Exercise III: Economics applications 1. Assume that you invest an amount of money So at a nominal: annual interest rate r. This means informally, that you get an extra amount of money TS after a year. However, this interest may be compounded several times a year: for instance, if the interest is compounded monthly, this means that after a month, you will get rSnf 12. The next month however, you will get a proportion 1112 of what you have on the account at this time, namely Sn +Ta5'uf12, and will therefore end up with an amount Sn++ So+ r51:- r ( r511) 12 12 12 L {a} Assume that you initially invest an amount S and that the interests are com pounded n times per year. Explain quickly why, have it years, you will have an amount 1" n! s, = sn(1+ ] . n {h} Assume that you invest $lflfll} at rate of 5% for 5 years. Compute 5,; when the interests are compounded annually ,H' monthly ,3 daily. Hound to the nearest integer. {c} It is mathematically easier to consider interest compounded continuously, which means compounded every very small time interval. lGive a precise interpretation of this concept, then compute St in this case. {d} Is it better to get a rate of 4.9% compounded continuously, or of 5% com pounded annually? 2. Assume that you are set to receive some amount ofmoney A in k years. This amount is worth less than A today: indeed, if you got A today, you could invest it at an interest rate r and obtain All + rjk after it years. Therefore, the present mine of getting an amount A in F: years is A [1+r}\"' A perpetuity is a financial instrument that allows you to get an amount A each year forever, starting at the end of the current year {so for I: = 1}. 1What is the present value of such a perpetuity