# Understanding. In Cash Value

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1 13 LMR MaY 2012 Understanding In Cash Value Life Insurance Understanding Interest Rates in Cash Value Life Insurance

3 15 LMR MaY 2012 \$1,000 \$900 \$800 \$700 \$600 \$500 \$400 \$ \$ \$300 \$200 \$ \$ \$100 \$0 \$41.95 \$ FIGURE 1. Present Discounted Value (PDV) of \$1000 Bond at Different Discount Rates sixty-five years, when the man will happen to be 100 years old. Now the question is, how much should the man value that promise right now? Another way of putting it is to ask, if the man can sell this IOU from the company, how much would he be able to fetch for it in the marketplace? Let s take risk out of the analysis entirely, and assume that no one has any doubt whatsoever that the company will be around in sixty-five years, and that it will indeed honor its promise to pay \$1,000 at that time. Clearly the IOU or what we will call a bond from now on isn t currently worth the full \$1,000, because a dollar today is more valuable than a dollar that will only be delivered decades in the future. That means we have to discount that future \$1,000 payment. In order to calculate a total discount for the entire period, it EVEN WITH THIS SIMPLE BOND EXAMPLE, we can illustrate a distinction that comes up in the analysis of life insurance: calculating the present value of an asset using either a prospective or a retrospective approach. Understanding Interest Rates in Cash Value Life Insurance

4 16 LMR MaY 2012 \$1,000 \$900 \$800 \$700 \$600 \$500 \$400 \$300 \$ \$ \$1.98 \$7.28 \$0.46 \$200 \$ \$100 \$ FIGURE 2. End-of-Period Value of \$1000 Bond With Various Discount Rates and Implied Level Annual Payments is standard practice to assume the man uses an average annualized discount rate. Figure 1 shows the present discounted value (PDV) of the bond, at various points in the man s life, at three different discount rates. Figure 1 shows three different trajectories for the present discounted value of the \$1,000 bond, as the owner of the bond ages. At a 2% discount rate, that future \$1,000 payment is valued more highly in earlier time periods it is discounted less than in the other scenarios. That s why the red line is consistently higher over the man s life, until finally in the 100th year the other lines finally catch up to it. Notice that if we fix the ultimate payout, then there is tradeoff between the height of the PDV at any time, and the rate of its growth. In other words, the red line is always higher than the other two lines, but the bond s value in that trajectory grows the most slowly (at only 2% per year). In contrast, the green line is consistently below the other two lines, yet the bond s market value grows very quickly here (8% per year). Even with this simple bond example, we can illustrate a distinction that comes up in the analysis of life insurance: calculating the present value of an asset using either a prospective or a retrospective approach. In our example above, consider the market value of the bond at age 80. At a discount rate of 2%, Figure 1 tells us that the value is \$ There are two (equivalent) ways of arriving at this figure. In the prospective approach, we look at future events and use them to determine the present value. In this simple case, the only cash flow that will occur is a payment of \$1000 to the man, which will happen (from his perspective at age 80) in twenty years. If we divide that \$1,000 payment by ((1.02)^20), we end up with \$ To repeat, this is the prospective approach. On the other hand, we could reach the same figure by the retrospective approach. The original market value of the bond what the man would have had to pay for it at auction was A Understanding PRIMER ON THE MEC Interest RULES Rates in Cash Value Life Insurance Photo from Flickr by: stevendepolo

5 17 LMR MaY 2012 \$ That original investment then grew at a 2% compounded annual rate for forty-five years, so its present value is \$ times ((1.02)^45), or \$ Thus we see that the prospective and retrospective approaches yield the same current market value, at least if we assume nothing relevant changes during the man s lifetime regarding the discount rate or the cash flows associated with the bond. 1 Baby Step 2: Level Contributions for a Certain Payout Now let s introduce another layer of complexity, inching us closer to our ultimate goal of a whole life insurance policy. In this baby step, the financial institution is still promising to pay the man \$1,000 when he reaches age 100. In this scenario, however, he is obligated to make a level stream of annual payments to the company from age 35 onward, in order to remain eligible for the \$1,000 payout. We ll further assume that initially the bond has zero value. So now the question is, what does the level payment need to be, at each of the hypothetical discount rates, in order to make the bond start at \$0 at age 35, 2 and end up at \$1,000 by age 100? By playing with an Excel spreadsheet, one can zoom in to find that the level payments are \$7.28, \$1.98, and 46 cents if we use discount rates of 2%, 5%, and 8%, respectively. Figure 2 shows the trajectories of their market values in this new setting. Here too we can use either the prospective or retrospective method to calculate the market value at any particular age, though the calculations are trickier. The retrospective method is quite intuitive, since the asset in this case behaves just like a savings account with a conventional bank, which is growing at interest while the man continually pumps in more saving each year. For example, using the 5% discount rate, pumping \$1.98 in at the beginning of each year to add to the previous year s end-of-period market value, and then letting the whole sum grow 5% during the current year, will lead to and end-of-period value of \$ at age 80. We can get the same result by forgetting the past, and just focusing on the future (i.e. by using the prospective approach). If the man at the end of his 80th year evaluates the future cash flows, he sees that he will receive \$1,000 from the financial institution in 20 years (I assume the payment comes at the end of the year). Thus the benefit is worth \$ just as we reckoned in the earlier section, when calculating the PDV of a simple bond. However, in our new scenario, this number would be overstating the value of the asset. In order to get his hands on that \$1,000 payment when he is 100 years of age, the man must continue to make his level \$1.98 contributions for the next twenty years, as well. From his vantage point at the end of his 80th year, the present value of that stream of contributions discounting Now, PUT YOURSELF IN THE POSITION of the financial institution which at this point we might as well start calling the insurance company. If the man wants to pay a level premium, what do you charge him to make sure you cover yourself? Understanding Interest Rates in Cash Value Life Insurance

7 19 LMR MaY 2012 TABLE 1. Values at Various Ages Using Net Amount at Risk (NAR) Approach (premium=\$11.22, bop=beginning of period, eop=end of period) For example, suppose our man dies at age 80. You the insurer can pay his beneficiary the \$1,000 out of the \$10 term premiums collected from everyone in the pool of customers that year, leaving the \$ (which had been accumulating from the \$1.98 portion of the premiums since age 35) free and clear. The longer the man lives and each year, he has a 99% probability of continuing on for another the larger the fund grows. So if \$1.98 is too low a premium, and \$11.98 is too high, how do you the insurer figure out the exact actuarially fair amount to charge the man, for what is now an ordinary whole life insurance policy that completes at age 100? Baby Step 4: Introducing Net Amount at Risk (NAR) Approach Actuaries have a very elegant solution to this pricing problem. The mistake we made in the previous section was to charge the break-even term premium for the full \$1,000 every year. Instead, all you as the insurer need to do is charge the term premium on the current difference between the death Understanding Interest Rates in Cash Value Life Insurance

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