Interest Rates That Vary With Time• Find the present equivalent value given a future value and a varying interest rate over the period of the loan ik : the interest rate of the kth period• FN P = ----------------- eq.(4-31) N (1 + ik)• (P/F,i%,N)= 1/(1+i)N eq.(4-2) k+1
Nominal And Effective Interest Rates• Nominal Interest Rate (r ) For rates compounded more frequently than one time a year, the stated annual interest rate.• Effective Interest Rate ( i ) For rates compounded more frequently than one year, the actual amount of interest paid.• i = ( 1 + r / M )M - 1 M : the number of compounding periods per year• Annual Percentage Rate ( APR ) percentage rate per period times number of periods. APR = (r/M) x M
Method1 Method2This kind of problems can be solved with two ways :1. get the effective interest rate which is compatible with the period forexample i = 16.14 per year compounded yearly and N = 10 yearsOR2. use the effective interest rate per quarter = 6%(nominal per year) / 4(compounded times per year) = 1.5% per quarter compounded quarterlyand N = 10 years * 4 = 40 quarter periods
Continuous Compounding and Discrete Cash Flows• Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval.• Given nominal per year interest rate r, compounding per year M one unit of principal = [ 1 + (r / M ) ] M• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp• Given lim (p ∞) = [ 1 + (1 / p) ] rp = er• ( F / P, r%, N ) = (1+i)N = e rN Then i = e r - 1
• End of Chapter 4 PART 3• See you next lecture with a very important revision !!!• Don’t, miss it !!!!!