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Chapter 17: Problem 6

A high-pressure life insurance salesman was heard to make the following argument: "At your age a \(\$ 100,000\) whole life policy is a much better buy than a similar term policy. Under a whole life policy you'll have to pay \(\$ 2,000\) per year for the first four years but nothing more for the rest of your life. A term policy will cost you \(\$ 400\) per year, essentially forever. If you live 35 years, you'll pay only \(\$ 8,000\) for the whole life policy, but \(\$ 14,000(=\$ 400 \cdot 35)\) for the term policy. Surely, the whole life is a better deal." Assuming the salesman's life expectancy assumption is correct, how would you evaluate this argument? Specifically, calculate the present discounted value of the premium costs of the two policies assuming the interest rate is 10 percent.

Short Answer

Expert verified
Answer: The term policy is a better deal considering the present discounted value of their costs.

Step by step solution

01

Understand the payment structure of both policies

The whole life policy requires a payment of \(2,000 per year for the first four years, while the term policy requires a payment of \)400 per year for 35 years.
02

Calculate the present value of the whole life policy premium

We can calculate the present value of the whole life policy's premium using the present value formula, which is: PV = ∑ (C / (1 + r)^t) where PV is the present value, C is the cash flow (annual cost), r is the interest rate, and t is the time (number of years). For the whole life policy, there are four cash flows of $2,000 each: PV_whole_life = (2000 / (1 + 0.1)^1) + (2000 / (1 + 0.1)^2) + (2000 / (1 + 0.1)^3) + (2000 / (1 + 0.1)^4) Calculate the individual terms and sum them up: PV_whole_life = (2000 / 1.1) + (2000 / 1.21) + (2000 / 1.331) + (2000 / 1.4641) PV_whole_life ≈ 1818.18 + 1652.89 + 1502.63 + 1365.94 PV_whole_life ≈ $6,339.64
03

Calculate the present value of the term policy premium

For the term policy, there are 35 cash flows of $400 each: PV_term_policy = ∑ (400 / (1 + 0.1)^t) for t = 1 to 35 Calculate the individual terms and sum them up: PV_term_policy ≈ 400(1/(1.1)^1) + 400(1/(1.1)^2) + ... +400(1/(1.1)^35) Since this is an arithmetic-geometric progression with 35 terms, we can use the formula for the sum of an AGP: PV_term_policy ≈ 400(1 - ((1 / 1.1)^35) / (1 - 1 / 1.1)) PV_term_policy ≈ 400(1 - (1 / 28.1023) / (1 - 1 / 1.1)) PV_term_policy ≈ 400(1 - 0.03557 / 0.09091) PV_term_policy ≈ $1,607.17
04

Compare the present values of both policies

Now we can compare the present values of the premiums for both policies: PV_whole_life ≈ $6,339.64 PV_term_policy ≈ $1,607.17 Since the present value of the term policy premium is significantly lower than the whole life policy premium, we can conclude that the term policy is a better deal considering the present discounted value of their costs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Microeconomic Theory
Microeconomic theory is the branch of economics that studies the behavior of individuals and firms in making decisions about the allocation of limited resources. It examines how these decisions affect the supply and demand for goods and services, which determines prices, and allocates resources through the market mechanism. In the context of the given exercise, microeconomic theory helps explain how consumers value insurance policies based on their costs and benefits over time.

When evaluating the insurance policies described in the exercise, we apply microeconomic principles to assess the cost and benefit associated with each policy type. Consumers seek to maximize their utility or satisfaction, which in this case may hinge on the amount paid versus the security provided by the insurance. The method employed to compare the policies, the present discounted value, is a microeconomic tool that factors in the time preference of money.
The Time Value of Money
The time value of money is a financial principle that reflects the idea that money available now is worth more than the same amount in the future due to its potential earning capacity. This core concept of finance holds that provided money can earn interest, any amount of money is worth more the sooner it is received.

In the exercise, the time value of money is central to determining which insurance policy is more beneficial. Calculations of present discounted value demonstrate how money paid or received over time should be adjusted to its value today, considering an interest rate. This calculation helps us to compare future cash flows—like the premiums of insurance policies—on a like-for-like basis as of today.
Policy Premium Calculation
Calculating policy premiums, especially for insurance, involves determining the present value of all future payments to be made or received over the life of the policy. Insurers use various factors, including the expected life span of the policyholder, the amount of coverage, administrative costs, and the potential interest rate.

In the exercise, the present discounted value of the two types of insurance policies (whole life and term) are calculated by determining what the series of future premium payments would be worth in today's dollars. This calculation allows individuals to make an informed comparison between the upfront cost and the long-term benefits of an insurance policy.
Interest Rate Impact
The interest rate is crucial in determining the present discounted value of future cash flows because it dictates the rate of return that can be earned on investment. A higher interest rate will decrease the present value of future cash flows as it increases the opportunity cost of holding or spending money in the future rather than investing it at the present time.

In the problem provided, an interest rate of 10 percent is used to calculate the present value of the insurance policy premiums. It significantly affects the comparison of the total costs of each policy. As interest rates change, so will the present values, which could lead to a different conclusion regarding which policy is more cost-effective. Understanding the impact of the interest rate on present value calculations is essential when evaluating long-term financial decisions such as choosing an insurance policy.

