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Uncertainty and Discounting

Uncertainty and Discounting. 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

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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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