Chapter 11: Time and Uncertainty

What will you learn in this chapter? • Why money is worth more now than in the future. • How compounding works over time. • How to calculate the present value of a future sum. • What the costs and benefits are of a choice using expected value. • How risk aversion makes a market for insurance possible. • What the importance is of pooling and diversification for managing risk. • What challenges adverse selection and moral hazard pose for insurance.

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1© 2014 by McGraw-Hill Education 1 Chapter 11 Time and Uncertainty © 2014 by McGraw-Hill Education 2 What will you learn in this chapter? • Why money is worth more now than in the future. • How compounding works over time. • How to calculate the present value of a future sum. • What the costs and benefits are of a choice using expected value. • How risk aversion makes a market for insurance possible. • What the importance is of pooling and diversification for managing risk. • What challenges adverse selection and moral hazard pose for insurance. © 2014 by McGraw-Hill Education 3 Value over time • When a decision requires weighing uncertain future costs and benefits, two complications are faced: – The value of money changes over time, causing an inaccurate direct comparison of current costs and benefits to future costs and benefits. – The future is uncertain, causing future benefits and costs to be only approximate estimates. 2© 2014 by McGraw-Hill Education 4 Timing matters • When costs and benefits of a choice occur at different times, this profoundly affects the choice. • Consider the following scenario: You have won a competition and can choose one of the following prizes: Option A: $100,000 now. Option B: $105,000 ten years from now. • Which would you choose and why? © 2014 by McGraw-Hill Education 5 Interest rates • When considering money today versus future money, individuals consider the opportunity cost of waiting until the future to receive the money. – The interest rate tells how much today’s money is worth in the future. – Depositing $100,000 in a bank at a 5% annual interest rate is worth in one year: $100,000 + ($100,000*5%) = $105,000 • Future money can be equated to the present. In the above example, $105,000 in one year is worth $100,000 today. © 2014 by McGraw-Hill Education 6 Compounding • When analyzing the value of money over a time period longer than one year, compounding the interest payments is necessary. – This causes a process of accumulation, as interest is paid on interest that has already been earned. • The future value of depositing $100,000 in a bank at a 5% annual interest rate for two years earns: $100,000(1 + .05) = $105,000 in 1 year $105,000(1.05) = $100,000(1.05)2 = $110,250 in 2 years 3© 2014 by McGraw-Hill Education 7 Active Learning: Computing future value What is the future value of depositing $100,000 in a bank at a 5% annual interest rate for ten years? © 2014 by McGraw-Hill Education 8 Present value • Interest rates are used to compare the present value and future value of a sum. • Individuals have a preferred interest rate that reflects their opportunity cost of waiting for money in the future versus receiving it today. • Suppose that an individual has a preferred interest rate of 8% annually. – The future value of $100,000 in 10 years is $215,893. • The present value of $215,893 in 10 years is $100,000. © 2014 by McGraw-Hill Education 9 Present value • If the future value is known, then given an individual’s preferred interest rate, his or her present value of any sum can be determined. • Rearranging the earlier formula: ܸܲ = ி௏ଵା௥ ೙ • Present value translates future costs or benefits into the equivalent amount of value today. • This information enables us to directly compare future amounts with the present sums. 4© 2014 by McGraw-Hill Education 10 Active Learning: Comparing present and future values Suppose you have a preferred interest rate of 9% annually. You just won the lottery and have two options: – Option A: Take $1,000,000 today. – Option B: Take $5,000,000 in 20 years. • What is the present value of Option B? Which option will you prefer? © 2014 by McGraw-Hill Education 11 Present value • Sometimes benefits and costs accrue over several years. • To calculate the present value of a flow of money in the future, add up the present value of each amount in the future. © 2014 by McGraw-Hill Education 12 Present value • Consider that many people expect to earn additional income every year after earning a college degree. • Suppose an individual expects to earn an additional $20,000 each year after starting their first job in 5 years and working for 30 years. Their preferred annual interest rate is 5%. • What is the present value of this future flow of income? ܸܲ = $ଶ଴௄(ଵ.଴ହ)ఱା $ଶ଴௄(ଵ.଴ହ)లା⋯ା $ଶ଴௄ଵ.଴ହ యరୀ$ଶହଶ,ଽଷଽ • The present value of an extra $600,000 spaced out evenly over 30 years is $252,939. • Thus, if the present value of the cost of attending college is less than $252,939, the individual will attend. 