Life insurance calculations use compound interest. Premium and reserve computations assume that companies keep funds continuously invested until those funds are paid out in settlement of claims. The company�s computations further assume that interest earnings are added to the original principal and reinvested. To understand the relationships, it will be helpful to examine four basic compound interest series.

The first series shows the amount to which a principal sum of $10,000 invested for a number of years (or other units of time) will increase over time. This applies compound interest to accumulate interest earnings. Figure 14-2 illustrates this concept. If you invest $10,000 at 5.5 percent for one year, the combined amount of principal and interest at the end of the year, according to the simple formula previously stated, will be $10,000 x 1.055, or $10,550. If the $10,550 is then invested for another year at 5.5 percent, the combined amount of principal and interest at the end of the second year will be $10,550 x 1.055, or $11,130. This is equivalent to multiplying $10,000 by 1.055² where the exponent�indicated in this example by the superscript "2"�denotes the number of times the base�in this case 1.055�is multiplied by itself. If the sum of $11,130 is again invested for another year at 5.5 percent, the principal and interest at the end of the third year will be $11,130 x 1.055, or $11,742, which is the equivalent of multiplying $10,000 by 1.055³. If this process continues, the accumulated principal and interest will always equal the sum obtained by multiplying $10,000 by 1.055 raised to an exponent equal to the number of years of compound interest earned.

The future value of any principal sum invested for any period of time at any rate of interest can be computed by the same process. If S represents the future value, or sum, at the end of the period, i the rate of interest, P the principal invested, and n the number of years, the general formula becomes S = P (1 + i)ⁿ. If the principal is $1, as is shown in most compound interest tables, the formula is S = (1 + i)ⁿ. Table 14-1 shows the accumulated amount of 1 at the end of each of 30 years at various rates of compound interest. A student who wants to compute the future value of $500, for example, at the end of 23 years with interest at 5.5 percent compounded annually looks up the factor (the number in the table opposite 23 years in the column headed 5.5 percent), 3.4262, and multiplies that factor by 500. This produces the answer of $1,713.10.

Instead of reading 3.4262 from a compound interest table, the y^x function of a calculator could be used. To solve the same problem, a calculator that uses so-called algebraic notation, such as the TI BA-II Plus, would require the following set of key strokes:

Pressing the �=� key then produces the answer, the factor 3.4262. For calculators like the HP 12-C, using so-called �reverse Polish� notation, the future value factor is found with the following series of key strokes:

Those with ready access to a computer spreadsheet, such as Lotus 1-2-3, would enter the formula (1.055)^23 into a spreadsheet cell and press ENTER to obtain the needed solution.

Likewise, future value problems can be solved using a financial calculator. For most calculators with financial function keys, the future value of an initial $500 principal at the end of 23 years with interest at 5.5 percent compounded annually requires the following key strokes to produce the $1,713.10 answer:

Despite the fact that different strokes are needed to solve mathematical formulas, the key strokes for financial functions are similar for algebraic notation and reverse Polish calculators (although the former may require that a CPT [computer] key be pressed before the FV key). If you do not obtain the correct answer, check your calculator to be certain it is set for beginning-of-year computations.

The spreadsheet solution with an initial $500 principal amount is a simple extension of the formula (1.055)^23. Enter the formula (1.055)^23*500 to verify the $1,713.10 answer.

If a higher rate of interest is used in the compounding process, or if a longer period of accumulation is used, a higher future value results. (While this is a somewhat obvious statement in the context of future value, it is important to make this explicit statement here because similar relationships that the student will encounter later in this chapter are less obvious.)

The second series, called present values, calculates the value today that is equivalent to a given sum due at a designated time in the future. This process is called discounting. The process of discounting is particularly vital to life insurance company operations since the companies deal heavily in future promises. Their contracts provide for benefits to be paid in the future. Companies finance these benefits through premium income and future interest earnings. Their very solvency depends on establishing an equivalence between future benefit payments and receipts from premiums and investments. This equivalence is established through the discounting process whereby all values are reduced to a common basis. That common basis is present value.

