Example 7.

Suppose that you are age 25 today and plan on retiring at age 65. Your personal and family health history is such that you forecast that you will live to age 85. However, you might pass away as young as 75 or as old as 95.

In retirement, you would like to have purchasing power of $60,000 (i.e., real dollars) before taxes. Suppose, for our example, that you anticipate receiving $20,000 in inflation-adjusted Social Security payments each year. Hence, your portfolio will need to provide $40,000 in real dollars each year. How much will you need to have saved (in real dollars) by retirement?

To answer this question, we need to choose an appropriate forecast for the real rate of return on your portfolio if you do not buy a life annuity. If you do plan to buy a life annuity at retirement, we need to forecast the market interest rate at the time you retire, because the insurance company will apply this rate to determine your annuity payment.

This forecasting task is nearly impossible with any accuracy. Hence, we consider three scenarios that are consistent with historical rates of return. Table 1 is based on historical real rates of return on well-diversified portfolios composed of 50% large company U.S. stocks and 50% 5-year U.S. Treasury bonds, where the portfolio is rebalanced back to this mix once a year. The historical data is from 1926-2015.

Down market scenario: historical returns have been this low or worse for the given holding period about 17% of the time.

Average market scenario: average historical returns for the given holding period.

Up market scenario: historical returns have been this high or better for the given holding period about 17% of the time.

Notice that shorter periods have greater uncertainty about the returns.

Table 1. Forecast real annual rates of return based on historical experience

	Years in Retirement (investment holding period)
Market scenario	10 years	20 years	30 years
Down markets	1.1%	2.8%	3.8%
Average markets	4.9%	4.9%	4.9%
Up markets	8.7%	7.0%	6.0%

To illustrate, lets evaluate the scenario of a 10-year life in retirement during average market performance. To determine the real dollar portfolio that you need at retirement to sustain $40,000 of consumption per year (in real dollars), we work with an annuity due with 10 payments. Let V represent your portfolio value. At retirement, you take out $40,000 immediately for your first years expenses. That leaves nine more payouts. We can write the formula as

VCC111rNr

where C = $40,000, N = 9 years, and r is the return from Table 1.

Apply the financial function keys. For example, let r = 4.9%.

Press 2nd, then CLR TVM.

TYPE 40000, then press PMT.

Type 4.9, then press I/Y.

Type 9, then press N.

Press CPT, then press PV. Rounded to the dollar, the display shows -285,583.

Press the +/- key.

Press +.

Type 40000, then press =.

In this scenario, the required portfolio at retirement must total $325,583 (in real dollars).

Take a few minutes now to apply the same procedures to calculate portfolio required at retirement for the other scenarios. Then compare your results to those in Table 2.

Table 2. Real dollar portfolio that you need at retirement to sustain $40,000 of consumption per year in real dollars

	Years in Retirement (investment holding period)
Market scenario	10 years	20 years	30 years
Down markets	$380,973	$623,229	$735,726
Average markets	$325,583	$527,377	$652,446
Up markets	$282,763	$453,424	$583,629

If you live only 10 years in retirement and you experience up markets during retirement, then you only need about $282,763 (real dollars) in your portfolio at retirement. However, if you survive 30 years in retirement but you experience down markets, then you need about $735,726 (real dollars) in your portfolio at retirement.

Table 2 illustrates the combined effects of longevity risk and market risk. For example, if you have a portfolio of $527,377 (real dollars) at retirement and plan to take payouts of $40,000 (real dollars), then you will fall short if

you live longer than 20 years after retirement (longevity risk) and/or

during your retirement the market performs worse than average (market risk).

Now we are ready for stage 2. Suppose that your target in real savings is $527,377. We simplify by treating your real annual savings for retirement as an ordinary annuity. We can write the formula as

FVAC1rN1r

where

FVA is your target portfolio at age 65: in this case, FVA = $527,377;

C is the annual real dollar contribution to your retirement account;

N = years to retirement: if you begin saving at age 25, then N = 40 years;

r = market rate of return forecast for your savings period.

We want to know the annual contribution. Thus, solve the last equation for C:

We know FVA and N. We need to forecast a real annual rate of return for your portfolio for the next 40 years. Suppose that your portfolio consists of a mix of 50% large company stocks (held in a well-diversified portfolio) and 50% 5-year Treasury bonds, where you rebalance back to this mix each year. Table 3 lists estimates based historical returns in the U.S. For 40 year holding periods, look at the last column in Table 3. Suppose that we assume that the average markets forecast is appropriate. Then, applying equation (2):

C5273771046401046481030

Because these types of calculations involve so much uncertainty, listing the solution to the penny exaggerates the precision of our answer. Hence, it is sufficient to report our result to the dollar: $4,810 (in real dollars each year).

