1. Chapter 7 The Costs and Benefits of a franchise to a city 1

2. Overview Here we want to study issues related to what a city gets when it has a professional sports team in its town. Chapter 6 set up some of the discussion. A point to consider about cities is that while they have the same economic problem as all other economic entities (how to use resources optimally given its budget and taste and preference for goods and services), cities do not typically act as a profit maximizing entity would act. 2

3. Financing stadiums When you look at the mechanics of building a stadium, how to pay for the facility is a major question that must be answered. Sometimes the franchise can get the city, or maybe even a collection of local governments, to pay for some or all of the facility. The method of finance here is similar to a home mortgage. There are money flows now and in the future. Here I would like to review some basic finance ideas. I start with a set of notion that will be used in an end of period convention to account for time. In this set up now is time 0 and then we talk about the end of each of n periods of time. P will present a current amount now. F will represent a future amount n periods down the road, and A will represent a constant amount that occurs at the end of each of n periods. 3

4. Time line 0 1 2 3 4 … n In finance there are really two parties involved. You have one party making payment(s) and the other party receiving payment(s). The time line above starts at 0 and we look at the end of each of n periods. 4

5. Single Payments F F F P P P … Compound amount – sometimes called the future amount. If P is put in an account today at interest rate r at the end of 1 year the account will be what was started with plus what was started with times the interest rate. (first picture) F = P + Pr = P(1 + r) If left in the account another period, at the end of two periods F = P(1 + r) + P(1 + r)r = P(1 + r)(1 + r) = P (1 + r)^2, and after n periods F = P (1 + r)^n. 5

6. Single Payments Example: Say you put 10 bucks in an account that earns 3% each year. At the end of 4 years you have F = 10(1 +.03)^4 = 10( 1.1255) = 11.26. Notice the term (1 +.03)^4 includes the interest rate of.03. The whole term is often called the future value factor and many finance books have tables where you can look up the factor. Excel would be useful to make the calculation as well. If the focus is on achieving a certain future amount with the present value unknown then the formula becomes P = F/(1 + r)^n 6

7. Single payments 7 Say you want $20 in two years and you can earn 4% each year. You need a present amount P = 20/(1 +.04)^2 = 20/1.0816 = 20(.9245562) = 18.49. The term (1 + r)^2 is called the present value factor, in this case the single payment present value factor. It can be found in a table or in Excel.

8. Annuities or Uniform series 8 F F F A A A A A A A … With an annuity there is a payment made at the end of each of n periods into an account earning rate r per period. When the payment is made at time n we then see the account has total value = F. My first picture on the left says put A in an account at the end of the first period. At the end of the first period the account will have F = A.

9. Annuities 9 In the middle picture on the previous screen we see the same amount A is put into an account at the end of the first and end of the second period. Here F = A + A(1 + r). In this equation the first A after the equal sign is the A that occurs at the end of period 2 and has no time to earn interest. The A at the end of the first period earns rate r for one period so the account is F. In this context F is called the uniform series compound amount. If A is put into an account at the end of each of n periods then at the end of the nth period the account will be F = A + A(1 + r) + A(1 + r)^2 + A(1 + r)^3 + … + A(1 + r)^(n-1). Let’s do some math to show how this can be simplified.

10. Annuities 10 Multiply both sides of the formula by (1 + r) to get (1 + r)F =(1 + r)[ A + A(1 + r) + A(1 + r)^2 + A(1 + r)^3 + … + A(1 + r)^(n-1)] or (on right side multiple each part by (1 + r) ) (1 +r)F = A(1 + r) + A(1 + r)^2 + A(1 + r)^3 + A(1 + r)^4 + … + A(1 + r)^n. Remember F = A + A(1 + r) + A(1 + r)^2 + A(1 + r)^3 + … + A(1 + r)^(n-1) SO, (1 + r)F – F = rF = A(1 + r)^n – A, or F = A[{(1 + r)^n – 1}/r]. Note the term in brackets has an interest rate term in it and the whole term is an interest factor – it may be called the uniform series compound amount factor. The value is often put into a table in finance books. This means for various values of r and n you can just look up the amount in a table instead of putting the whole thing into your calculator.

11. 11 If the focus of the uniform series is the future then we just saw F = A[{(1 + r)^n – 1}/r]. But, if the focus is on the uniform series we have A = F[r/{(1 + r)^n – 1}] and the term in brackets is called the uniform series sinking fund factor. The term sinking fund comes from the idea that if you put A away, or sink, at the end of each of n periods you will get F. Sometimes the focus is on the present instead of the future. Since in a single payment we saw F = P(1 + r)^n, we can sub this above to have A = P[r(1 + r)^n/{(1 + r)^n -1}] where the term in brackets is the uniform series capital recovery factor. If one starts with P and can earn r each period then A can be withdrawn at the end of each period and at time n have nothing left.

12. 12 If the focus is on the present we have P = A[{(1 + r)^n – 1}/{r(1 + r)^n}], where the term in brackets is called the uniform series present worth factor. The Excel file I have you print out is set up so that if you change the interest rate at the top then all the other numbers change and you should be able to solve problems. When cities finance large projects like building a stadium they often sell bonds and pay back the amount over time. Say a stadium costs $250 million to finance. They sell that much in bonds right now and get the funds. Let’s say that they have decided to pay 3.5% interest each year and they will pay the bond off with a uniform series of payments for 30 years. The bondholders would be looking at a capital recovery idea. If they put down P = $250,000,000 they want A each year when r =.035 is incorporated into the problem for n = 30 years.

13. 13 So, A = 250,000,000(factor from table or Excel) If you go to the Excel file I have and change the interest rate to.035 and then look at where n = 30 under the capital recovery problem you see the factor is 0.0544. A = 250,000,000(.0544) = $13,600,000. So, each year the city will have to pay out 13.6 million to pay off the bond. The city will either have to charge rent and get other revenue from the sports team, or raise taxes, or cut spending elsewhere. The authors point out that most cities do not get enough money from the sports team to make these kinds of payments. In other words, the city is not getting paid back what they put out!

14. 14 Before we turn to ask why cities would do this without being paid back, let’s do a few other problems to see how the Excel spread sheet can be used. 1) Say you have 10 bucks and can earn 7%. How much will you have in 10 years? F = P(single payment future value factor) = 10(1.9672) = 19.67. 2) Say you want to have 20 bucks in 5 years and you can earn 12% interest each year. What do you need to have today? P = F (single payment present worth factor) = 20(.5674) = 11.35 3) Say you put away $1000 a year and you can earn 5%. How much do you have in 30 years? F = A (uniform series compound amount factor) = 1000(66.4388) = 66,438.80 4) Say you want $1,000,000 in 30 years and you can earn 10% a year. How much to you put away each year in a uniform series? A = 1,000,000(sinking fund factor) = 1,000,000(.0061) = 6,100.

15. 15 5) Say you think you can pay $400 a month for a house. If you could get a 30 year mortgage at 6% that would really mean you would make payments for 360 months and the interest rate would be 6%/12 =.5% or.005 per month. You could afford to buy what amount of house? P = A(uniform series present worth factor) = 400(181.7476) = 72,699.04. This really assumes nothing down. If you have to put 20% down, 72,699.04/.80 = 90,873.8 is the house you could buy. Your 20% down would be 90,873.8(.2) = 18,174.76 and you finance 90,873.8 – 18,174,76 = 72.699.04 Well, there you go! You know some finance.