The discounting process implicitly recognizes that a dollar due one year from now is worth less than a dollar due now. And a dollar due 5 years from now is worth less than a dollar due in one year. Money in hand can be invested to produce more money or capital. The difference in value between money in hand and money due in the future depends on the rate of return that can be obtained from invested capital. Also the longer the period before the future money will be received, the greater the expected interest earnings and the greater the difference between the value of the present and future capital.

The present value of an amount due at a specified date in the future is that principal sum that, if invested now at an assumed rate of interest, would accumulate to the required amount by the due date. For example, $1 invested at 5.5 percent compound interest will accumulate in 10 years to $1.71. This is the meaning of the statement that $1 is the present value at 5.5 percent compound interest of $1.71 due in 10 years. This suggests how easily the present value of a sum due in the future can be obtained by reversing the accumulation process described above. The higher the rate of return available on invested capital, the greater the difference between the value of present and future money, and vice versa.

As explained earlier, the amount to which a given sum will accumulate is found by multiplying the principal by 1 plus the rate of interest raised to an exponent equal to the number of interest-compounding time units in the period (years in our example). If, instead, the amount at the end of the period is given, the beginning principal may be found by dividing the future amount by 1 plus the rate of interest raised to the proper exponent. For example, $1 invested for one year at 5.5 percent interest will accumulate to $1.055 by the end of the year. If the process is to be reversed, the principal is found by dividing $1 by 1.055. If the amount due 2 years from now is $11,130, the present value of the amount at 5.5 percent compounded annually is $11,130 ÷ (1.055)² = $11,130 ÷ 1.1130 = $10,000. Similarly, if $10,000 is due one year from now, it is worth only $10,000 ÷ 1.055, or $9,479, today, based on an interest rate of 5.5 percent.

If $10,000 is to be paid 2 years from now, its value now is only $10,000 ÷ (1.055)² = $10,000 ÷ 1.1130 = $8,985. Figure 14-3 illustrates this derivation. Conversely, $8,985 invested at 5.5 percent compound interest will, at the end of 2 years, amount to $10,000.

The present value of 1 at 5.5 percent compound interest can be derived for any number of years and arranged in tabular form for convenient use by use of this general formula,

In standard actuarial notation the present value of 1 due n years from now is represented by the symbol vⁿ where v = 1/(1+i) and n = number of years:

These values are shown for 30 years in table 14-2 or can be computed easily using a financial calculator or computer spreadsheet.

Once you have a table of values for vⁿ, the present value of any amount due at the end of any number of years is determined by multiplying the future amount by the appropriate value of vⁿ. Thus the present value of $1,000 due 15 years from now is, at 5.5 percent compound interest, $1,000 x 0.4479, or $447.90.

Once again, the financial calculator can produce the same answer. For the present value of $1,000 to be received at the end of 15 years with interest at 5.5 percent compounded annually requires the following key strokes:

Pressing the PV key, or the CPT and PV keys, yields an answer of $447.93 (the 3 cents difference between table value and the calculation value is due to the rounding in the table values). The spreadsheet user would enter the formula (1/1.055)^15*1000 to obtain this solution.

One final note about present values is important for anyone working or investing in the financial marketplace. Sometimes banks and other commercial organizations use approximation methods that produce results slightly different from those obtained by the precise determination of present values described here. For example, interest may be deducted in advance. For small amounts and short terms the differences produced by the alternate methods of discounting are inconsequential.

The relationship between interest rates and discounting period to present value is an inverse one. The magnitude of the difference between present and

future values depends on the interest rate that is used for discounting. Assuming a higher rate of interest means that more interest is made or lost over time. That means the present values of future amounts are smaller for higher interest rates.

For example, the present value of $100 to be paid 20 years from now based on 3 percent interest is $100 x .5537, or $55.37. The present value assuming 5.5 percent interest is $100 x .3427, or $34.27. A higher rate of interest means a lower present value of a specific future value. Similarly, the longer the discounting period, the lower the present value. For example, the present value of $100 to be paid 30 years from now based on 3 percent interest is $100 x .4120, or $41.20, compared to $55.37 for a 20-year discounting period.

The third compound interest series deals with the future value of periodic equal annual payments. This stream of annual payments is called an annuity.

