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Don’t Discount the Importance of Patience April 9, 2013

Posted by tomflesher in Micro, Teaching.
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Uncertainty is one explanation for why interest rates vary. Tolerance for uncertainty is called risk aversion, and it can be pretty complicated. (We’ll talk about it a little bit later on.) Another big concept is patience. Willingness to wait is also pretty complicated, but that’s our topic for today.

It’s easy to imagine some reasons that people would have different levels of patience. For one, you’d expect a healthy thirty-year-old (named Jim) to be more patient than a ninety-year-old (named Methuselah). What if someone (named Peter) offered us a choice between $100 today or a larger amount of money a year from now? How much would it take for Jim and Methuselah to take the delayed payoff? Would they take $100 a year from now? A lot can change in a year:

  • There could be a whole bunch of inflation, and the $100 will be worth less next year than it is now. Boom, we’ve lost.
  • We could put the money in the bank and earn a few basis points of interest. Boom, we’ve lost.
  • We could die and not be able to pick up the money. Boom, we’ve lost.
  • Peter could die and we wouldn’t be able to collect. Boom, we’ve lost.

Based on these, we’ll want a little bit more money next year than this year in order to be willing to take the money later instead of the money now. Statistically, though, Jim is more likely than Methuselah to be there to pick up the money.

Neither would take any less than $100 next year, but that’s just a lower limit. According to Bankrate.com, Discover Bank is paying 0.8% APY, which means that the $100 would be worth .8% more next year – just by putting the money in the bank, we can trade the risk that the bank goes bust (really unlikely) for the risk of Peter dying. That’s an improvement in risk and an improvement in payoff, so there’s no reason to take any less than $100.80. Again, though, this is a lower bound. Peter still has to pay for making them wait. That’s where the third point comes into play.

Methuselah is probably not going to live another year. It’s much more likely that he’ll get to spend the $100 than whatever he gets in a year; in order to make it worth the wait, the payoff would have to be huge. Methuselah views money later as worth a lot less than money today. He might need $200 to make it worth the wait. Jim, on the other hand, might only need $125. He has more time, so he’s much more patient.

This level of patience is called a discount rate and is usually called β. You can do this sort of experiment to figure out someone’s patience level. You’d then be able to set up an equation like this, where the benefit is the $100 and the cost is what you give up later:

100 = \beta \times Payoff

Methuselah, then, would have

100 = \beta \times 200

so β = 1/2.

Jim would have the following equation:

100 = \beta \times 125

so β = 4/5.

Based on this, we can say that Methuselah values money one year from now at 50% of its current value, but Jim values money one year from now at 80% of its current value. Everyone’s discount rate is going to be a little bit different, and different discount rates can lead people to make different choices. If Peter offers $100 today or $150 tomorrow, Jim will wait patiently for $150. Methuselah will jump at the $100 today. Both of them are rational even though their choices are different.

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Really Interesting (Or Nominally Interesting, At Least) December 4, 2012

Posted by tomflesher in Finance, Teaching.
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Interest rates describe how much money you’ll have at the end of a year if you lend to someone. Mostly, you “lend” money to a bank by putting it in a savings account, but you might lend to the government by buying Treasury bonds or to your no-good brother by floating him $100. Currently, my bank pays 0.01%, although some commercial money market accounts pay around 1%; Treasuries pay about 0.18% for one-year bonds; my brother is currently paying me 10% (and the Mets are paying Bobby Bonilla 8%). Why the differences?

Borrowers need to pay the lender for two things: giving up the right to use his money for a year, and the risk that he won’t get his money back. The first element is pretty important for the lender: Patient Patricia will take a lower interest rate because she doesn’t need to buy stuff today – she’s willing to wait, especially if she can make a little money for waiting. On the other hand, Antsy Andrew wants to head right out and buy stuff, so asking him to wait a year for his stuff will cost a lot of money. Plus, I know there’s some inflation most years, so my brother will have to at least cover that.

That explains part of the difference between interest rates. A Treasury bond takes your money and keeps it for a full year, but my bank’s savings account allows me to withdraw my money at will. I’m not giving up much use of my money, so I don’t need to be paid much. When the Mets paid Bobby Bonilla 8% interest, they expected high inflation, when inflation turned out to be low.

It’s also really likely that, when I go to cash in my bond or take out my ATM card, the money’s going to be there. The government’s not going bankrupt1 and my bank deposits are insured. My no-good brother, though, might lose his job tomorrow. He’ll probably have the money to pay me, but he might not. That worries me, and I want him to pay for making me nervous. (In the real world, this also means I’ll get more money up front in an installment payment plan.)

That boils down to an important identity known as the Fisher Effect:

Nominal interest ≈ Real return + Expected inflation

When I expect inflation, that affects how much I’d rather have money now than later. My real return is how much I have to get from my no-good brother to compensate me for the risk that I’ll lose all my money. We can estimate inflation as being nearly zero today, so real returns (compensating for risk) explain almost all of the variation in interest rates.

But what about those banks paying 1%, when other banks (and the government) are paying much less? Banks need to hold reserves. A bank that’s nervous about how much money it has on hand will be willing to pay higher rates in order to get more deposits – in other words, if you need it, you’ll pay for it.

Note:

1 Okay, it might go bankrupt, but it’s reeeeaaaally unlikely to happen this month.