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 30year 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 15year 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 15year
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.
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