Winthrop University
College of Business Administration
Sports Finance Notes Dr. Pantuosco
Solving for the Present Value
A dollar received today is worth more than a dollar received in the future.
Why?
If spent, your dollar today will buy more than your dollar will a year from now.
If invested, your dollar could have earned interest.
A dollar in the future is worth less than a dollar today. Also, a dollar today is worth more than a dollar in the future.
If you borrow $100 today from your friend Nate and promise to return his $100 three years from now, Nate loses and you win. If the interest rate over that period is 10 percent, then you are really only giving Nate $75.13. His purchasing power with that $100 decreases. What Nate really needs just to breakeven is $133.10.
In another example, if you gave Joe $1000 10 years ago, you could have invested that money. If you earned a 12 percent over that time period, you would now have $3105.80.
Two factors influence the difference between the present value, which is money put into today’s dollars, and the values in the future; these factors are interest rates (or rates of return) and inflation rates (price changes). If prices never went up and no one charged, or earned interest on their money (assuming that everyone simply buried their excess in their back yard) the present value would be the same as the future value. Or, a dollar today is the worth the same as a dollar tomorrow.
As inflation rises and interest rates rise, the present values decrease. In other words, dollars become worth less if you hold on to them.
Inflation and interest rates are encompassed in what finance people call a “discount rate.” In the examples above the discount rate in Nate’s example was 10 percent, and in Joe’s example it was 12 percent.
The discount rate is the opportunity cost of lending your money. You could have invested it yourself and earned interest.
At one time in Brazil the interest rate was over 1000 percent! What do you think happened to discount rates? What do you think people did with their money?
Currently, in the United States conservative people can earn about 6 percent, while risk takers can earn a bit more, let’s say 11 percent. Risk takers also maintain a better chance of losing money. Inflation rates hover around 3 percent. Lenders build their expectations of inflation into their interest rate. Fisher’s Theory
The Time Value of Money
It is necessary to understand the time value of money in order to determine whether or not an investment is financially sound.
Is it worth it for the city of New York to build a new stadium for the Yankees? The projected cost of the stadium is $1 billion. The city will contribute $200 million.
Assuming the life of the asset is 25 years (many places are building new stadiums about every 20 years), then how much money would the Yankees have to bring into the city above and beyond what they currently generate to justify the investment?
Assume the interest rate is 6 percent. Answer $15.6 million per season. Assuming they go to the ALCS, it comes out to $178,000 per game.
If the stadium helps them to increase their tax revenues by $20 million per season for the next 25 years then is purchasing the stadium a good idea?
This same technique of determining the value of an investment can be applied to players.
Is it worth it for the Arizona Cardinals to pay Matt Leinart $50 million for 6 years? Unlike other teams, Arizona has a lot of upside potential. They do not fill their stadium, and their record of getting into the post season is poor.
What information do we need to determine whether or not Leinart is worth it?
There are five components necessary to find the price of a bond. Four of these components are given and the fifth one has to be solved for.
· Present value is the bond’s current price. This is what the bond sells for given today’s market conditions.
· Future Value is the par value of the bond, typically this will be $1000.
· n is the number of payments the bond holder will receive. For example, a bond that matures in 5 years has a semi-annual payment. How many payments will the bond holder receive? 10 (2 a year for 5 years)
· r is the current market interest rate.
· PMT is the amount of the payment. For example, a bond with a 12 percent coupon rate paid semi-annually, and a par value of $1000, has payments of $60.
If the objective is to determine a bond price, we are solving for PV. There are three ways to solve for the bond’s price.
1. Using the mathematical formula.
2. Using the present value tables.
3. Using a financial calculator.
Solve for a bond’s price given the information below.
The par value of the bond is $1000.
The coupon rate of the bond is 12 percent, paid annually.
The bond matures in 3 years.
The market interest rate is 8 percent.
1. Using the mathematical formula approach.
PV = PMT/(1+r)n + PMT/(1+r) n+1+ PMT/(1+r)n+2+ Par Value/(1+r)n+2
PV = 120/(1+.08) + 120/(1+.08)2 + 120/(1.08)3 + 1000/(1.08)3
PV = 111.11 + 102.91 + 95.24 + 793.65
PV = 1102.91
2. Using the present value tables.
A steady stream of equal payments is called an annuity. Payments are considered annuities. Therefore, for the payments use the PVIFA table.
The par value is a one time payment. Therefore, for the par values present value use the PVIF table. When using the tables you need to know the number of payments “n” and the market interest rate “r”.
PV = PMT*(PVIFAn,r) + FV *(PVIFn,r)
the future value is the par value. The number of payments “n” is 3. The current market interest rate “r” is 8.
PV = 120*(2.577) + 1000*(.794)
PV = 309.24 + 794.00
PV = 1103.24
3. Using a financial calculator.
FV = 1000 the par value
n = 3 the number of payments
PMT = 120 the amount of the coupon payment (par value * coupon rate)
r = 8 the current market interest rate
PV this is what we are solving for
Plugging these numbers into the calculator
PV = 1103.08
Notice there is a small rounding discrepancy between the techniques.