An annuity with payments at the beginning of the period is called an annuity due. Premium payments on most life insurance policies, although they include a mortality factor as well as interest, are one common example of beginning-of-period payments. Figure 14-4 shows the future values based on a 5.5 percent interest rate of four $1 payments made at the beginning of the year. Adding together the future values of the four payments�$1.055 + $1.113 + $1.1742 + $1.2388 gives us the $4.5810 future value of all four payments.

The first number in the table 14-1 extract is the same as the first figure in the 5.5 percent column of table 14-3. To obtain the second number in table 14-3�2.168 �add the first two numbers above. The third number in the table 14-1 extract, 1.1742, is added to 2.1680 to produce 3.3423, the third number in table 14-3 in the 5.5 percent column except for a difference due to rounding. Add 1.2388 to this to yield 4.5811, the fourth number in the 5.5 percent column in table 14-3, and so on. To repeat, the values in table 14-3 assume payments of $1 occur at the beginning of each year.

An annuity with payments at the end of each period is called an ordinary annuity. A homeowner�s mortgage with its payment due at the end of the first month is a common example of an ordinary annuity. Figure 14-5 illustrates a
4-year ordinary annuity. Figure 14-5 can be compared with figure 14-4 to better understand the difference between this and a 4-year annuity due. Three of the four payments align perfectly, with a payment of $1 at the end of year 0 in table 14-5 corresponding exactly to a payment of $1 at the beginning of year 1 in table 14-4 and so on.

Suppose you need to find the amount to which $1,000 per year, payable for 5 years at the end of each year, will accumulate by the end of the last period at 5.5 percent interest. Since the first payment is not due until one year from now, interest will be earned for only 4 years. From table 14-3, the amount of $1 per annum in advance at 5.5 percent interest for 4 years is $4.5811. The last of the five payments, however, will be made at the end of the period and will not earn any interest. This last payment of $1 added to $4.5811 yields $5.5811, the amount of $1 per annum payable at the end of each year over a 5-year period at 5.5 percent interest. Since $1 payments made in this manner will amount to $5.5811, then $1,000 payments made in the same manner will, obviously, amount to $1,000 x 5.5811, or $5,581.10.

Most financial calculators can be set directly to do either beginning-of-year or end-of-year calculations of the future value of an annuity. The end-of-year calculator solution to finding the future value of $1 per year for 5 years and annual compounding of interest entails the following keystrokes:

When the user presses the FV (or CPT and FV) button, the calculator produces the answer of 5.5811, which is 1 plus the 4th-year value from table 14-3.

Computer spreadsheet functions usually assume end-of-year payments. The typical spreadsheet function to obtain the same answer is formatted @FV(payment, interest rate, term of years). This problem is solved with the formula @FV(1,0.055,5). Spreadsheet users enter the interest rate as a decimal rather than as a percent.

Test your spreadsheet or calculator�s financial functions to be certain that you understand whether it is performing computations that assume payment at the beginning of a year or the end of a year. Enter the three factors specified above for 5 years of payments at 5.5 percent interest. If the answer is 5.5811, then the computation assumes end-of-year payments. If the answer is 5.8881, then the computation assumes beginning-of-year payments. The HP-12C calculator is easily set for beginning or end-of-period payments through the function and the BEG or END key. The BA-II Plus calculator is set by using the 2nd, BGN, and SET keys.

Another way to check your calculator or spreadsheet assumption about timing of payments is to solve for the future value of a single payment of $1 at an interest rate greater than zero. If the payments are assumed to be at the end of the period, the calculated FV will be $1. If the payment is assumed to be made at the beginning of the period, the calculated FV will be greater than $1 because of the interest earned during the period.

Also be certain you understand how interest is being compounded. Many manual calculations assume annual compounding. Some calculators, including the HP-12C and the BA-II Plus, can be set for monthly, weekly, or other compounding intervals. The interest rate must be appropriately adjusted to coincide with the length of the compounding period.

The fourth and final compound interest series determines the present value of a number of equal annual payments at various rates of interest. This series is used by life insurance companies to find amounts payable under the settlement options that liquidate insurance proceeds over several years. An understanding of this series is particularly important here so that we can extend it in the next chapter to include computations that combine these values with life contingencies.

