e^{- r T}} . \end{aligned}$ Therefore, delaying to time $d$ increases my payments by a factor of $\frac{X_{d}}{X_{0}} = \frac{1 - e^{- r T}}{e^{- r d} - e^{- r T}} .$ The chart below shows how $X_{d} / X_{0}$ grows with the proportion of time $d / T$ I delay saving. Part of this growth comes from having less time remaining: if my savings earn no interest, then the factor $lim_{r \to 0} \frac{X_{d}}{X_{0}} = \frac{T}{T - d}$ equals the ratio of time until retirement and time spent saving. Raising $r$ raises $X_{d} / X_{0}$ because I forgo more opportunities to earn interest on my interest the longer I delay. This is especially true when I’m far from retiring (i.e., $T$ is large).

Thanks to Michael Boskin for inspiring this post.