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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.

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Budgets and Opportunity Cost January 19, 2012

Posted by tomflesher in Finance, Micro, Teaching.
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In the previous entry, we talked a little about opportunity cost. In short, it’s what someone gives up in order to acquire something else. What does that have to do with budgeting?

Start with the plain-sense definition of a budget. Most people think of a budget – quite correctly – as a list of planned expenses by category. I might plan out my week’s spending as:

  • $30: gas
  • $200: planned monthly expenses (rent, insurance, utilities)
  • $50: groceries
  • $40: a meal out
  • $30: savings
  • $50: petty cash (Starbucks, forgot my lunch, that sort of thing)

That comes out to $400. What does that tell me? Well, for it to be a good budget, I’d better not make any less than $400 a week! Otherwise, I’m planning to spend more than I make, and that’s going to get me into trouble shortly. (If I spend myself into a deficit, I’ll need to plan to get out at some point.) It also tells me that I expect to make no MORE than $400 this week. After all, I have a spot for savings and a spot for petty cash, so I have places for overflow. Petty cash is what some people call slack,where anything “extra” would go.

Let’s simplify just a little. Let’s say I have that same $400 income, but I can only spend it on two things: coffee and sweaters. Sweaters cost $25 each, and coffee costs $2 per cup. I have a couple of choices – in fact, an infinite number of them – for how I can spend that money. For example, if I want to spend all my money on coffee, I can buy

\frac{400}{2} = 200

cups of coffee. If I want to spend all my money on sweaters, I can buy

\frac{400}{25} = 16

of them. Of course, there’s no reason to spend all my money on one of those two goods. I can buy any combination, subject to the budget constraint that I don’t create a deficit: that is, that

2*(cups\:of\:coffee) + 25*(sweaters) \le 400

Think back to when we talked about opportunity cost: if I have the same budget either way, then think about how many cups of coffee I have to give up to get a sweater. There are a couple of ways to do this, but the easiest is to compare the prices: for $25, I get one sweater or 12.5 cups of coffee, so the opportunity cost of one sweater is 12.5 cups of coffee. Similarly, for $2, I get one cup of coffee or 1/12.5 = .08 sweater, so my opportunity cost of buying a cup of coffee is 8% of a sweater. Note that this is the same as dividing the total quantities I can buy:

16\:sweaters = 200\:cups

1\:sweater = \frac{200}{16}\:cups

1\:cup = \frac{16}{200}\:sweaters

Basically, we can use a budget constraint to determine opportunity costs by imagining that we spend our entire budget on each of two goods and then comparing the quantities, or simply by comparing prices. Opportunity costs represent budgeting decisions on a much smaller – some might say marginal – scale.

Opportunities and What They Cost December 23, 2011

Posted by tomflesher in Finance, Micro, Teaching.
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One of the fundamental concepts in economics is that of opportunity cost. In order to understand opportunity cost, though, we need to take a step back and think about costs in a more general way.

The standard example goes like this: I like books, so I want to turn my passion into a job and open a bookstore. To do so, I need to rent a storefront and buy some inventory; for now, I’m going to run the bookstore myself. I’ll be open eight hours a day, so I’ll quit my current job at the box factory and do the bookstore job instead. Easy peasy, right? I even have $100,000 in the bank to get myself started.

Let’s look at the costs. The first thing to consider is the actual money I’m laying down to run this business. Say rent is $1,000 per month. If I don’t want to bother with discounting – and for now, I don’t – then that means I’ve spent $1,000 x 12 = $12,000 on rent this year. Then, imagine that in order to meet demand and still have a decent inventory, I need to spend $43,000 on books. That brings me up to $55,000 worth of cash that I’m laying out – my explicit costs, otherwise known as accounting costs, are $55,000.

But accounting costs don’t show that I had to give up my job at the box factory to do this, and I could have made $45,000. They also don’t account for the interest income I’m giving up by pulling my money out of the bank to live on it. Even if interest rates are only 1.25 APR (that’s annual percentage rate), I’m losing $1,250 in interest income. So, for simplicity (I don’t want to bother with compounding, either), let’s assume that in January I take out this year’s $45,000 in salary and keep it under my mattress. I buy my inventory and pay my rent. I’ve given up $1,250 in interest income and $45,000 in salary, so even though I haven’t laid out that cash, I have to count it as a cost. Accountants won’t write down what I gave up on a balance sheet, but my implicit costs, or opportunity costs, are $46,250.

Total costs are simple – just add implicit and explicit costs. My total costs for starting the business are $55,000 + $46,250 = $101,250.

The key to understanding opportunity cost is that it’s a measure of what you gave up to make a choice, and so it shows how much something is worth to you. If I offer you a free Pepsi, your opportunity cost is zero, so you might as well take it. If I offer you a Pepsi for your Dr. Pepper, we can infer two things:

  1. I value Dr. Pepper at least as much as Pepsi, and
  2. I can figure out how much you value Dr. Pepper, relatively, based on your decision. If you accept, then you must value Pepsi at least as much as Dr. Pepper; if not, you must value Dr. Pepper more (and would thus be rational, since Dr. Pepper is superior to Pepsi).

Opportunity cost is very useful for determining preferences. We’ll talk about that a little more later on. In the meantime, just remember this distinction:

  • Explicit costs are things you paid money for
  • Opportunity costs are how much you’d value your best alternative
  • Total costs are the sum of explicit and opportunity costs.

