The Time Value of Money,9,1-2,Chapter Outline,Time value associated with money Determining future value based on number of periods over which funds are to be compounded at given interest rate Present value based on current value of funds to be received Tables for future and present values, their need in computations, determination of yield Compounding or discounting occurring on a less than annual basis,1-3,Relationship to the Capital Outlay Decision,Determine whether future benefits are sufficiently large to justify current outlays Mathematical tools help in making capital allocation decisions,1-4,Future Value Single Amount,Measuring value of an amount that is allowed to grow at a given interest over a period of time is necessary Assuming that the worth of $1,000 needs to be calculated after 4 years at a 10% interest per year, we have:1st year$1,000 X 1.10 = $1,100 2nd year.$1,100 X 1.10 = $1,210 3rd year$1,210 X 1.10 = $1,331 4th year$1,331 X 1.10 = $1,464,1-5,Future Value Single Amount (contd),A generalized formula is:Where FV = Future value PV = Present value i = Interest rate n = Number of periods;In the previous case, PV = $1,000, i = 10%, n = 4, hence;,1-6,Future Value of $1(FVIF),1-7,Future Value Single Amount (contd),In determining future value, the following can be used:Where = The interest factorIf $10,000 were invested for 10 years at 8%, the future value would be:,1-8,Present Value Single Amount,A sum payable in the future is worth less today than the stated amount The formula for the present value is derived from the original formula for future value:The present value can be determined by solving for a mathematical solution to the formula above, thus restating the formula as: Assuming,1-9,Present Value of $1(PVIF),1-10,Relationship of Present and Future Value,1-11,Future Value Annuity,A series of consecutive payments or receipts of equal amount The future value of each payment can be totaled to find the future value of an annuity Assuming, A = $1,000, n = 4, and i = 10%,1-12,Future Value of an Annuity of $1(FVIFA),1-13,Compounding Process for Annuity,1-14,Present Value Annuity,Calculated by discounting each individual payment back to the present and then all of these payments are added up Assuming that A = $1,000, n = 4, i = 10%, we have:,1-15,Presentation of Time Value Relationship,Requires various comparison which include: The relationship between present value and future value The relationship between the present value of a single amount and the present value of an annuity Future value related to future value of annuity,1-16,Annuity Equaling a Future Value,Assuming that at a 10% interest rate, after 4 years, an $4,641 needs to accumulated:For n = 4, and i = 10%, is 4.641. This A equals $1,000,1-17,Annuity Equaling a Present Value,Determining what size annuity can be equated to a given amount:Assuming n = 4, i = 6%:,1-18,Relationship of Present Value to Annuity,1-19,Annuity Equaling a Present Value (contd),Determining the necessary repayments on a loan:Assuming n 20, i = 8%, Total payments ($4,074 for 20 years)$81,480 Repayment of principal. 40,000 Payments applied to interest.$41,480,1-20,Payoff Table for Loan (amortization table),1-21,Review,1-22,Yield Present Value of a Single Amount,To calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:We see that for n = 4 and = 0.683, the interest rate or yield is 10%,1-23,Yield Present Value of a Single Amount (contd),Interpolation may also be used to find a more precise answerDifference between the value at the lowest interest rate and the designated valueThe exact value can be determined thus:,1-24,Yield Present Value of an Annuity,To calculate the yield on an investment of $10,000, producing $1,490 per annum for 10 years:Hence:,1-25,Special Considerations in Time Value Analysis,Certain contractual agreements may require semiannual, quarterly, or monthly compounding periods In such cases, to determine n, multiply the number of years by the number of compounding periods during the year The factor for i is determined by dividing the quoted annual interest rate by the number of compounding periods,1-26,Cases,Case 1: Determine the future value of a $1,000 investment after 5 years at 8% annual interest compounded semiannually Where, n = 5 X 2 = 10; i = 8% / 2 = 4%Case 2: Determine the present value of 20 quarterly payments of $2,000 each to be received over the next 5 years, where i = 8% per annum Where, n = 20; i = 20%,1-27,Patterns of Payment,Time value of money evolves around a number of different payment or receipt patterns Assume a contract involving payments of different amounts each year for a three-year period To determine the present value, each payment is discounted to the present and then totaled(Assume 8% discount rate),1-28,Deferred Annuity,Assume, a contract involving payments of different amounts each year for a three year period An annuity of $1,000 is paid at the end of each year from the fourth through the eighth year To determine the present value of the cash flows at 8% discount rate,