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 rate i

• # of periods n

• future value FV

• present value PV

• cash flow PMT

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

Simple versus compound interest

• Simple interest involves computing interest only on the original principal, not on any accrued interest. Compound interest involves calculating interest on interest.

Future Value – Simple Interest

Example 1

Invest $1 for 3 years @ 12% per annum.

Period Beg. Amt. Interest End. Amt

1 1.00 0.12 1.12

2 1.00 0.12 1.24

3 1.00 0.12 1.36

Future Value – Compound Interest

Example 2

Period Beg. Amt. Interest End. Amt Formula

1 1.0000 0.12 1.1200 1.12**1

2 1.1200 0.13 1.2544 1.12**2

3 1.2544 0.15 1.4049 1.12**3

n =3, i = 12, PV = 1, FV = ?

Future Value

Example 3Invest $5 at the end of each year for 4 years @ 12%. What is the FV? 5 5 5 5now 1 2 3 4

5 x 1.00 = 5.005 x 1.12 = 5.605 x 1.2544 = 6.27205 x 1.4049 = 7.0245 4.779 23.897 This is the same as the future value of an

ordinary annuity

n =5, i = 12, Pmt = 5, FV = ?

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 4What is the PV of a 10 dollars received 1 year from

today assuming 12% interest?

? $10Now 1

Note that $8.93 grows to $10 in 1 year @ 12%8.93 x 1.12 = 10

n =1, i = 12, V = FV, PV = ?

Present Value

Example 5What is the PV of $4 received 3 years from today and $4

received 2 and 1 year from today at 5% interest? 4 4 4Now 1 2 3

4 x .9524 = 3.8104 x .9070 = 3.6284 x .8638 = 3.455 2.7232 10.893

n =3, i = 5, PMT = 4, PV = ?

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 6What is the FV of a $75,000 deposit made every 6 months

for 3 years using an annual rate of 10%?

0 1 2 3 4 5 6 7 8 9 10

75 75 75 75 75 75

[((1.05**6)-1)/.05] x 75,0006.80191 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 7

How many periods does it take for $130 to grow to $261.48 @ 15% per annum?

n = ?, i = 15, PV = 130, FV = -261.48

Solving for i

Example 8

At what annual interest rate will $175 grow to $377.81 in ten years?

n = 10, i = ?, PV = 175, FV = -377.81

Find the required payment

Example 9Compute the required semi-annual payment in order to have $14,000 at the end of 5 years @ 8%

14,0000 1 2 3 4 5 6 7 8 9 10

x x x x x x x x x x

n=10, i=8, PMT = ?, FV = -14,000

Car payments

Example 10

What would be your monthly car payment on a $15,000 4 year loan @ 10%. Payments are made at the end of each month.

PV = 15,000 n = 48 i = 10%; pmt =

Car payment

Example 10 (continued)

Instead of a 4 year loan, compute the payment for a 5 year (60 payment) loan.

PV = 15,000 n = 60 i = 10%; pmt =

Car payment

Example 10 (continued)

Leave the loan at 5 years, but lower the interest rate to 8%. Compute the payment.

PV = 15,000 n = 60 i = 8%; pmt =

Car payment

Example 10 (continued)

With the 5 year, 8 % loan, assume the maximum payment you can afford is $275. How much of a loan can you afford?

n = 60 i = 8% pmt = 275; PV =

Car payment

Example 10 (continued)Go back to the $15,000, 5 year, 10% loan.

How much of the 12th payment applies toward principal? Interest? What is the remaining balance?

Do the same for the 36th payment?

Do the same for the 13th – 24th payments combined?

Present Value of an Annuity

Example 11You win a $4,000,000 lottery that pays $200,000 per year for 20 years. What is the present value of the lottery assuming a rate of 10%?

n = 20, i = 10, PMT = 200,000; PV =

Uneven cash flows

• Up to this point we have assumed cash flows are the same each period. This is common for mortgage and lease payments. Things are not nearly as tidy when you need to determine if a project makes financial sense. Typically you will experience cash flows from revenues and expenses that vary each period.

Uneven cash flows

• This is the situation firms face when attempting to decide if a new location makes economic sense. Luckily this situation can still be handled with your financial calculator. You will be using a few new keys:

• CFj, Nj, IRR/YR, and NPV

Internal rate of return

• IRR/YR is used to compute the internal rate of return. This represents the interest rate that the project is earning over its life. This is similar to solving for i in the previous problems.

Net present value

• Sometimes you may know that you need a minimum return (internal rate of return) to take on a new location. You can then use this interest rate in the calculation and then compute the present value of all the combined cash flows. The summary number is the net present value of the project. If the project is earning a return greater the the required IRR, the NPV will be positive, otherwise it will be negative.

IRR and NPV example

Example 16You wish to determine the IRR and NPV of a project with the following

projected cash flows:

At inception: -10,000

End of year 1: 3,000

End of year 2: 1,000

End of year 3: 3,000

End of year 4: 8,000

Determine the IRR and then the NPV if the required return is 15%

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?)