Continuous Compounding Discrete Cash Flow Continuous Compounding (DC) Continuous Cash Flow Continuous Compounding (CC)

Discrete Cash Flow Continuous Compounding (DC) i = er - 1 then follow the DD case. Suppose that one has a present loan of $1,000 and desires to determine what equivalent uniform end-of-year payments, A, could be obtained from it for 10 years if the nominal interest rate is 20% compounded continuously i = er - 1 = e0.2 - 1 = 0.2214 A = P(A/P,r,n) = 1,000(A/P,0.2214,10) = 1,000*0.2214*7.389/6.389 = 256 Sep. 9 1997

Another Example i = er - 1 = e0.035 - 1 = 0.0356 An individual needs $12,000 immediately as a down payment on a new home. Suppose that he can borrow this money from his insurance company. He must repay the loan in equal payments every six months over the next eight years. The nominal interest rate being charged is 7% compounded continuously. What is the amount of each payment? i = er - 1 = e0.035 - 1 = 0.0356 A = P(A/P, 0.419, 16) = 12,000*0.0356*1.75/0.75 = 997 Sep. 9 1997

Continuous Cash Flow Continuous Compounding If there is a continuous cash flow A each year for n years with nominal interest rate per year r. then: (P/A, r, n) = (F/A, r, n) = er n - 1 rern ern - 1 r Sep. 9 1997

Example Sol: F = A(F/A,8%,5) = $500 x 6.1478 = $3,074 What will be the future equivalent amount at the end of five years of a uniform, continuous cash flow, at the rate of $500 per year for five years, with interest compounded continuously at the nominal annual rate of 8%? Sol: F = A(F/A,8%,5) = $500 x 6.1478 = $3,074 Sep. 9 1997

Example What is the future equivalent of $10,000 per year that flows continuously for 8.5 years if the nominal interest is 10% compounded continuously? Sol: F = A(F/A,5%, 17) = 133,964.5 or F = A = 10,000 ern - 1 r e0.1(8.5) - 1 0.1 Sep. 9 1997

Discussion Examples Suppose the 8% nominal interest rate. If compounding occurs monthly, what is the effective annual interest rate? Sep. 9 1997

Example With the minimum number of interest factors, find the value of X below so that the two cash flow diagrams are equivalent when the interest rate is 10% per year. Sep. 9 1997

Example Set up an expression for the value of Z on the left-hand cash flow diagram that establishes equivalence with the right-hand cash flow diagram. The nominal interest rate is 12% compounded quarterly. Sep. 9 1997

Example A student decides to make semi-annual payments of $500 each into a bank account that pays an APR (nominal interest) of 8% compounded weekly. How much money will this student have accumulated in this bank account at the end of 20 years? Assume only one (the final) withdrawal is made. Sep. 9 1997

Example Consider an end-of-year (EOY) geometric gradient, which lasts for eight years, whose initial value at EOY one is $5,000, and f = 6.04% per year thereafter. Find the equivalent uniform gradient amount over the same time period if the initial value of the uniform gradient at the end of year one is $4,000. The nominal rate is 8% compounded semi-annually. What is iCR? P0? and What is G? Sep. 9 1997

Example An individual makes five annual deposits of $2,000 in a savings account that pays interest at a rate of 4% per year. One year after making the last deposit, the interest rate changes to 6% per year. Five years after the last deposit the accumulated money is withdrawn from the account. How much is withdrawn? Sep. 9 1997

Example Some future amount, F, is equivalent to $2,000 being received every 6 months over the next 12 years. The nominal interest rate is 20% compounded continuously. What is the value of F? Sep. 9 1997

Example What is the value of P that is equivalent to A = $800/yr ($800 flowing continuously each year) for 11.2 years? The nominal rate of interest is 10%, continuously compounded. Sep. 9 1997

Discrete Cash Flows Continuous Compounding i = ( 1 + r/M)M - 1 If M ->  then i = er - 1 F/P = (1+i)n => F/P = ern P/F = e-rn F/A = (ern - 1) / (er - 1) Sep. 9 1997

Example Suppose that one has a present loan of $1,000 and desires to determine what equivalent uniform end-of-year payments, A, could be obtained from it for 10 years if the nominal interest rate is 20% compounded continuously. A = P (A/P, r, n) = P[ern (er - 1)]/ (ern - 1) ern = e0.2 (10) er = e0.2 Sep. 9 1997

Homework # 3 - Chapter 3 Problem Number: 93, 95, 98, 99, 101, 102, 105, 106 Due Date: Oct. 8 (Thursday) Sep. 9 1997

Engineering Economy