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```							                   Chapter 2 - The Basics of Time Value of Money

The “time value of money” describes the reality that a dollar today is not worth the same
as a dollar one year from today when interest rates are positive. This is true because you
could invest your dollar today and have more than one dollar in a year. Thus, we typically
say that a dollar today is worth more than a dollar at some future point in time.

Cash flows from different points in time (different months, years, etc), can not simply be
added together because they have different “time values.” Therefore, cash flows from
different time periods must all be discounted or compounded to the same point in time
before they can be added together.

This handout will cover both lump sum and annuity time value of money computations. A
lump sum payment is a one time payment while an annuity is a series of equal, periodic
payments such as a loan. At the end of the handout, you will find practice problems with
answers so you can practice doing time value of money computations.

Learning about the time value of money is the basis of many financial concepts. Most
importantly, learning how to do time value of money computations will prepare you to
perform more complicated valuation calculations such as valuing a share of stock, a bond,

Abbreviations:
PV = present value, the value of a cash flow today (also the principal of a loan)
FV = future value, the value of a cash flow at some future point in time
r = the rate of interest in decimal form (i.e. 5% equals .05 in decimal form)
t = is the time period
PMT = an equal, period cash flow such as the payment on a loan

Lump Sum Formulas

Compounding – Computing Future Value
When there is a known cash flow today (PV), and you want to know what it will be worth at
some point in the future, you will compound that present value (PV) to a future value (FV)
using the following formula:

FV = PV(1+r)t

Example: Future Value
Suppose your great, great grandmother put \$1.00 in a savings account 150 years ago. If it
earned 4% annual what would the dollar be worth today? HINT: FV=PV (1+r) t

Using the formula:

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
FV=PV(1+r)t
FV=1(1.04)150

Note: In order to take 1.04 to 150 th power on a calculator, you can do the following:
1. Enter 1.04
2. Hit the y x key
3. Enter 150
4. Hit the = key
5. You should get 358.92 which you then multiply by 1 for a final answer of
FV=358.92

Using a financial calculator:
If you have a financial calculator with the PV, FV, and PMT keys, and you wish to use it
for these computations, the key strokes are listed below.
1     PV
4     %i
150 N
CPT FV 358.92

Discounting – Computing Present Value
If you know that you a certain amount of money at some point in the future, that is the
future value (FV). You can figure out how much you would need to invest today (PV) at a
particular rate so the money will grow to the desired future value (FV) at the designated
future date. We call this “discounting” the future value back to the present.

PV= FV
(1+r)t

Example: Present Value
What if you knew that the account had \$500 in it today, and it earned 4% annually for 150
years, what did grandma originally put in the account?
Using the formula:

PV= FV                      PV= 500              PV= 500             PV=\$1.39
(1+r)t                      (1.04)150           358.923

Using the financial calculator:
500    FV
4      %i
150    N            CPT PV 1.39

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
Example: More Present Value
You need \$20,000 in 3 years to pay tuition at USD and you can earn 8% per year on your
money. How much would you need to put away today to have \$20,000 in 3 years?
Using the formula:
PV= FV             PV= 20,000       =         20,000      = \$15,876.65
(1+r)t                (1.08) 3             1.2597

Using the financial calculator:
20000        FV
8            %i
3            N            CPT PV 15,876.65

Example: Want to be a millionaire?
If you are currently 45 years old and you can earn 8% on your investments, how much will
you have to invest today to accumulate \$1 million by age 65?
Using the formula:
PV= FV              PV= 1,000,000       =     1,000,000    = \$214,546.23
(1+r)t                 (1.08)20              4.661

Using the financial calculator:
1,000,000           FV
20                  N
8                   %i             CPT PV \$214,548.21

Example: A series of cash flows
What is the present value of three payments of \$500 each if the first payment occurs in
one year, the second payment in two years, and the third payment in three year? Assume
the interest rate (or discount rate) is 6%.

To solve this problem, you must do three separate present value computations and sum
the results of the three computations as illustrated below:

PV = 500    + 500 + 500           =      471.70 + 445.00 + 419.81 = \$1,336.51
(1.06)1  (1.06)2 (1.06)3

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
Annuity Formulas

An annuity is a series of equal periodic cash flows.

Annuity Present Value
The annuity present value is used to discount a series of future, equal, periodic cash
payments to a present value. The annuity present value formula can be used to compute
the present value of a series of equal, periodic payments. The formula can also be used to
compute the payments on a car loan or a mortgage.

 
 1  ( 11r )t
PVannuity = PMT x 
     r                  
                        

Example: A series of cash flows revisited
What is the present value of three payments of \$500 each if the first payment occurs in
one year, the second payment in two years, and the third payment in three year? Assume
the interest rate (or discount rate) is 6%.

This problem is really an annuity problem because the future cash flows are an annuity, a
series of equal, annual payments of \$500 each. You can solve for the present value using
the PV annuity formula above.

 
 1  ( 11r )t
PVannuity = PMT x 
     r                  
                        

PVannuity =

 1  ( 1 .1 )3
500 x 
06
 = \$1,336.41
     .06          
                  

Interest Rates

Interest can be compounded (paid) on an annual basis (annual compounding) or more
frequently, such as on a monthly basis. The Annual Percentage Rate (APR) is the annual
rate that this charged by the lender or promised by the borrower. If interest is
compounded monthly (car loans, home loans, credit card payments, etc), we convert the
APR to a monthly rate. To convert the APR to a monthly rate, simply divide the APR by
12.