Bobby Bonilla’s Deferred Deal: A Case Study August 2, 2011

Posted by tomflesher in Finance, Macro, Teaching.
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Note: This is something of a cross-post from The World’s Worst Sports Blog.  The data used are the same, but I’m focusing on different ideas here.

In 1999, the Mets owed outfielder/first baseman Bobby Bonilla $5.9 million dollars. They wanted to be rid of Bonilla, who was a bit of a schmoe, but they couldn’t afford to both give him his $5.9 million in the form of a buyout and meet payroll for the next year. No problem! The Mets sat down with Bonilla and arranged a three-part deal:

  1. The Mets don’t have to pay Bonilla until 2011.
  2. Starting in 2011, Bonilla will receive the value of the $5.9 million as an annuity in 25 annual installments.
  3. The interest rate is 8%.

Let’s analyze the issues here.

First of all, the ten-year moratorium on payment means that the Mets have use of the money, but they can treat it as a deposit account. They know they need to pay Bonilla a certain amount in 2011, but they can do whatever they want until then. In their case, they used it to pay free agents and other payroll expenses. In exchange for not collecting money he was owed, Bonilla asked the Mets to pay him a premium on his money. At the time, the prime rate was 8.5%, so the Mets agreed to pay Bonilla 8% annual interest on his money. That means that Bonilla’s money earned compound interest annually. After one year, his $5.9 million was worth (5,900,000)*(1.08) = $6,372,000. Then, in the second year, he earned 8% interest on the whole $6,372,000, so after two years, his money was worth (6,372,000)*(1.08) = (5,900,000)*(1.08)*(1.08) = $6,881,760. Consequently, after ten years, Bonilla’s money was worth

5,900,000 \times (1.08)^{10} = 12,737,657.48

The prime rate didn’t stay at 8.5%, of course, as the Federal Reserve Economic Data (FRED) webpage maintained by the St. Louis Fed shows. This is a monthly time series of prime rates, which are graphed to the right. The prime rate  can be pretty volatile, so it was a bit of an odd choice to lock in a rate that would be effective for 35 years. As it turns out, the prime rate fluctuated quite a bit. Taking the annualized prime rate by dividing each monthly rate by 12 and adding them together, we can trace how much a bank account paying the full prime rate would have paid Bonilla:

\begin{tabular}{c||cc} Year& Annualized interest rate & Current Value \\ \hline 2000& 0.09233 & 6444766.67 \\ 2001& 0.06922 & 6890851.93 \\ 2002& 0.04675 & 7212999.26 \\ 2003& 0.04123 & 7510355.16 \\ 2004& 0.04342 & 7836429.74 \\ 2005& 0.06187 & 8321243.53 \\ 2006& 0.08133 & 8998038.00 \\ 2007& 0.08050 & 9722380.06 \\ 2008& 0.05088 & 10217006.15 \\ 2009& 0.03250 & 10549058.85 \\ 2010& 0.03250 & 10891903.26 \\ \end{tabular}

So since his agreement with the Mets paid him $12,737,657.48 and he could have invested that money by himself to earn around $10,891,903.26, Bonilla is already better off to the tune of (12737657.48 – 10891903.26) or about 1.85 million.

In addition, we can measure the purchasing power of Bonilla’s money. Using FRED’s Consumer Price Index data, the CPI on January 1, 2000, was 169.3. On January 1, 2011, it was 221.062. that means the change in CPI was

\frac{221.062 - 169.300}{169.300} \times 100 = 30.57 \%

So, the cost of living went up (for urban consumers) by 30.57% (or, equivalently, absolute inflation was 30.57% for an annualized rate of around 3%). That means that if the value of Bonilla’s money earned a real return of 30.57% or more, he’s better off than he would have been had he taken the money at the time.

Bonilla’s money increased by

\frac{12,737,657.48 - 5,900,000}{5,900,000} \times 100 = 115.89 \%

Bonilla’s money overshot inflation considerably – not surprising, since inflation generally runs around 3-4%. That means that if Real return = Nominal return – Inflation, Bonilla’s return was

115.89 - 30.57 = 85.32 \%

Because Bonilla’s interest rate was so high, he earned quite a bit of real return on his money – 85% or so over ten years.

Bonilla’s annuity agreement just means that the amount of money he holds now will be invested and he’ll receive a certain number of yearly payments – here, 25 – of an equal amount that will exhaust the value of the money. The same 8% interest rate, which is laughable now that the prime rate is 3.25%, is in play.

In general, an annuity can be expressed using a couple of parameters. Assuming you don’t want any money left over, you can set the present value equal to the sum of the discounted series of payments. That equates to the formula:

PV = PMT \times [\frac{1 - \frac{1}{(1 + r)^t}}{r}] \\ 12737657.48 = PMT \times [\frac{1 - \frac{1}{(1.08)^25}}{.08}] \\ PMT = 1193248.20

where PV is the present value of the annuity, PMT is the yearly payment, r is the effective interest rate, and t is the number of payments. Bonilla will receive $1,193,248.20 every year for 25 years from the Mets. It’s a nice little income stream for him, and barring a major change in interest rates, it’s one he couldn’t have done by himself.