## Wednesday, June 5, 2013

### The 30-60-90 Approach to Retirement Planning, Part 2: Considering Inflation and Investment Returns

According to the "30-60-90" approach to retirement planning, since the time it takes to accumulate funds for retirement and the retirement period are both 30 years, the amount that you save in any given month or year will finance your retirement expenses in 30 years. In this post, we'll discuss a simple way to estimate how much you need to save today to be able to finance what you intend to spend in 30 years considering the effects of inflation and investment returns.

Let's say that you estimate that on any given month, you'll need 32,000 pesos at today's prices to support your chosen lifestyle. If we consider inflation, then you have to save more than 32,000 this month so that in 30 years, you'll be able to buy what 32,000 can buy today (maybe you should read this phrase one more time, it can be confusing)--but how much more? The inflation rate is the percent increase in the prices of basic goods and services every year. Specifically, if the price a good or service at time t is Price(t) and the average annual inflation rate is g, then the price of the good after n years is

Price(t + n) = Price(t)*(1 + g)^n

(I hope you're not turned off by the math. Honestly, using a bit of math is unavoidable in practical financial management. I always try to make technical discussions as simple as possible, so I hope you'll bear with me.)

For example, let's say that today, or t = 0, 32,000 pesos, or Price(0), can buy a certain amount of goods and services. In 30 years, or  n = 30, how much money do you need to be able to buy the same amount of goods and services if the average annual inflation rate, g, is 4.5%? Using the above equation,

Price(30) = 32,000*(1.045)^30 = 119,850

Which means that 32,000 today will be able to buy as much stuff as 119,850 in 30 years. Does this mean you have to save 119,850 today in order to to finance your target lifestyle? Well, yes, if you plan on keeping your savings in a piggy bank or under the mattress--if your retirement savings will earn zero or very little interest. But if you keep your retirement savings in an interest-earning vehicle, you won't have to save as much. In fact, if you invest in vehicles that provide returns that beat inflation, then you can even save an amount that is less than your target expense. But how much less?

To take investment returns and the time value of money into account, we need to use the present value concept. If you need an amount Price(t + n) in n years and invest your savings at time t in an instrument that earns a rate of return i per year, then the amount that you have to save and invest at time t is

Savings(t) = [Price(t + n)]/[(1 + i)^n]

Using the same example above, in order to accumulate 119,850 in 30 years by investing in an instrument that earns an average annual return of 7%, then you have to save

Savings(0) = 119,850/[(1.07)^30] = 15,744

Less than half of our original retirement expense estimate of 32,000.

Taking inflation and investment returns simultaneously by combining the two equations above, we get

Savings(t) = [Expenses(t)*(1 + g)^n]/[(1 + i)^n] = [Expense(t)]*[(1 + g)^n]/[(1 + i)^n

Where "Expenses(t)" is the estimated monthly or annual expense at time t. With the 30-60-90 approach, t = 0 and n = 30, so

Savings = Expenses*[(1 + g)/(1 + i)]^30

This equation shows that if your annual investment return is the same as the inflation rate, or i = g, then Savings = Expenses, or you have to save an amount equal to your projected future expense at today's prices (32,000 in the example above). If you invest such that i > g, like in the above example, then your savings requirement will be less than your estimated expenses (e.g., 15,744 vs. 32,000). Finally and most importantly, if your annual investment return is less than the inflation rate, such as if you invest in savings deposits, time deposits, or not at all, then you would need to save more than your estimated periodic expenses.

(1 + i)/(1 + g) is a special quantity in finance and economics that is referred to as the real rate of return on investments, for which we'll henceforth use the symbol r. It is the rate of return of an investment at constant prices, or at g = 0. Approximately, r = i - g, so that

Savings = Expenses/(1 + r)^30

To check, at i = 7% and g = 4.5%, r = 2.5%. If your target monthly expense is 32,000, then Savings = 32,000/(1.025)^30 = 15,256. Not exactly the same as the earlier result of 15,744, but close enough for all intents and purposes.

This last equation shows that as long as you invest your retirement savings in an instrument with a consistently positive real rate of return, then you can save an amount that is less than your estimated retirement expenses. But which instrument can reliably provide a positive real rate of return? Low-cost equity funds, particularly in long-term horizons such as 30 years. I'll discuss this in more detail in a future post, but if you want to look into it now, I suggest reading Jeremy Siegel's Stocks for the Long Run.

In the Philippines, the average annual inflation rate in the past decade is 4.5% and the average one-year change in the PSEi from 1994 to 2013 is around 8.5%. Assuming a PSEi dividend yield of 2%, the average annual return of the market is 10.5%, resulting in an average real rate of return r of 6% per year. For a conservatism, however, we can use a lower estimate for r, such as 5%.

Savings = Expenses/(1.05)^30

Savings = Expenses/4.3

For further simplification, you may want to round the divisor to 4, which is equivalent to r = 4.7%.

Savings = Expenses/4

To summarize, using the 30-60-90 approach and assuming that the average real rate of return of an equity fund is 4.7%, you need to save an amount equal to your estimated expenses divided by 4. It does not end here, though, because in order to realize your estimated real returns, you have to religiously invest your retirement savings in a low cost equity fund and withdraw no earlier than 30 years after.

Finally, I must clarify that the savings amount given by "Savings = Expenses/4" is only for the first month or period of the earning period, or at age 30 in the 30-60-90 framework. In the succeeding years, the savings amount must be adjusted by the annual inflation rate.

For example, if Expenses = 32,000 per month, then

Savings at age 30: 32,000/4 = 8,000 per month
Savings at age 31: 8,000*1.045 = 8,360 per month

...

Savings at age 55: 8,000*1.045^25 = 24,043 per month

But what if your situation does not adequately fit the 30-60-90 scenario, like if you're just starting to save for retirement at age 40? In a follow-up post, I'll show how you can adjust the savings formula to better reflect your situation.

***

EDIT: 2 July, 2013

I checked my numbers again, and the average one-year change in the PSEi from 1994 to 2013 that I got was 8.6%, not 12%. I will make the necessary changes in the above discussion to reflect this difference.