## Thursday, June 13, 2013

### The 30-60-90 Approach to Retirement Planning, Part 3: Adjusting the Savings Formula for Different Saving Periods

In Part 2, we have seen that using the 30-60-90 approach, and assuming an annual real investment return of 4.7%, you should save an amount equal to

Savings = Expenses/4

Where "Expenses" is your estimated monthly or yearly retirement expenses at today's prices.

But what if you don't closely fit the 30-60-90 scenario? What if, for whatever reason, you decide to start saving for retirement much later, like say, age 40? How can we adjust the savings formula above to better reflect your decisions?

Starting to save for retirement later than age 30 will obviously result a higher savings amount since you'll have less earning years to prepare for the same amount of retirement expenses. If you start at age 40, for example, then your savings for the entire year should be enough not just for year 60--your first year of retirement--but also a portion of your expenses in the following year. Furthermore, whereas in the 30-60-90 scenario all your retirement fund deposits have a 30-year horizon, starting later shortens your investment horizon correspondingly and exposes your retirement portfolio to greater risk.

To adjust the savings formula in order to reflect a variable saving period y, take the following. Assuming constant prices, you expect to spend an amount E every year starting on your 60th birthday--the beginning of retirement--for 30 years until you turn 89. You plan to finance your retirement by contributing an amount S every year to your retirement fund, starting on your (60 - y)th birthday, for y years until age 59. If your retirement fund earns a real rate of return r, then the future value of all S payments should equal the present value of all E expenses on your 59th birthday. In equation form, using the formula for future value and present value of an annuity, we get

S*[(1 + r)^y - 1]/r = E*[1 - 1/(1 + r)^30]/r

Simplifying,

S = E*[1 - 1/(1 + r)^30]/[(1 + r)^y - 1]

(I apologize, this equation can't be simplified any further.)

As an example, if E = 32,000, r = 4.7%, and y = 20, then

S = 32,000*[1 - 1/1.047^30]/[1.047^20 - 1]

S = 32,000*0.4967 = 15,894

Or almost double the savings amount if you start at 30 years old, or just half of the expense estimate. If you use y = 30 as in the 30-60-90 scenario, you'll actually get the original equation S = E/4.

Finally, you can also use the above formula for different saving and retirement periods. If z = the number of years of retirement, just replace "30" by so that

S = E*[1 - 1/(1 + r)^z]/[(1 + r)^y - 1]

So if you're now 30 years old and you plan to retire by 50 and you retain the planning horizon of up to 90 years old, then y = 20 and z = 40. Using the same E and r,

S = 32,000*[1 - 1/1.047^40]/[1.047^20 - 1]

S = 32,000*0.5584 = 17,868

Remember that these estimates are only for the amount that you need to save in your first year (age 60 - y). For subsequent years, you need to adjust for inflation, like in the Part 2, but this time with a slightly different factor

Savings in Year t = (Savings in Year t)*(1 + g)^(z/y)

So that if you start saving at age 40 (y = 20), retire at 60 (z = 30), and the annual inflation rate is g = 4.5%, then

Savings at age 40: 15,894 per month
Savings at age 41: 15,894*(1.045^1.5) = 16,979 per month

...

Savings at age 55: 15,894*1.045^(1.5*15) = 42,791 per month

As always, figuring out the savings amount is just the first step. To meet your target real rate of return, you should invest your retirement savings in a low-cost equity fund and only redeem your units/shares at retirement and as needed.