5© 2014 by McGraw-Hill Education 13 Present value • Knowing how to calculate present value can be useful in making other decisions when the benefits and opportunity cost occur at different times. – If you want a certain level of income when you retire, how much should you save into your retirement fund now? – If you run a business, what value of future sales would be needed to make it worthwhile to invest in a new piece of machinery? • Comparing the present value of costs and benefits leads to informed decision making. © 2014 by McGraw-Hill Education 14 Risk and uncertainty • The previous examples assumed certain future costs and benefits. • Many decisions are based on weighing uncertain future costs and benefits against today’s costs and benefits. – Risk is a special class of uncertainty in which the costs or benefits of an event or choice are uncertain, but calculable. • Evaluating risk requires analysis of different possible outcomes. © 2014 by McGraw-Hill Education 15 Expected value • Even when future events are uncertain, often the set of outcomes are known. • There are costs and benefits as well as a likelihood that the outcome will be realized. • By combining outcomes with likelihoods, a single cost or benefit estimate can be calculated. ܧܸ = ଵܲ ଵܵ + ଶܲܵଶ++ ௡ܲܵ௡ • The expected value of a choice, EV, is equal to the sum of each possible event, S, weighted by its probability of occurring, P. 6© 2014 by McGraw-Hill Education 16 Expected value Our previous analysis of the decision to attend higher education can be extended by assuming that additional future earnings is uncertain. • Using the probabilities and outcomes, the expected value of attending college and not attending college can be calculated. Lifetime earnings by education level College degree 25% 25% 50% No college degree 50% 0%50% $0.9 million $1.5 million $2.4 million © 2014 by McGraw-Hill Education 17 Expected value • The value of not attending college is: ܧܸ = .5 × $0.9ܯ + .5 × $1.5ܯ + (0 × $2.4ܯ) = $1.2ܯ • The value of attending college is: ܧܸ = .25 × $0.9ܯ + .25 × $1.5ܯ + (.5 × $2.4ܯ = $1.8ܯ • Unlike the earlier estimates, these expected values incorporate the risk of lower income. • Using these estimates, one can make a choice based on expected future income. • The expected benefit of attending college is $600,000, the difference in the expected values. © 2014 by McGraw-Hill Education 18 Propensity for risk • Although individuals have varying tastes for taking on risks, people are generally risk-averse with their financial decisions. (Someone who does have a high tolerance for risk is risk-seeking.) • When faced with two options, with equal expected value, individuals typically prefer the one with lower risk. • Imagine you are given two options based on a coin toss: – Option A: Heads, receive $100,001. Tails, receive $99,999. – Option B: Heads, receive $200,000. Tails, receive $0. • While both have an estimated value of $100,000, most people prefer Option A because it has less risk. 7© 2014 by McGraw-Hill Education 19 Propensity for risk • The previous example suggests that many worry about worst-case outcome. • Even if two possible outcomes have the same expected value, the one with lower risk will typically be chosen. • This implies that individuals must be compensated for taking on risk. – The expected value of Option B would have to be greater than $100,000 before most individuals would accept the risk of winning nothing. • How much higher would depend on individuals’ personal taste for risk. © 2014 by McGraw-Hill Education 20 Insurance and managing risk • Risk averse individuals cope with risk in many ways. – If possible, avoid the risk altogether. – If unavoidable, buy insurance. • An insurance policy is a product that lets risk averse individuals (or companies) pay to reduce some uncertainty. • Insurance is an agreement in which: – An individual pays a regular fee. – An insurance company covers costs associated with a specific event occurring. © 2014 by McGraw-Hill Education 21 Insurance and managing risk • The cost of insurance is typically greater than its expected value. – Most people are risk-averse enough to find insurance worth the extra expense. • Individuals are generally willing to pay for insurance because the costs of the worst-case events are typically quite large. • If the cost of insurance was equal to its expected value, then insurance companies would not make any profits. 8© 2014 by McGraw-Hill Education 22 Pooling and diversifying risk • Insurance does not reduce risk. • Insurance reallocates costs from individuals to insurance companies. • Why are insurance companies better able to handle the same risk? – Insurance companies pool individuals together, called risk pooling. – Insurance companies use risk diversification in which risks are shared across many different assets or people. © 2014 by McGraw-Hill Education 23 Problems with insurance • There are two big inherent problems with insurance: adverse selection and moral hazard. • Both occur due to individuals and insurance companies having different information (asymmetric information sets). • Adverse selection occurs when higher risk individuals are drawn towards insurance. • Moral hazard occurs when individuals behave riskier once they become insured. • If insurance companies had the same information set as their clients, adverse selection and moral hazard would be eliminated. © 2014 by McGraw-Hill Education 24 Summary • The present value formula provides a way of comparing current costs and benefits to future costs and benefits. – An interest rate links the future value to the present. • Sometimes these costs and benefits are known and other times they are uncertain, but calculable. – When they are calculable, an expected value of each outcome can be determined. • When worse-case events are high cost, individuals tend to avoid these activities or buy insurance to reduce risk.