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Chapter 11 - Time and uncertainty. After studying this chapter you will be able to understand: 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?...
Chapter 11 Time and Uncertainty © 2014 by McGraw-Hill Education 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 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 © 2014 by McGraw-Hill Education 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 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 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 year $105,000(1.05) = $100,000(1.05)2 = $110,250 in years © 2014 by McGraw-Hill Education 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 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 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: = • 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 © 2014 by McGraw-Hill Education 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 10 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 11 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 $20,000 each year after starting their first job in years and working for 30 years Their preferred annual interest rate is 5% • What is the present value of this future flow of income? = ($ ) ($ ) ⋯ $ $ , • 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 © 2014 by McGraw-Hill Education 12 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 13 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 14 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 + +…+ = • The expected value of a choice, EV, is equal to the sum of each possible event, S, weighted by its probability of occurring, P © 2014 by McGraw-Hill Education 15 Expected value Our previous analysis of the decision to attend higher education can be extended by assuming that additional future earnings is uncertain Lifetime earnings by education level $0.9 million No college degree 50% 50% 0% College degree 25% 25% 50% $1.5 million $2.4 million • Using the probabilities and outcomes, the expected value of attending college and not attending college can be calculated © 2014 by McGraw-Hill Education 16 Expected value • The value of not attending college is: + × $1.5 + (0 × $2.4 ) = × $0.9 = $1.2 • The value of attending college is: = 25 × $0.9 + 25 × $1.5 + (.5 × $2.4 = $1.8 • 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 17 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 © 2014 by McGraw-Hill Education 18 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 19 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 20 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 © 2014 by McGraw-Hill Education 21 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 22 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 23 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 © 2014 by McGraw-Hill Education 24 ... present value of costs and benefits leads to informed decision making © 2014 by McGraw-Hill Education 13 Risk and uncertainty • The previous examples assumed certain future costs and benefits • Many... 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... selection and moral hazard would be eliminated © 2014 by McGraw-Hill Education 23 Summary • The present value formula provides a way of comparing current costs and benefits to future costs and benefits