Time Value of Money

The Starting Point NPV analysis allows us to compare monetary amounts that differ in timing. We can also incorporate risk into the analysis, however we will not concern ourselves with this complication at this time. Two items need to be determined before you start the NPV analysis, future cash flows and interest rates. Forecasting these is often more an art than a science, however in many situations these are either known or can be estimated.

Items needed to solve these problems You will need to know all but one of the following: interest ratei # of periodsn future valueFV present valuePV cash flowPMT

Methods to solve the problems A decent business calculator (e.g., HP10BII) A formula Tables A spreadsheet package (e.g., excel)

The following are useful formulas Future value of a single sum FV = PV * (1+i)**n Present value of a single sum PV = FV * 1/(1+i)**n

Future Value – Compound Interest Example 1 PeriodBeg. Amt. Interest End. Amt Formula ** ** **3 n =3, i = 12, PV = 1, FV = ?

Future Value Example 2 Invest $5 at the end of each year for 5 12%. What is the FV? now x 1.00 = x 1.12 = x = x = x = This is the same as the future value of an ordinary annuity

Present Value In each of the cases so far we wished to determine what a dollar would be worth in the future. We can also go the other direction. Often we wish to know what future sums are worth today. This is called present value (PV)

Present Value Example 3 What is the PV of a 10 dollars received 1 year from today assuming 12% interest? ? $10 Now 1 Note that $8.93 grows to $10 in 1 12% 8.93 x 1.12 = 10

Present Value Example 4 What is the PV of $3 received 3 years from today and $3 received 2 and 1 year from today at 5% interest? Now x.9524 = x.9070 = x.8638 =

Non- Annual Periods So far we have computed FV of a single sum and an annuity and also PV of a single sum and an annuity. Each are basically the reverse of the other. Each has been computed with one compounding period per year. Often the compounding period is shorter.

Future values with non-annual deposits Example 5 What is the FV of a $75,000 deposit made every 6 months for 3 years using an annual rate of 10%? [((1.05**6)-1)/.05] x 75, x 75,000 = 510,143 n=6, i = 10, pmt = 75,000, FV = ? Note: Be sure to set your calculator to 2 payments per year.

Other Items to Solve For N = how long will it take a sum to grow to a certain FV at a given interest rate i = what interest rate is required to grow a certain sum to a given FV in a given length of time PMT = what payment is required to pay off a loan at a given interest rate in a set amount of time

Solving for n Example 6 How many periods does it take for $130 to grow to 15% per annum? n = ?, i = 15, PV = 130, FV =

Solving for i Example 7 At what annual interest rate will $175 grow to $ in ten years? n = 10, i = ?, PV = 175, FV =

Find the required payment Example 8 Compute the required semi-annual payment in order to have $14,000 at the end of 5 8% 14, x x x x x x x x x x n=10, i=8, PMT = ?, FV = -14,000

Tips 1.Draw time lines 2.Put in all the knowns 3.Be sure to use the period interest rate 4.Make sure the answer passes the smell test (e.g., is the present value < the future value?)

Bond example Suppose you have two cash flows, one an annuity and the other a lump sum. You can calculate the present value of the combined sums. What is the PV of an annuity that pays $50 every 6 months for 5 years plus a lump sum payment of $1000 at the end of five years. Use the following discount rates: –10%, 6%, 12%