The Time Value of Money Module 24

Opening Why is a dollar today, worth more than a dollar a year from now? Would you lend a friend $100 now for a repayment of $115 in 4 years?

The Time Value of Money 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Money in the present is worth more than money in the future The interest rate represents a payment to help compensate the lender, who would otherwise be repaid in dollars worth less than those borrowed.

Why We Charge Interest Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Opportunity Costs If we decide to loan a friend money, we face the opportunity costs of what we could use that money for So, interest is what compensates us for the delayed gratification

The Time Value of Money Purchasing Power 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Purchasing Power We know that due to inflation, the purchasing power of the dollar will most likely decrease. Charging interest allows us to maintain that purchasing power

Present and Future Value 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Present value The value of a future sum in today’s dollars How much is $1 received a year from now worth today? Future Value The value in the future of money invested today How much is $1 invested today going to be worth in a year?

Present and Future Value 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Present value The present value of $1 received t years from now is: $1/(1+r)t t= years r= interest rate Future Value The future value of $1 invested today in t years is: $1 × (1+r)t

Present and Future Value 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Example If our friend offered to give us $115 in four years for $100 today, how do we know it’s worth it? Depends on the prevailing interest rate: At 4% interest rate, the present value of $115 in four years is $115/(1+.04)4 = $98.30 At 2% interest rate, the present value of $115 in four years is $115/(1+.02)4 = $106.48

Present and Future Value 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Example You could also compare the future value to the $115: At 4% interest rate, the future value of $100 invested in four years is $100 × (1+.04)4 = $116.99 At 2% interest rate, the present value of $115 in four years is $100 × (1+.02)4 = $108.24

Choosing Between Options 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present& Future Value - Explanation - Calculation 3) Choosing Between Options If you are given a choice of several options where up front costs and future payoffs might vary, you choose the one that maximizes net present value

Choosing Between Options 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options A hotel manager needs to replace all of the light bulbs and has heard about these new compact fluorescents. Each CFL costs $15 and each regular incandescent bulb costs $1. The hotel has 1000 fixtures. The energy bill will be $800 if you use the CFL’s and $4800 if you use the incandescent. They will both last 4 years. Which is the better choice?

Choosing Between Options 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options It seems like the CFLs are the best bet, but what else could we do with that $14,000?

Choosing Between Options 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options As interest rates get higher, the present value of the savings realized from switching to CFL’s decreases

Choosing Between Options 1) Why we charge interest - Opportunity Costs - Purchasing Power 2) Present & Future Value - Explanation - Calculation 3) Choosing Between Options Suppose that a person only lives 2 years and has 2 choices. She can go to school in next year, or go straight into the work force. If a person immediately starts working, she will earn $20,000 in both years 1 and 2. If a person goes to school in year 1, she must pay $5,000, but she would earn $47,500 in year 2. If the interest rate is 5%, calculate the present value of a person who goes to school and a person who does not. School: PV = -5,000/(1.05) + 47,500/(1.05)2 = $38,322.00 Workforce: PV = 20,000/(1.05) + 20,000/(1.05)2 = $37,188.21