The Time Value of Money

Chapter 6

 

 

Time lines help keep track of payments and disbursements

-100      +100           +200           +300

 |-------------|-------------|-------------|

 0                   1                   2                   3

5%

 

Simple Interest Versus Compound Interest:

u     With COMPOUND Interest, interest is earned on interest paid previously

u     This chapter deals with compound interest

Future Value of a Sum

5%            ?

 |-------------|-------------

-100

 

One Period Model

FV1  = PV0  (1 + i)

FV1  = 100 (1.05) = 105

 

If deposit is left to earn for more than one period:

FV2  = FV1 (1.05) = 105 (1.05) = 110.25

substitute FV1 = 100 (1.05) into the above

 

FV2  = 100 (1.05) 
(1.05) = 110.25

So the General Multi-Period Model is:

 

FVN  = PV0 (1 + i)N 

 

Example

If you deposit $2000 into an
account earning 6% per year, what will be in the account after 10 years if interest is paid annually?

 

Solution

FV10 = $2000 (1.06)10

FV10  = $2000 (1.7908) = $3, 581.69

Non-Annual Compounding

1.  Divide the annual interest rate by the number of payment periods per year to find the interest rate per period.

 

2.  Multiply the number of years by the number of periods per year to find the number of compounding periods.

 

FVN  = PV (1 + i/m)mn

m = number of times compounding  
   occurs per year

n =  number of years

 

Example

What is the future value of a
$2000 deposit after ten years if
it earns 6% per year and is compounded semi-annually?

 

Solution

FV10 = $2000 (1 + .06/2)2 x 10

FV10 = $2000 (1.8061) = $3,612.22

 

As the Number of Compounding Periods Increases, the Future Balance Increases

FV10  with annual = $3,581.69

FV10  with semi-annual = $3,612.22

 

Effective Interest Rate

Eff Rate = (1 - 1/m)m - 1

based on equation for multiple compounding periods

 

Compounding                Effective
   Interval               Equation 6.3        Rate

 

Annual         FV = (1.12)1 – 1    12.00%

Semiannual  FV = (1.06)2 – 1    12.36

Quarterly     FV = (1.03)4 – 1    12.55

Monthly       FV = (1.01)12 – 1   12.68

Weekly        FV = (1.0023)52 – 1         12.73

Daily  FV = (1.0003288)365 – 1  12.7475

Continuously         FV = e.12 – 1         12.7596

Using Financial Tables

n   4%         5%    6%    7%    8%    9%    10%

3   1.125      1.158 1.191 1.225 1.260 1.29   1.331

4   1.170      1.216 1.162 1.311 1.260 1.412 1.464

5   1.217      1.276 1.338 1.403 1.469 1.539 1.611

 

Example: using FV to compute future balance

How much must you invest at 6%
to have $10,000 in ten years with
semi-annual compounding?

 

Solution

FV0  = PV0 (1 + i/m)nm

FV0  = PV0 (1 + .06/2) 10 x 2

$10,000 = PV0 (1.8061)

PV0  = $5,536.71

 

Example: Inflation Adjustment

How much will you need in 35
years when you retire to have the same purchasing power provided
by $35,000 today if inflation averages 4%?

 

Solution

FV = 35,000 (1.04)35

       = 35,000 (3.9461) = $138,113.11

 

Future Value of Annuity

 

u     An annuity is a series of equal payments made at regular intervals.

u     Usually solve FVAs using a calculator or tables since equation is complex.

 

                10%

 |----------|----------|----------|

            100         100       100

                             __________   110

               _________________   121

                                                  = 331

 

 

Table Approach

FVannuity = 100 (FVIFA10%,3)

 

FVannuity  = 100 (3.3100) = 331

 

 

Example

How much will be in your retirement account if you
deposit $2,500 per year for
35 years and it earns 10% compounded annually?

 

Solution

FV = 2,500  (FVIFA10%, 35)

FV = 2,500 (271.0244) = 677,561.00

 

Ordinary Annuity versus Annuity Due

Ordinary annuity cash flows occur at the end of the period. 

