The mathematics of retirement planning

Step 1: Capital accumulation

Step 2: Retirement

Step 3: Piecing 1 & 2 together

Assumption 1.1: Worker lives with average real income 'Y' until retiring in 't1' months;

Assumption 2.1: Worker monthly redeems a real fixed amount 'A' from a given amount saved for 't2' periods;

Now, we are going to join steps 1 and 2 equating 'S = P'.

Assumption 1.2: During the working period, you save amount 's', fraction of your income, every month;

Assumption 2.2: The remaining unredeemed capital will continue to yield the same return 'r' until fully consumed;

Note that we were careful to represent 'S' & 'P' as $ amounts

Assumption 1.3: Your savings are fully invested in fixed income assets with a real monthly return 'r'

Assumption 2.3: Savings will be fully consumed at a gradual rate by a certain life expectancy.

of the same period, which is the month when retirement begins.

Defining S as the accumulated future value of savings after interest on the investment. Then S is:

Let's think about the saver's counterparty. This bank/financial institution

will have a debt to repay to such a worker/family whose present value P is:

Let us simplify this algebra:

S is a finite geometric series (GS) of ratio (1+r). The sum of the n terms can be represented as:

Again, we have a case of a GS. But this time its ratio is 1/(1+r). Applying the SGS formula again we have:

Where 'a1' is the first term of the GS,

In a final step, we can make some useful definitions using

'q' is its ratio and 'n' is the qty of terms

'A' and 's' as shares worker/household real incomes:

We can then substitute the terms of the generic SGS into our capital accumulation formula

Once again, we can simplify our equation as follows:

So our final financial planning equation for retirement is:

We can finally simplify the equation as follows:

So that 'α' is the retiree/family income as a fraction of their real average income

and 'β' the share of average real income saved during the period of capital accumulation.

If we want to find out the value of the amortization installments of this debt, we can solve for A:

As in practice the real interest rate 'r' varies over time and each worker/family

Debt amortization equation. Alternatively, we can

start their savings phase at different ages for different retirement goals, it is

think of 'A' as the value of a pension benefit to be

reasonable to consider different parameters 'r', 't1', 't2' and 'β' in order to find

consumed from savings 'P'

reliable intervals for the retirement replacement rate of income 'α'