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Most popular questions from this chapter

This problem focuses on the interaction of the corporate profits tax with firms' investment decisions. a. Suppose (contrary to fact) that profits were defined for tax purposes as what we have called pure economic profits. How would a tax on such profits affect investment decisions? b. In fact, profits are defined for tax purposes as \(\pi^{\prime}=p q-w l-\) depreciation where depreciation is determined by governmental and industry guidelines that seek to allocate a machine's costs over its "useful" lifetime. If depreciation were equal to actual physical deterioration and if a firm were in long-run competitive equilibrium, how would a tax on \(\pi^{\prime}\) affect the firm's choice of capital inputs? c. Given the conditions of part (b), describe how capital usage would be affected by adoption of "accelerated depreciation" policies, which specify depreciation rates in excess of physical deterioration early in a machine's life but much lower depreciation rates as the machine ages. d. Under the conditions of part (c), how might a decrease in the corporate profits tax affect capital usage?

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The calculations in Problem 17.4 assume there is no difference between the decisions to cut a single tree and to manage a woodlot. But managing a woodlot also involves replanting, which should be explicitly modeled. To do so, assume a lot owner is considering planting a single tree at a cost \(w,\) harvesting the tree at \(t^{*},\) planting another, and so forth forever. The discounted stream of profits from this activity is then \\[ V=-w+e^{-r t}[f(t)-w]+e^{-r 2 t}[f(t)-w]+\cdots+e^{-r n t}[f(t)-w]+\cdots \\] a. Show that the total value of this planned harvesting activity is given by \\[ V=\frac{f(t)-w}{e^{-r t}-1}-w \\] b. Find the value of \(t\) that maximizes \(V\). Show that this value solves the equation \\[ f^{\prime}\left(t^{*}\right)=r f\left(t^{*}\right)+r V\left(t^{*}\right) \\] c. Interpret the results of part (b): How do they reflect optimal usage of the "input" time? Why is the value of \(t^{*}\) specified in part (b) different from that in Example \(17.2 ?\) d. Suppose tree growth (measured in constant dollars) follows the logistic function \\[ f(t)=50 /\left(1+e^{10-0.1 t}\right) \\] What is the maximum value of the timber available from this tree? e. If tree growth is characterized by the equation given in part (d), what is the optimal rotation period if \(r=0.05\) and \(w=0 ?\) Does this period produce a "maximum sustainable" yield? f. How would the optimal period change if \(r\) decreased to 0.04 ? Note: The equation derived in part (b) is known in forestry economics as Faustmann's equation.

Suppose an individual has \(W\) dollars to allocate between consumption this period \(\left(c_{0}\right)\) and consumption next period \(\left(c_{1}\right)\) and that the interest rate is given by \(r\) a. Graph the individual's initial equilibrium and indicate the total value of current-period savings \(\left(W-c_{0}\right)\) b. Suppose that, after the individual makes his or her savings decision (by purchasing one-period bonds), the interest rate decreases to \(r^{\prime}\). How will this alter the individual's budget constraint? Show the new utility- maximizing position. Discuss how the individual's improved position can be interpreted as resulting from a "capital gain" on his or her initial bond purchases. c. Suppose the tax authorities wish to impose an "income" tax based on the value of capital gains. If all such gains are valued in terms of \(c_{0}\) as they are "accrued," show how those gains should be measured. Call this value \(G_{1}\) d. Suppose instead that capital gains are measured as they are "realized"-that is, capital gains are defined to include only that portion of bonds that is cashed in to buy additional \(c_{0} .\) Show how these realized gains can be measured. Call this amount \(G_{2}\) e. Develop a measure of the true increase in utility that results from the decrease in \(r,\) measured in terms of \(c_{0}\). Call this "true" capital gain \(G_{3}\). Show that \(G_{3}

The notion that people might be "shortsighted" was formalized by David Laibson in "Golden Eggs and Hyperbolic Discounting" (Quarterly Journal of Economics, May \(1997,\) pp. \(443-77\) ). In this paper the author hypothesizes that individuals maximize an intertemporal utility function of the form \\[ \text { utility }=U\left(c_{t}\right)+\beta \sum_{\tau=1}^{\tau=T} \delta^{\tau} U\left(c_{t+\tau}\right) \\] where \(0<\beta<1\) and \(0<\delta<1 .\) The particular time pattern of these discount factors leads to the possibility of shortsightedness. a. Laibson suggests hypothetical values of \(\beta=0.6\) and \(\delta=0.99 .\) Show that, for these values, the factors by which future consumption is discounted follow a general hyperbolic pattern. That is, show that the factors decrease significantly for period \(t+1\) and then follow a steady geometric rate of decrease for subsequent periods. b. Describe intuitively why this pattern of discount rates might lead to shortsighted behavior. c. More formally, calculate the \(M R S\) between \(c_{t+1}\) and \(c_{t+2}\) at time \(t .\) Compare this to the \(M R S\) between \(c_{t+1}\) and \(c_{t+2}\) at time \(t+1 .\) Explain why, with a constant real interest rate, this would imply "dynamically inconsistent" choices over time. Specifically, how would the relationship between optimal \(c_{t+1}\) and \(c_{t+2}\) differ from these two perspectives? d. Laibson explains that the pattern described in part (c) will lead "early selves" to find ways to constrain "future selves" and so achieve full utility maximization. Explain why such constraints are necessary. e. Describe a few of the ways in which people seek to constrain their future choices in the real world.
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