Table 14-4 showing the present value of annuity payments is derived from the values presented in table 14-2 in the same manner that table 14-3 was derived from the values in table 14-1. As in table 14-2, the values in this series are for annual payments of $1 due at the end of each year (ordinary annuity).

(Adjustments similar to those described in the previous section can be made to derive the present value of payments due at the beginning of each year.) Remember that in table 14-2 at 5.5 percent interest, the present value of $1 payable one year from now is $0.9479. The present value of the second payment of $1 due 2 years from now is $0.8985; the payment due in 3 years is worth $0.8516 now. The present value of all three payments is, of course, the sum of the present values of each taken separately, or $2.6980, as depicted in figure 14-7.

The present value factors in table 14-4 are based on the assumption that each $1 payment is made at the end of the period. Many life insurance calculations involve payment streams that are made at the beginning of the period, such as premium payments. When the number of payments, the interest rate, and amount of each payment are all the same, the present value of an annuity due is higher than the present value of a corresponding immediate annuity. This is purely a result of the earlier timing of the payments.

In order to convert ordinary-annuity present values factors into annuity due present value factors, first, find the ordinary-annuity factor for the appropriate interest rate and for one payment period less than the total number of payments to be made. Then add 1.000 to that factor to represent the $1 payment at the beginning of the stream. The result is a present value factor representing the specific annuity due payments of $1 per period.

Figure 14-8 depicts a 3-year annuity due of $1 per year. The first payment is not discounted because it is made now and worth the full $1 amount. The second payment is one year away and is worth $0.9479 now when discounted at 5.5 percent interest. The third payment is worth $0.8985 now. All three $1 payments have a present value equal to the sum of their individual present values ($2.8464 in this example). The difference between the present value of the 3-year annuity due and the corresponding ordinary annuity does not seem important. However, consider the amount involved if each annual payment is for $10 million instead of $1. The difference in timing amounts to $1,484 million, which is one year of additional interest earned when each payment is made one year earlier.

By the process of cumulative addition, the present value of a series of annual payments of $1 for any number of years can be found. Table 14-4 shows these values at various rates of interest for durations up to 30 years. The value for 2 years in table 14-4 is the sum of the first two values in table 14-2. The value for 3 years is the sum of the first three values in table 14-2 (2.6979, which is the same present value depicted in figure 14-7), and so on.

Annuity values, typically based on a present value of $1, are used in several different ways by life insurance companies. For example, determining what payment amount each $1 of death benefit can buy if paid to the beneficiary in 20 annual end-of-year installments under the fixed-period settlement option earning 5.5 percent interest requires the present value of $1 per annum for 20 years, which is $11.9504. This number is then divided into $1 to yield $0.08368. Stated differently, at 5.5 percent compound interest, $1 will provide 8.4 cents at the end of each year for 20 years. Therefore, each $1,000 of death benefit would provide $1,000 x 0.08368, or $83.68 per year for 20 years.

table 14-4 to reflect the present value of a series of periodic payments due at the beginning of the year. The adjustment involved is simple and very similar to the adjustment made for future values.

Consider, for example, a series of annual payments of $1 due at the beginning of each year for 25 years. The first payment is due immediately and is worth $1. Twenty-four additional payments remain, the first of which is due one year from now. The present value of the entire series is obtained by adding $1 to the present value of $1 per annum (end of year from table 14-4) for one year less, in this case, 24 years. The result at an interest rate of 5.5 percent is $13.1517 + $1, or $14.1517.

Financial calculators and spreadsheets are also used to perform present value computations. The financial calculator solution to valuing a temporary annuity that pays $1 per year at the end of each of 20 years at 5.5 percent interest requires input similar to that used for future value computations:

Then press the PV (or CPT and PV) button. The present value of $1 per annum for 20 years is $11.9504 as before. If this is divided into 1, the annual payment for 20 years purchased by $1 is $0.08368. The typical computer spreadsheet function to obtain the same answer is formatted @PV(payment, interest rate, term of years). This problem is solved with the formula @PV(1,0.055,20).