1. Application to Economics: Present & Future Value; Income Streams (3/6/06) If I invest P dollars at an annual rate of r, compounded continuously, then the future value B of P after t years is B = P e r t. From this it is follows that the present value P of a future amount B is P = B e - r t. For example, if someone will pay me $1000 five years from now, and if the ongoing investment rate is 7%, then its present value is P = $1000 e –(0.07)(5) = $705.

2. Income Streams In most businesses (and homes), income arrives continuously, not in discrete packets. If the rate at which it arrives is a function P(t) in dollars/year, then between time t and time t +  t, the present value of that money will be P(t)  t e – r t. Adding up over the period from 0 years to M years gives

3. Example of Income Streams If income arrives at a rate of $5000/year over a ten year period, and if the investment rate averages 7%, the Present Value of this income stream is Likewise, the Future Value (in ten years) is

4. Remarks The reason the last example (future value) needed the exponent (10 – t) is that that is the amount of time (not t) that interest can accrue (e.g., at t = 2, interest accrues for 8 years). If the income stream in constant, then the exponent can simply be t to get future value.

5. Problems for Wednesday Suppose I have a constant income stream of $10000/year which I can invest at a 6% annual rate over the next 8 years. – How much would I have to have right now to earn the same amount over the 8 years (i.e., what is the present value of my income stream)? – How much will I have at the end of the 8 years (i.e., what is the future value)? Answer the same questions assuming my income stream is not constant, but rather will improve steadily, say P(x) = $2000x per year.

6. Looking ahead… On Wednesday, we will first meet in the classroom, then go to the lab. We will meet from 12:20 to 1:15 on Friday and I will hand out Hand-in #2. We will meet in the classroom on Monday, March 20 and there will be no assignment for that day. (But work on Hand-in #2!) Hand-in #2 is due on Wed, March 22. There will be additional homework for that day.