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
In addition to using the PV annuity formula to solve for a present value, we can use it solve
for the annual or monthly payment on a loan if the present value (principal) is known.

Example: Car Loan
You want to take advantage of excellent cash rebates GM is currently offering on their
Yukon XL’s. After rebate, the cost of the vehicle is \$32,000. If you put no money down on
a 5-year, 8% interest loan, how much will your monthly payments be?

Because car payments and mortgage payments are typically made on a monthly basis, we
need to adjust r and t to do this computation. We need to divide r by 12 to get a monthly
interest rate, and we need to multiply t by 12 to get a number of months.

Using the formula:
1  1  
    ( 1 12 )tx12  

rr

PVannuity = PMT x 
                     
r
       12            
                     

This time, we will solve for PMT because the principal, PV annuity, is already known.

1  1  
                        = PMT x  1  1.4899   = PMT x  1  0.6712  = PMT x  .3288 
 ( 1 .12 )5 x 12  

08                               1
32,000 = PMT x                                    .006667 
                                                        


.08
12              
                                  .006667                .006667 
                        
32,000 = PMT x 49.318

PMT = 32,000/49.318 = \$648.85

Using the financial calculator:
32000                         PV
5*12 = 60                     N
8/12 =.6667                   %i
CPT PMT                648.85

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
Example: Home Mortgage
You have just decided to purchase a new home for \$200,000. If the interest rate on a 30-
year mortgage is 6%, and you put 5% down, how much will your monthly payments be?

The principal that you will actually borrow is 200,000*(1.-05) = 190,000.

Using the formula:
1  1  
                        = PMT x  1  6.0226   = PMT x  1  0.16604  = PMT x  .83396 
 ( 1 .12 )30x 12  

06                               1
190,000 = PMT x                                    .005 
                                                        


.06
12             
                                       .005               .005 
                        
190,000 = PMT x 166.792

PMT = 190,000/166.792 = \$1,139.14

Using the financial calculator:
190000                            PV
30*12 =      360                  N
6/12    =    0.5                  %i
CPT PMT              \$1,139.15

After you have computed the payment on a mortgage or loan, you can multiply the dollar
amount of the monthly payment by the number of months the payment will be made to
compute the total of all of the payments. If you subtract the principal from that total, you
will get an estimate of the total dollars of interest that will be paid on the loan.

Total payments = 360 * 1,139.14 = 410,090.40

Interest = Total payments – principal = 410,090.40 – 190,000 = 220,090.40

Shocking, isn’t it! Most of us would probably prefer not to know the total interest that will
be paid on a 30 year mortgage loan. However, you can significantly decrease the interest
paid over the life of the loan by making extra principal payments when at all possible.

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
Time Value of Money Practice Problems

These problems are designed to enable you to practice solving the time value of money
problems discussed on the previous pages. The answer for each problem is giving in
order that you might check your work.

1. You are to receive \$6,000 five years from now. At an annual rate of 10%, what is this
worth today?
3,725.53

2. You receive \$4,000 today. If the account you invest it in earns 10% per year, what will
the balance in the account be at the end of five years?
6,442.04

3. You are offered the option of receiving \$4,000 today or \$6,000 five years from now.
Assuming an annual interest rate of 10%, which of the two would you choose? (Base
your choice only on the TVM.)
4,000 today

4. You need \$15,000 each year for the next 3 years. If you earn 9% interest, how much
must you invest today?
37,969.42

5. You want to buy a car that costs \$14,000, you will put 10% down, and a bank will
finance the balance at a rate of 1% per month for 60 months. What is your monthly
payment?
280.28

6. A car dealer says a car will cost you only \$179.99 per month. He won't tell you the price
of the car so you resort to computing it yourself. To make you a well-informed buyer,
he tells you the payment of \$179.99 per month is based on a sixty-month loan with an
8% APR. Based on this, the cost of the car is ____?
8,876.82

7. A car dealership offers you a car for \$8,000. If you can finance it for 60 months at an
APR of 12.6%, what is your monthly payment?
180.39

8. You want to buy a house. You can finance \$148,000 of the purchase
price with a local bank at a fixed 7% APR with monthly payments for
30 years. Compute your monthly payment on this loan.
984.65

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004
9. Using the information from the previous question, if you make all 360
payments on time, what are your total payments on the mortgage? How
much is interest? How much is principal?
Total payments = 354,474
Principal = 148,000
Interest = 206,474

10. Use the information from the previous two questions but assume an APR of
6% instead. What is your monthly payment on the 30 year loan? How
much do your total payments amount to? How much is interest? How
much is principal?
Monthly payment = 887.33
Total payments = 319,440.52
Principal = 148,000
Interest = 171,440.52

11. You borrow \$12,000 on a car loan with an APR of 8.75% and monthly
payments over 5 years.
A.     What is your monthly payment? 247.65
B.     If you make all of your payments on time, how much interest do you pay over
the life of the loan? 2,859

_____________________________________________________________________________________________
Prepared by Angeline M. Lavin, Ph.D., CFA                                 May 2004

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