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Uncertainty and Discounting

This article explores the concepts of uncertainty and discounting, particularly in relation to probability theory and decision-making. It differentiates between the frequentist and Bayesian views of probability and provides practical examples using coin tosses and dice. The discussion extends to expected outcomes and preferences in risk-return scenarios, illustrating how these principles apply to financial markets, interest rates, and the pricing of carbon in climate change strategies. Understanding these concepts is crucial for economic decision-making amid uncertainty.

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Uncertainty and Discounting

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  1. Uncertainty and Discounting

  2. Uncertainty • Having incomplete knowledge of future outcomes when more than one is possible • Closely linked to probability • Chance of an outcome happening • Frequentists • View probability as the long run average of a repeated random variable • Bayesians • View probability as a degree of belief that an outcome will occur given evidence

  3. Example: Coin Toss • ½ or 50% chance of heads • ½ or 50% chance of tails • Expect on average if repeated many times to get roughly ½ heads and ½ tails • Example: Dice • 1/6 chance of each face • Example: Chance Lebrone resigns with Cavs • How do we assign probability to this • Maybe strength of belief • Say 35% • Note: The sum of the probabilities must = 1

  4. Expected Outcomes • We want to define an expectation of a random event • For example say get $1 for a heads and $2 for a tails, if I flip a coin what do I expect to get? • 50% I get $1 and 50% I get $2 • Define: Expectation to be • 0.5 x $1 + 0.5 x $2 = $1.50 • This is my expected outcome • Expected outcome = Pr(1) x 1 + Pr(2) x 2

  5. Example: You get $100 dollars if you pull a spade out of the deck of cards and $0 otherwise. What is your expected outcome? • 25% you will pull a spade, 75% you won’t if a random fair deck • So • 0.25 x $100 + 0.75 x $0 = $25 • So the expected payment received is $25 • This is how we “should” put values on uncertain outcomes

  6. Now which would you prefer? • #1: 50% you get $100 and 50% you get $200 • #2: 50% you get 0 and 50% you get $310 • Lets see the expected payments • #1: 0.5 x 100 + 0.5 x 200 = $150 • #2: 0.5 x 0 + 0.5 x 310 = $155 • If you chose #1 you are risk adverse • You don’t like risk, even though #2 has a higher expected payout • Your risk nature is something that would be incorporated in your utility function

  7. This simple idea is how risky assets in financial markets are priced • This simple idea is used in studies that try to estimate the correct price of carbon • This process is what leads to differences in countries bond market yields • This is why the interest rate Greece is now paying on its debt is about 10 times that of Germany’s • People think they might default so demand a higher premium to lend money • Why people with bad credit pay higher interest rates

  8. Bond Example • Say “risk free rate of return” = 1% • Greece wants to sell gov’t bonds (1 year note) • People think there is a 10% chance they will default • What percentage will Greece pay on its bonds? • Say it is a €100 note

  9. Expected payment = 101 • 10% you get 0 • So what must you get if they do pay? • 0.10 x 0 + 0.90 x Y = 101 • 0.90Y = 101 • Y = 101/0.90 • Y = 112.2 • So Greece will pay 12% interest

  10. Time Discounting • Start with the assumption that now is better than later • So having $100 now is better than $100 in a year • But how much better? • Depends on the “discount rate” • Called discount rate because you discount the future

  11. First we have to look at how things grow in value over time • Invest 100 at 5% interest • In one year you will have • 100 x (1 + 0.05) = 105 • In two years you will have • 100 x (1 + 0.05) x (1 + 0.05) = 100 x 1.052 = 110.25 • In three years will have • 100 x 1.053 = 115.76 • Compound Interest

  12. That was how money grows • We want to do the reverse • Instead of “what will this dollar be worth in 3 years” we want to ask “what is a dollar 3 years from now worth now?” • So we reverse the math

  13. 1 year • $100 = Y x 1.05 • Y = 100/1.05 = 95.23 • 2 year • $100 = Y x 1.052 • Y = 100/1.052 = 90.7 • 3 year • $100 = Y x 1.053 • Y = 100/1.053 = 86.4

  14. General Formula • Finding Present Value • PV = Value/(1 + interest rate)t

  15. Both In Climate Change Problem • Both of these are issues in the climate change problem • Huge divergences in the “price of carbon” that is optimal • Remember externalities and the price should be the external cost

  16. The costs are uncertain • Probability on possible events make big differences • The costs are in the future • What rate we use to discount make huge differences

  17. Say $100, but 100 yrs from now • Say 5% vs 2% • PV = 100/(1.05)100 = 76 cents • PV = 100/(1.02)100 = $13.80

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