This blog mini-series demonstrates the impact of fraud on retirement. Studying (and planning for) retirement is complex because it involves moving beyond simple time value of money (TVM) calculations to accurately gauge longevity risk as a function of anticipated rates of returns on one's retirement assets. In other words, whether one can safely pass away with money still in the retirement account largely depends on the performance of one's portfolio. This first post explains the inadequacy of TVM and the consequent necessity of developing models that account for uncertainty in retirement planning.

*Time Value of Money: A Primer*

TVM can be a powerful tool to financial plan for one's future. Yet, it is limited by static assumptions that cannot capture the complexity of asset performance when those assets are activate traded in a capital market. TVM calculations contain five variables:

PV = present value (the amount your money is worth today)

N = the number of compounding periods within a period of time (usually within one year)

I = interest rate (often synonymous as expected rate of return)

PMT = payment; the amount that is systematically added or subtracted from the account

FV = future value (the amount your money is worth at the end of the period)

Usually, time value of money is expressed in years (since we tend to equate rates of return with years or the number of years until we retire, etc.). For example, we might say that $1 in today's dollars is the same as $1.22 in 10 years (assuming an inflation rate of 2%). Any financial calculator can perform TVM functions as can Microsoft Excel. It is quite handy to be able to quickly calculate how long a current deficit will last until funds within an account are exhausted. It can also tell you how long you must save for a specific goal such as a trip or a new car.

Nevertheless, whether you are using TVM to compute the accumulation (saving for a goal) or decumulation (withdrawing funds from your account in retirement), both fail to capture the complexity of oscillations in asset value within a capital market setting. TVM *does *showcase the power of compounding. Each period during the accumulation phase when you are working, you are adding the interest your assets make to the principal. The next period, the interest rate, being multiplicative, is applied to an even larger principal than the previous period. Put differently, TVM calculations accurately model exponential growth of one's assets over time where a linear calculation cannot.

*Why TVM Matters: Compound vs. Simple Interest*

Suppose you have a choice between two deferred annuities. Annuity C will credit your account 5% *compounding* interest each year. By contrast, Annuity S will credit your account 10% *simple *interest each year. Simple means a snapshot is taken of the original account value, and 10% of that initial value is added to your account each year. Compounding means the interest is added to the principal, which together combine to form a new principal upon which future interest will be calculated. Now, suppose you put $100,000 into each annuity at Year 0. After 10 years, which one will have more money? The table below shows a year-by-year comparison:

After 10 years, the simple interest annuity (Annuity S) is now valued at $200,000 because 10% simple interest--based on the initial value, or $100,000--has been added to the account each year for 10 years. By contrast, Annuity C's value is $162,889 (rounded). The important thing to note is that the difference between the two values continues to widen; In Year 1, the gap is only $5,000, but by year 10, it is $37,000. Now, see what happens when another 10 years pass:

Years 14 and 15 show the widest gap between the two annuities at $42,000. However, beginning in Year 16, Annuity C begins to catch-up. By Year 20, Annuity C has narrowed the lead to $35,000. Now, look at Years 21 - 30:

Year 26 is the final period in which it makes sense to have invested in Annuity S over Annuity C. Beginning with year 27, the value of Annuity C is greater than S. Not surprisingly, Annuity C's lead over Annuity S continues to widen. This is because of the compounding effect that is present within Annuity C but not Annuity S.

Figuring out how much money you will possess within a simple-interest annuity is straightforward since it's a linear function:

A = P(1 + rt)

A = accrued amount (basically, the value of the account, or the sum of principal + interest)

P = principal amount

r = rate of interest per time period

t = time period (usually measured in years)

Thus, if we plug-in the values from the example above, we have:

$400,000 = $100,000(1 + .10*30)

The formula for compound interest is a bit more involved and can be done using the TVM equation:

FV = PV * [1 + (i/n)] ^ (n*t)

FV = future value

PV = present value

i = interest rate

n = number of *compounding *periods per year

t = number of years

Again, plugging-in the values of Annuity C above:

$432,194 = $100,000 * [1 + (0.05/1)] ^ (1*30)

*The Glaring Limitation of TVM*

To reiterate, time value of money is a powerful tool **when static assumptions hold**. Unfortunately, as we step into the real world, we know that inflation varies from quarter to quarter, people invest different amounts into their retirement plans each month, and capital markets oscillate in terms of annual growth. TVM is necessarily uni-directional; Within the same calculation, you cannot have net cash inflows and net cash outflows. Year 1 cannot produce a positive rate of return while Year 2 yields a negative one. This is one of the major limitations of TVM. Interest rate in particular (for the purposes of retirement planning) is the Achilles' heel of TVM.

Capital markets (such as stocks and bonds) are couched in uncertainty. Share prices of a company's stock fluctuates over time and is particularly sensitive to the news media. Dividends are not guaranteed especially within a specified period of time (even for preferred stockholders). While bond interest rates are often fixed, many of these instruments contain provisions whereby the yield can be cut short such as with callable bonds (where the issuer may "call" the bonds by paying the creditors a specified amount to pay-off the debt ahead of schedule) or in the case of convertible bonds (where the investor may convert the bond into common stock shares of the issuing company). Also, many investors do not purchase bonds directly but instead invest in bond funds where a fund manager trades bonds on a daily basis. In this scenario, the price of specific bonds are more important than collecting interest in a buy-and-hold strategy, and investors own shares of the bond fund itself rather than owning the individual bonds directly. Bond prices vary inversely with interest rates, and the degree to which this is the case is a bond's *duration *(the bond's price sensitivity to changes in interest rates).

Given this uncertainty, how does one go about calculating or predicting the future retirement account value based on shifting returns of stocks and bonds? This is where Monte Carlo analysis (MCA) enters the picture. By running thousands of simulations using historical data, MCA can closely approximate the likelihood of the account maintaining a positive value at the end of the retiree's life. The next post in this mini-series will discuss the mechanics of Monte Carlo analysis and demonstrate the application of the MCA procedure to retirement planning specifically.

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