          Annuity due cash flows occur at the beginning of each period.

     To convert an ordinary annuity to an annuity due, multiply by (1 + i).

Present value of lump sum

100

|-------------|-------------|

            ?  Ώ

 

We already know that:

FVN = PVO  (1 + i)N

Convert to PV equation

PVO  = FVN  / (1 + i)N

 

Example

You will receive $50,000 in 5 years from a trust fund.  If you can earn 8% on investments, how much is this future payment worth today?

 

Solution

PVO  = 50,000 / (1 + .085) = $34,029.16

 

PV of Annuity

u     Usually computed using tables
or calculators

 

Example

You won the lottery and will receive $50,000 per year for the next 20 years.  If you can invest to earn 10%, how much is the stream of lottery payments worth today?

 

Solution

PV = pmt (PVIFAi, N)

PV = 50,000 (PVIFA10, 20)

PV = 50,000 (8.5136) = $425,680.00

 

Perpetuity

u     A cash flow that continues forever

u     PVperp  = Pmt / I

 

Example

What is the value today of a company expected to generate
net revenues of $10,000 per
year forever, assuming a 15% interest rate?

 

Solution

PV = 10,000 / .15 = $66,666.67

 

Example:
Deferred Annuities

What is the PV of the following
cash flow stream?

 

                                                100        100            100

 |------------|------------|------------|------------|----------- -|

 0              1               2               3               4               5    

 

Solution

PV2 = $100 (PVIFA3, 10%) (PVIF2, 10%)

        = $100 (2.4869) (.8264)

        = $205.52

 

LOAN PAYMENTS

u     These are computed using the present value of annuity method.

 

 

Example

What is the annual payment due on a 5 year $10,000 loan if IR = 10%?

 

Solution

PV = pmt (PVIFA)

10,000 = pmt (PVIFA10%, 5)

pmt = 10,000 / 3.7908 = $2,673.96

 

GROWTH RATES

u       To compute the compounded
    growth rate use the FV              equation.

u       The beginning value is the PV.

u       The ending value is the FV.

 

Example

You bought stock for $15.00
per share 5 years ago.  It is currently selling for $45.00 per share.  What is the compounded average growth rate?

 

Solution

FV = PV (1 + g)N

45   = 15  (1 + g)5

(45 / 15) 1/5 - 1 = g

g = .2458 = 24.58%

 

 

u     Use future value to find the balance resulting from an interest-earning deposit.

u     Use future value of an annuity
 to find the payment needed to achieve a known future balance.

u     Use either present or future value (without an annuity) to compute growth rates.

u     Use present value on all
loan calculations.

u     Use present value to value assets.

u     Use present value to evaluate investments.

 

 

 

 

Tables, Calculators, Equations, and Spreadsheets

u      Tables preceded modern calculators/computers

u      Tables pre-calculated results for equations

u      Most calculations today done by calculators or computers

u      Calculators and computers merging

 

Calculators

u       Constant memory

u       Multiple functions of keys

u       Algebraic or Reverse Polish Notation

u       Clearing the calculator

u       Changing Sign

u       Financial Tour

 

Calculator Financial Tour

u      n

u      I/Yr

u      PV                 

u      PMT

u      FV

u      P/Yr

u      xP/Yr

 

Calculating the Future Value of a Lump Sum

u       Enter the proper payments per year. You first must type in the appropriate number (eg. monthly would be 12, quarterly would be 4, etc.), then push the cream colored        key, then the P/YR        key .

u       The order of the next three steps is not important, but I recommend that you follow across the financial tour of your calculator from left to right. If you do this, then the next step would be to enter in the proper number of years.

 

Future Value of a Lump (Continued)

u      Next, You first type the appropriate number of years, then push the cream colored       key, then the xP/YR        key.

F    Now enter the appropriate interest rate per year. This is done by entering the appropriate annual interest rate as a whole number, not as a decimal (the calculator will convert it to decimal automatically), then pressing the I/YR         key.

 

 

Future Value of a Lump Continued

u      Next enter the lump sum amount you are beginning with, then press the PV        key.

