# 4 Retirement Math

## 4.1 A lump sum for fixed retirement expense

Suppose you plan to deposit a lump sum to an interest-bearing account for retirement expenses, how much do you need to save?

Let the real interest rate (monthly) be \(r\) (inflation-adjusted) and assume the interest payments are made monthly and are tax-free. Let your life expectancy minus retirement age be \(live\) months. Let your inflation-adjusted monthly expense be \(expense\). Let your monthly balance be \(balance(n)\) where \(balance(0)\) is the beginning balance (i.e., what we are trying to figure out) and \(balance(live)\) at your death should be 0 (assume you leave nothing for your children).

We have

\[balance(n) = (1+r)balance(n-1) - expense\]

It’s a first order inhomogeneous linear difference equation and its solution is given by

\[balance(n)=\frac{expense}{r}\left[1-(1+r)^{n-live}\right]\]

(an exponential decay function of \(n\)).

Suppose \(r = \frac{0.03}{12} = 0.0025\) (\(1.0025^{12}=1.030416\), so slightly larger than a 3% annual percentage yield), \(live = (85 - 67)\times12 = 216\) (US life expectancy in 2021 was 76, full retirement age for Social Security is 67 for people born after 1960), \(expense = 4345\) (https://www.retireguide.com/retirement-planning/average-spending/), then \(balance(0)=\frac{4345}{0.0025}(1-(1.0025)^{-216})=724,500\). The interest payments saved you \(216\times4345-724500=214020\), i.e. 22.8% over 19 years.

### 4.1.1 Interest rate and saving

Let’s consider how changes in the interest rate affect the amount of the lump sum saving

\[balance(0)=\frac{expense}{r}\left[1-(1+r)^{-live}\right]\]

As \(r\) increases, \(\frac{expense}{r}\) decreases, \((1+r)^{-live}\) decreases (because \(-live < 0\)), \(\left[1-(1+r)^{-live}\right]\) increases, so the effect of increase in \(r\) appears ambiguous when common sense tells us you can save less. To mathematically resolve this question, we turn to the partial derivative:

\[\frac{\partial balance(0)}{\partial r}=\frac{expense}{r^2}\left[(live\cdot r+r+1)(1+r)^{-live-1}-1\right]\]

If \(\frac{\partial balance(0)}{\partial r} < 0\) holds for any reasonable \(r, expense,\text{ and } live\), then an increase in \(r\) results in a decrease in \(balance(0)\), meaning we need to save less. Because \(\frac{expense}{r^2}>0\), we only need to consider the sign of \(\left[(live\cdot r+r+1)(1+r)^{-live-1}-1\right]\).

Since

\[\left[(live\cdot r+r+1)(1+r)^{-live-1}-1\right]\biggr|_{r=0}=0\]

and

\[\frac{\partial\left[(live\cdot r+r+1)(1+r)^{-live-1}-1\right]}{\partial r}=-live(live+1)r(1+r)^{-live-2}<0\text{ for }r, live>0\]

we have

\[\left[(live\cdot r+r+1)(1+r)^{-live-1}-1\right]<0 \text{ for } r>0\]

which leads us to conclude that \(\frac{\partial balance(0)}{\partial r}<0 \text{ for } r>0\). Therefore, a higher interest rate decreases \(balance(0)\), i.e., requiring us to save less for retirement.

## 4.2 Fixed contribution for a lump sum

How much should you save every month to an interest-bearing account for the lump sum above?

Let your monthly contribution be \(contrib\), the real interest rate (monthly) \(r\), and the contribution period be \(work\) months. We could solve a difference equation as above or take advantage of a known summation formula \(\sum_{i=0}^nx^i=\frac{x^{n+1}-1}{x-1}\). Your accumulated contributions with compounded interests at the end is given by

\[balance=\sum_{i=1}^{work}contrib(1+r)^{work-i}=contrib\sum_{i=0}^{work-1}(1+r)^{i}=\frac{contrib}{r}\left[(1+r)^{work}-1\right]\]

Suppose \(r=0.0025, work=(67-25)\times12=504\) and desired \(balance=724,500\), then \(contrib=\frac{r}{(1+r)^{work}-1}balance=719\). The interests saved you \(\frac{724500-719\times504}{724500}=\) 50%.

## 4.3 Saving ratio assuming fixed expense throughout working and retirement

Combining the previous two sections:

\[\frac{contrib}{r}\left[(1+r)^{work}-1\right]=\frac{expense}{r}\left[1-(1+r)^{-live}\right]\]

\[\implies\frac{contrib}{contrib+expense}=\frac{(1+r)^{live}-1}{(1+r)^{work+live}-1}\]

At \(work=504, live=216, r=0.0025\), we have \(\frac{contrib}{contrib + expense}=0.142\) i.e. a 14.2% saving ratio during working. A higher interest rate requires less saving. Without interests, you need to save \(\frac{216}{504+216}=\) 17.7%.