## Tuesday, May 21, 2013

### Timing of market slumps and returns on lump sums and annuities.

Timing of market slumps and returns on lump sums and annuities

The question today has two parts.  The first part addresses the issue of how the timing of changes in the market impact returns on a lump sum.  The second part considers how the timing of market changes impacts returns on an annuity.

We consider two market scenarios for a 30-year investment horizon.  Returns for the first scenario are 8.0%/12 for the first 180 months and 0% for the second 180 months.  Returns for the second scenario are 0% for the first 180 months and 8.0%/12 for the second 180 months.

 Monthly Returns Scenario One Scenario Two First 15 Years 0.6667% 0.0000% Second !5 Years 0.0000% 0.6667%

Situation one:  A person invests a lump sum of \$10,000.  What is this person’s final wealth after 30 years under the two scenarios?

Analysis:   The wealth in both cases is W=\$10,000 x (1.006667)180 or \$33,069.

Situation two:  A person invests \$500 per month for 360 months.  What is the person’s final wealth after 30 years?

Step one involves the calculation of the future value of the annuity after 15 years.  This is done with the FV function in Excel.  The only difference across scenarios is that the input for the monthly interest rate is 0.667% for scenario one and 0.0% for scenario two.

Step two involves the calculation of the future value of the annuity after the second 15-year period.  There are two differences in the inputs to the FV function.  The interest rates are now 0.0% and 0.6667% for scenario one and scenario two respectively.  The present value of the annuity at the start of the second 15-year period is \$173,019 and \$90,000 for scenario one and two respectively.

 Impact of order of stock return movements on the future value of an annuity Rate of return Total Wealth at end of period Rate of return Wealth at end of period First 15 years 8.0% \$173,019 0.00% \$90,000 Total 0.0% \$263,019 8.00% \$470,642

The second period calculation can be decomposed.  First, take the terminal value of the first period annuity and compound it with second period rates of return.  The future value after an additional 15 years of the lump sum accumulated after the first 15 years is \$173,019 for scenario one and \$90,000*(1+0.00667)180 or \$297,623 for scenario two. Second, add in the future value of payments made during the second period.

 Future Value Decomposed Scenario One Scenario Two Value of first period annuity at 15 years \$173,019 \$90,000 FV of first period annuity at 30 years. \$173,019 \$297,623 FV of second period annuity at 30 years \$90,000 \$173,019 Total FV at year 30 \$263,019 \$470,642

All else equal, investors who invest a fixed amount per month want their best returns at the end of their investment horizon.   Poor returns at the end of an investment horizon affect both the accumulated lump sum and ongoing annuity contributions.

This result explains why on average the return loss from life cycle investing is so large compared to the reduction in risk.    The post below discusses these problems with life cycle investing.

This result also points to a major problem with 401(k) accounts and proposed private accounts as part of Social Security.  A large market down turn at the beginning of an investment plan is inconsequential because the lump sum component of the pot is small.  A large market down turn right before retirement is fatal to the nest egg.