 


u      Finally all that is necessary is push the FV           key.

 

 

 

Future Value of Lump Sum Example

u Now let's try a problem. Consider that you invest $1000 in a CD earning 8% annual interest for 3 years. What would your CD be worth at the end of the three years?

u First let's see what we know. We know that the CD worth $1000 today.  That indicates that the present value (PV) of that CD is $1000.  We know that the number of years (xP/YR) is 3.  We know that the annual interest rate is 8% (8 is the I/YR).  Finally, we may assume that the periods per year is one since we are not told it is monthly, quarterly, etc. (If the payments per year would have been, for example, monthly, the problem would have read, ". . . earning 8% annual interest, compounded monthly.")

 

Future Value Example Continued

u      First we must enter the appropriate payments per year by pushing 1, then the P/YR              key.

u      Next we will enter the number of years by pushing 3, then the cream colored       key, then the xP/YR       key.

u      Now we will enter the interest rate per year by pushing 8 and then pressing the I/YR         key.

 

Future Value Example Continued

u Next we enter the Present Value by pressing 1000, then the         key (to show that this is paid to the bank), and then the PV         key

u Finally, we press the FV       key to compute the answer. The display then shows 1259.71 if the display was set to show two decimal places. Note that the answer is positive. This indicates that the direction of the cash flow is in. (This is logical since we pay $1000 now in order to receive $1,259.71 in the future.)

 

 

 

 

 

 

Calculating the Present Value of a Lump Sum

u  Calculating the Present Value of a Lump Sum is almost the same as for a Future Value

u  Now let's try a problem. Consider that you buy a $1000 US Savings Bond that will mature in five years and paid 8% interest per year. How much should you pay today for the bond?

u  First let's see what we know. We know that the bond will be worth $1000 in five years. That indicates that the future value (FV) of that bond is $1000. We know that the number of years (xP/YR) is 5. We know that the annual interest rate is 8% (8 is the I/YR). Finally, we may assume that the periods per year is one since we are not told it is monthly, quarterly, etc. (If the payments per year would have been, for example, monthly, the problem would have read, ". . . paid 8% per year, compounded monthly.")

 

Present Value Example Continued

u      First we must enter the appropriate payments per year by pushing 1, then the P/YR              key.

u      Next we will enter the number of years by pushing 5, then the cream colored         key, then the xP/YR        key.

u      Now we will enter the interest rate per year by pushing 8 and then pressing the I/YR          key.

 

 

 

 


Present Value Example Continued

u       Next we enter the Future Value by pressing 1000 and then the FV      key.

u       Finally, we press the the PV         key to compute the answer.  The display then shows -680.58 if the display was set to show two decimal places. Note that the answer is negative. This indicates that the direction of the cash flow is out. (This is logical if we will be getting $1000 in the future, we must pay $680.58 now.)

Annuities

u       An annuity payment is a series of equal payments that reoccur every period such as making monthly payments on a loan.  To enter an annuity press the amount of the annuity and then press the Payment       key.

u       Most annuities occur at the end of the period such as a mortgage payment.

u       Annuities may occur at the beginning of the period such as rent payments that are paid in advance.  If you are working with payments in advance, you must first press the Begin        key to indicate the payment is at the beginning.

   

 

Calculating the Present Value of an Annuity

u      Calculating the present and future values of annuities are basically the same as the calculations for lump sums except that you now must enter the amount of the annuity.

u      Let's try a problem. Consider that you invest $1000 in a bank account at the end of every year for 3 years. The bank will pay 8% annual interest on the deposits. How much would your bank account be worth at the end of the three years?

 

Present Value of an Annuity Problem

u First let's see what we know. We know that there will be three deposits of $1000 at the end of each year for three years. That indicates that the payment (PMT) is $1000 since it is a recurring payment every period. We know that the number of years is 3. We know that the annual interest rate is 8% (8 is the I/YR). Finally, we may assume that the periods per year is one since we are not told it is monthly, quarterly, etc. (If the payments per year would have been, for example, monthly, the problem would have read, ". . . pay 8% annual interest, compounded monthly.")

