Elements of Life Insurance Policy

Life insurance protects an individual's beneficiaries against a loss of income. Pricing a life insurance policy is a time value of money issue that takes some practice to get the handle on. Insurance companies use the law of large numbers in their favor; that is, it is very difficult to predict the lifespan of any one individual, but with a sufficiently large population, this estimation can be made with surprising precision.

The company must take in enough premiums to cover the claims and may invest the premiums in the meantime before payout, earning what is called “the float”. This section will demonstrate the mechanics of life insurance pricing.

Mortality Estimates: Insurance companies must be able to dial in the risk of the individual being insured. Historical data is the starting point for any estimate, which is then refined based on the individual's demographic and personal attributes. Older, less healthy, and less risk-averse policyholders will pay high premiums due to their higher risk of dying and thus making a claim.

Net Premium: The premium is the periodic payment the policyholder makes to the insurer. It is the price of being insured. The present value of the premiums must equal the present value of the claim, or the discount rates will be adjusted until this is true.

Load: The load is similar to a sales charge for a mutual fund. It pays for the insurer's cost of doing business and all the work that goes into writing a policy.

$$
\text{Load} + \text{Net Premium} = \text{Gross Premium} $$

Net Payment Cost Index assumes the insured passes away at the end of the horizon and cash value is not considered. It involves 5 steps in its computation:

1. Compute the FV of premiums paid.
2. Compute the FV of the dividends.
3. Subtract the FV of premiums paid from the FV of dividends.
4. Annuitize this FV difference.
5. Divide the annualized FV difference by $1,000 worth of insurance purchase for standardization.

Illustration

Assume a 25-year time horizon and an 8% discount rate on a $100,000 policy. The policy has an annual beginning-of-year premium of $2,000 and an assumed annual end-of-year dividend of $500. Terminal year cash value will be estimated at $25,000 (at year 25)

1. The FV of premiums = 2,000 PMT, 8% I/Y, 25 N, FV = 157,908.83 (Hint: since premiums are made at the beginning of the period, must switch calculator to annuity due or [BGN] mode as it is sometimes called).
2. The FV of dividends = 500 PMT, 8% I/Y, 25 N, FV = 36,552.97.
3. Cost of insurance = 157,908.83 – 36,552.97 = 121,355.86
4. Annuitized difference = 121,355.86 = FV, 8% I/Y, 25 N, PMT = 1,660
5. For each $100 of insurance purchased, the cost will be \(\frac {1,660}{\left( \frac {100,000}{100} \right)} = $16.60\)

Net Surrender Cost Index assumes the individual terminates the policy at the end of the horizon and the cash value is received.

1. Compute the FV of premiums paid.
2. Compute the FV of the dividends.
3. Subtract the FV of premiums paid from the FV of dividends and then subtract the projected cash value of the policy.
4. Annuitize this FV difference.
5. Divide the annualized FV difference by $1,000 worth of insurance purchase for standardization.

Continuation of Illustration above:

1. The FV of premiums = 2,000 PMT, 8% I/Y, 25 N, FV = 157,908.83 (Hint: since premiums are made at beginning of the period, must switch calculator to annuity due or [BGN] mode as it is sometimes called).
2. The FV of dividends = 500 PMT, 8% I/Y, 25 N, FV = 36,552.97.
3. Cost of insurance = 157,908.83 – 36,552.97 – 25,000 = 96,355.86
4. Annuitized difference = 96,355.86 = FV, 8% I/Y, 25 N, PMT = 1,318.03
5. For each $1,000 of insurance purchase, cost will be \(\frac {1,318.03}{\left( \frac {100,000}{1,000} \right)} = $13.18\)

Question

Consider the following policy:

$$ \begin{array}{l|c}
\textbf{Insurance amount purchased} & \$150,000 \\ \hline
\textbf{Annual Premium} & \$4,990 \\ \hline
\textbf{Expected annual dividend} & \$1,190 \\ \hline
\textbf{Time to Maturity} & 30 \text{ years} \\ \hline
\textbf{Expected Surrender Value} & \$90,000
\end{array} $$

With a discount rate of 8%, the net surrender cost index per 1,000 of insurance is closest to:

1. $21.17.
2. $25.92.
3. $22.70.

Solution

The correct answer is A.

1. FV of premiums = 610,505.88 [BGN]
2. FV of dividends = 134,807.02
3. Cost of insurance = 610,505.88 – 134,807.02– 90,000 = 385,698.86
4. FV = 385,698.86, 8 = I/Y, 30 = N, PMT = 3,125.53 [BGN]
5. \(\frac { 3,175.99}{\left( \frac {150,000}{1,000} \right)} = 21.17\)

Solution B omits the expected surrender value:

1. FV of premiums = 610,505.88 [BGN]
2. FV of dividends = 134,807.02
3. Cost of insurance = 610,505.88 – 134,807.02 = 475,698.86
4. FV = 475,698.86, 8 = I/Y, 30 = N, PMT = 3,888.15 [BGN]
5. \( \frac { 3,888.15 }{\left(\frac { 150,000}{1,000} \right)} = 25.92\)

Solution C uses END mode for the annuitized cost of insurance:

1. FV of premiums = 610,505.88 [BGN]
2. FV of dividends = 134,807.02
3. Cost of insurance = 610,505.88 – 134,807.02 – 90,000 = 385,698.86
4. FV = 385,698.86, 8 = I/Y, 30 = N, PMT = 3,404.73 [END] INCORRECT
5. \(\frac {3,404.73 }{\left( \frac {150,000}{1,000} \right) }= 22.70\)

Reading 9: Risk Management for Individuals

Los 9 (g) Describe the basic elements of a life insurance policy and how insurers price a life insurance policy