Table 3. Forecast real annual rates of return (50% large company stock/50% 5-year Treasury bond portfolios; rebalanced annually back to the target mix) based on historical data

	Years of Saving (investment holding period)
Market scenario	20 years	30 years	40 years
Down markets	2.8%	3.8%	3.8%
Average markets	4.9%	4.9%	4.6%
Up markets	7.0%	6.0%	5.4%

If we wish to analyze several scenarios, then using the financial function keys in the BA II Plus allows us to carry out a series of these calculations much more quickly. For example, let r = 4.6%.

Press 2nd, then CLR TVM.

TYPE 527377, then press FV.

Type 4.6, then press I/Y.

Type 40, then press N.

Press CPT, then press PMT. Rounded to the dollar, the display shows -4,810.

In this scenario, you must contribute $4,810 in real dollars at the end of each year in order to reach your goal. Thus, for example, if inflation in your first year is 5%, then you would deposit $4,810 x (1.05) = $5,050.50 nominal dollars. (That is, you would write your check for $5,050.50.)

Take some time to apply the same procedures to calculate the annual contribution that you must make to reach your goal of $527,377 (real dollars) at retirement:

if you start saving at age 25, but you experience a down market or an up market over your savings phase (fourth column in Table 3);

if you start saving at age 35 (hence for 30 years until retirement) and for all three market scenarios (third column in Table 3);

if you start saving at age 45 (hence for 20 years until retirement) and for all three market scenarios (second column in Table 3).

Compare your results to those listed in Table 4.

Table 4. Annual real dollar contribution required to reach $527,377 (real dollars) at retirement at age 65.

	Years of Saving (investment holding period)
Market scenario	20 yrs (start at age 45)	30 years (start at age 35)	40 years (start at age 25)
Down markets	$20,029	$9,722	$5,817
Average markets	$16,119	$8,075	$4,810
Up markets	$12,864	$6,671	$3,957

One of the important lessons of Table 4 is that the longer you wait to begin saving, the more you have to save. Moreover, the relation is exponential, due to the effect of compounding of interest.

Keep in mind that Table 4 reflects savings required under various circumstances to achieve the retirement portfolio that would support $40,000 (real dollars) per year in consumption in retirement IF you only live 20 years in retirement AND markets perform about average during retirement. But that is not the worst case. In Table 2, the worst case is that you live at least 30 years in retirement and you experience down markets during that time. To protect yourself against that scenario, you need to target $735,726 (real dollars) in savings.

Change your target to $735,726 and recalculate the annual (real dollar) savings required to reach this goal under the scenarios in Table 3. Then compare your results to those in Table 5.

Table 5. Annual real dollar contribution required to reach $735,726 (real dollars) at retirement at age 65.

	Years of Saving (investment holding period)
Market scenario	20 years	30 years	40 years
Down markets	$27,942	$13,562	$8,115
Average markets	$22,486	$11,265	$6,711
Up markets	$17,947	$9,306	$5,521

Recall that the three types of uninsurable risk during your savings phase are inflation risk, market risk, and risk that you wont save enough.

Inflation risk is addressed by adjusting your contributions annually for inflation.

Risk that you wont save enough is the possibility your financial misbehavior leads to poor outcomes.

You have no control over market risk, so plan for the worst.

Which column in Table 5 applies is determined by your decision on when to begin saving for retirement. The row that applies depends on how much you feel like rolling the dice concerning your standard of living in retirement. For example, if you gamble on an up market over your savings phase, then you contribute less to your retirement account. However, you risk falling short of your goal if markets are average or worse.

HERE IS THE PROBLEM! Still DON'T UNDERSTAND!

Suppose that you are age 25 today and plan on retiring at age 65. You determined that you need to have saved $621533 in real dollars by the time you retire. How much must you contribute (in real dollars) each year to your retirement account to achieve your goal? Assume the following:

you make annual contributions;

each contribution is the same amount in real dollars;

the first contribution will be one year from today;

your last contribution will be at age 65;

your portfolio earns a real annual rate of return of 5.58%, compounded annually.

Do not round at intermediate steps in your calculation. Round your answer to the nearest dollar. Do not type the $ symbol.