Using the Rule of 72 to compute residual income?

Often I check an investment strategy's performance by summing the year-to-date realized gains/losses.

I take the sum of all closing(or market for unrealized) values CT, subtract the sum of all cost basises (sp?) OT, and divide that difference by the total cost basis OT. That gives me raw performance year-to-date. So in terms of percentage gain/loss:

``````performancePercYTD =  ((CT - OT) / OT - 1) * 100
``````

From there it can be roughly annualized into an APR performance `= performancePercYTD * 365.25 / day_of_year`

Question. Using the Rule of 72 how can one estimate:

• The doubling frequency in days
• Projected portfolio value in 90 days
• How long before at this pace, the account will reach \$10000
• How long before the first sustainable \$500/month withdrawal indefinitely will not deplete the principal, assuming the current growth rate continues?
• The reverse of the last question, for example, what minimum amount could be a sustainable monthly withdrawal in 24 months(730.5 days)?

It's an exercise I would like to script for daily computations, since of course variables change constantly. Thanks!

• I'm beginning to realize there is a fundamental flaw in the way I've calculated my portfolio growth rate. `OT` above is the sum of all costs, included the same dollar that has already been used to own & release many different equities over time. What I probably need to is figure out somehow what was the account's starting value on Jan 1st, and then account for all external deposits and withdrawals since, not related to trading or dividends or margin interest fees. Mar 19, 2012 at 16:19

If your investment is growing at the rate of x% per period, then after n periods, it will have grown by a factor of (1+x/100)^n. This formula also applies when n is not an integer. So, the investment will double in value for whatever n is the solution to the equation

2 = (1+x/100)^n

If you take logarithms on both sides and solve the equation, you can get the exact value of n. The Rule of 72 gives a very good approximation to the exact answer: it says that

the investment will double in value in 72/x periods.

If the period is an year, multiply 72/x by 365.25 to get the doubling period in days.

The projected value in 3 months cannot be obtained directly from the Rule of 72. If your annualized growth rate is x%, then after three months you should expect the value to have increased by a factor of

(1 + x/400).

It is impossible to say when the investment will have value \$100,000 without specifying the initial value. Taking the initial value to be P, we have to solve for n in the equation

100000 = P(1+x/100)^n

Once again, the Rule of 72 is not directly applicable, but we can fake it a bit. If P = 100, the increase is by a factor of 1000, slightly less than 1024 = 2^10. The investment will double in value each 72/x periods, and so increase by a factor of 1024 in 720/x periods. So, the exact value of n is somewhat smaller than 720/x periods in this example.

If your investment is generating x% gain each year, or (x/12)% gain each month, you can withdraw \$500 per month without depleting principal as long as your investment has value I at least as large as the solution to

I(x/1200) = 500, that is, I is at least 600,000/x

So, now that you know what I needs to be, if you start out with P, you can figure out how long it will take to reach I using the formulas above. From that point on, you can withdraw \$500/month in perpetuity without reducing the value of your investment.

• Thank you for a thorough answer. For simplicity I chose continuous compounding, being a portfolio with irregular and frequent asset&value changes, unlike CODs with steady interest rates etc. 72/x is a fine heuristic for quick mental estimates though for computers the actual formula Years = ln(2) / rate is not less convenient. In my question I used arbitrary number values for clarity although the variables will interest me more when coding. So in your reply to part2, by (1 + x/400) you meant (1 + (90/365.25) * x/100), correct? Mar 18, 2012 at 19:53
• You might want to use 91/365.25, but if you take 3 months to mean a quarter, then x/400 is pretty close to the number you want (technically, you need to use 1+y/100 where (1+y/100)^4 = 1 + x/100 but 1+x/400 is a good enough approximation to 1+y/100 and easier to compute. Mar 18, 2012 at 20:30