Econ 100A Chapter 16 Interest Rates Investments and Capital Markets 1 2 Comparing Money Today to Money in the Future Interest rates Choices over Time Invest in a project if return from investment return on best alternative Capital and other durable goods products that are usable for years If capital is rented rent up to the point where the marginal benefit marginal cost Ex Firm rents trucks until current marginal rental cost current marginal benefit marginal revenue product of the trucks If capital must be bought or built rather than rented firm must compare the current cost of the capital to the future higher profits it will make from using the plant Such comparisons involve stocks and flows stock quantity or value that is measured independently of time A firm owns an apartment building this year not an apartment building per year flow quantity or value that is measured per unit of time hotdogs per week Interest Rates Assume no inflation Consuming a 1 worth of candy today is better than consuming 1 worth in 10 years How much more you must pay in the future to repay a loan today is specified by an interest rate the percentage more that must be repaid to borrow money for a fixed period of time Deposit funds in a bank today Bank agrees to pay you an interest rate i 4 1 04 1 i one year from now for every dollar you loan it an individual s personal interest rate that person s discount rate a rate reflecting the relative value an individual places on future consumption compared to current consumption A person s willingness to borrow or lend depends on whether his or her discount rate is greater or less than the market interest rate Compounding interest on interest accumulation of interest You place 100 in a bank account that pays 4 at the end of a year Remove interest payment each year Based on Jeffrey M Perloff Microeconomics Addison Wesley Longman 1999 These notes are Jeffrey M Perloff you can take out the interest payment of 4 each year and leave your 100 in the bank to earn more interest in the future or you have convert your 100 stock into a flow of 4 a year payments forever Let interest accumulate must pay you interest on 104 at end of the second year bank owes you interest of 4 on your original deposit of 100 and interest of 4 0 04 0 16 on your interest from the first year for a total of 4 16 End of Year 1 your account contains 104 00 100 1 04 100 1 041 End of the Year 2 you have 108 16 104 1 04 100 1 042 End of Year 3 your account has 112 49 108 16 1 04 100 1 043 End of year t you have 100 1 04t Generally with compounding For every 1 you loan the bank it owes you 1 i dollars after 1 year 1 i 1 i 1 i 2 dollars after 2 years 1 i 1 i 1 i 1 i 3 after three years 1 i t dollars at the end of t years Frequency of Compounding The more frequently interest is compounded holding the interest rate constant the greater the payment at the end of a year Let a bank s annual interest rate be i 4 It pays interest two times a year get half a year s interest i 2 2 after six month 1 i 2 1 02 per dollars At the end of the year it owes 1 i 2 1 i 2 1 i 2 2 1 02 2 1 0404 U S Truth in Lending Act requires lenders to tell borrowers what is the equivalent noncompounded annual percentage rate APR of interest See Table 16 1 Interest Rates Connect the Present and Future future value FV present value PV interest Put PV dollars in the bank today and allow interest to compound for t years F V P V 1 i t 16 1 Table 16 2 Application Power of Compounding Manhattan 2 Dutch allegedly bought it in 1626 for about 24 worth of beads and trinkets If Native Americans had invested in tax free bonds with an APR of 7 the bond would now be worth over 2 0 trillion which is much more than the assessed value of Manhattan Island Alaska If US had invested the 7 2 million it paid Russia in 1867 in the same type of bonds for the next 127 years that money would now be worth only 50 9 billion Alaska s current value Present Value How much is a 1 in the future worth today Or much money PV must we put in the bank today at an interest rate i to get a specific amount of money FV in the future Want FV 100 at the end of a year i 4 From Equation 16 1 PV 1 04 100 Dividing both sides of this expression by 1 04 PV 100 1 04 96 15 General formula PV FV 1 i t 16 2 At high interest rates money in the future is virtually worthless today A dollar paid to you in 25 years is worth only 1 today at a 20 interest rate Stream of Payments Payments per period flow measure may be used instead of PV or FV stock measures PV of payments for a finite number of years Example you agree to pay 10 at the end of each year for 3 years to repay a debt i 10 PV 10 1 1 10 1 12 10 1 13 24 87 Generally future payment of f per year for t years at an interest rate of i the present value stock of this flow of payments is 1 1 1 16 3 P V f 1 2 t 1 i 1 i 1 i Table 16 4 shows that the present value of a payment of f 10 a year for 5 years is 43 at 5 38 at 10 and 30 at 20 annual interest 3 Payments Forever PV dollars in a bank account earning i produces a flow of f i PV at the end of year Dividing both sides by i f 16 4 PV i Thus PV of you d have to deposit 10 i in the bank to ensure a future payment of f 10 forever 10 a year forever 200 at 5 100 at 10 50 at 20 Future Value of Payments over Time How much will you have in your savings account FV at some future time if you save f each year Year 1 Put f in the account Year 2 Add a second f so you have the first year s payment plus its accumulated interest f 1 i 1 or f 1 1 i 1 in total Year 3 total is f 1 1 i 1 i 2 General Equation 16 5 FV f 1 1 i 1 1 i 2 1 i t Application Saving for Retirement It pays to start saving early take adv of compounding Two approaches …
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