 

Annuity Problem (Continued)

u      First we must enter the appropriate payments per year by pushing 1, then the P/YR              key.

u      Next we will enter the number of years by pushing 3, then the cream colored        key, then the xP/YR        key

u      Now we will enter the interest rate per year by pushing 8 and then pressing the I/YR         key.

.

Annuity Problem (Continued)

u      Next we enter the Payment by pressing 1000, then the           key (to show that this is paid to the bank), and then the PMT         key.

u      Finally, we press the the FV         key to compute the answer.  The display then shows 3246.40 if the display was set to show two decimal places. Note that the answer is positive. This indicates that the direction of the cash flow is in. (This is logical since we pay $1000 per year in order to receive $3,246.40 in the future.)

 

 

 

 

 

Solving for Other Factors

u  In solving for n or I/YR, it is critical that you be careful to indicate the direction of outgoing cash flows by hitting the +/-         key to make the amount negative.

u  n

–   To solve for n (number of periods) enter the rest of the problem information and then press n

u  number of years

–   After solving for n, simply hit the RCL key and then the n key.

u  I/Yr

–   To solve for I/YR enter the rest of the problem information and then press I/YR

Payments (PMT)

uLoan payment (mortgage payment)

–   What is the monthly mortgage payment on a $100,000 loan at 12% annual interest amortized over 30 years?

–   12            (monthly payments); 30            ;  12          (annual interest); 100000        (loan amount received initially)

–   To solve for the payment, push      ; PMT=-1028.61

uSinking fund

–   What is the monthly payment necessary to accumulate $15,000 to replace a roof in 20 years if the reserve fund pays 8% annually?

–   12              (monthly payments); 20              ;  8          (annual interest); 15000     (amount to be received in the future)

–   To solve for the payment, simply push      ; PMT=$25.46

Loan Amortization

u  Consider the monthly mortgage payment on a $100,000 loan at 12% annual interest amortized over 30 years from the previous slide.

–   If the interest rate is 12% over 12 months, then the monthly interest rate is 1%

–   For the first month:

•   $1,028.61 = the monthly payment

•   $1,000.00 = the first month’s interest (1%*100000)

•          28.61 = the first month’s principal reduction

•   $99,971.39 = remaining balance = $100,000 - $28.61

–   For the second month:

•   $1,028.61 = the monthly payment

•   $   999.71 = the second month’s interest (1%* 99,971.39 )

•          28.90 = the second month’s principal reduction

•   $99,942.49 = remaining balance = $ 99,971.39 - $28.90

The Amortization Function

u  First calculate the mortgage payment

–   12            (monthly payments); 30           ;  12        (annual interest); 100000     (loan amount received initially)

–   To solve for the payment, push      ; PMT=-1028.61

u  The press the cream colored key and the AMORT key       .  The display shows PEr  1-12 (The amortization schedule for the first 12 months of payments.)

u  Pressing the = key will give you the interest paid during the year.  The display briefly flashes Int then shows -11,980.47.

u  Pressing the = key again will give you the principle paid. The display briefly flashes Prin then shows –362.88

u  Pressing the = a third time will give you the remaining balance. The display briefly flashes bal then shows 99,637.12.

 

Remaining Balance

uEither use the amortization function shown on the previous slide or enter the number of years remaining on the loan and solve for PV.

uFor example, after solving for the mortgage in the previous slide, if you wished to know the balance on that loan after paying for one year, you would simply enter the years remaining—29           .  Then press PV to solve for the remaining balance.  The result shown is 99,637.12.

APR

u      The “true” interest rate charged by the lender

u      Calculate the mortgage payment ignoring the points.

u      Calculate the amount of the points

u      Subtract the amount of the points from the amount of the loan (the PV)

u      Reenter the result into PV

u      Solve for I/YR

 

Uneven Cash Flows

u       Calculating time value of money problems with uneven cash flows is considerably more difficult than for simple TVM problems. 

u       Net present value

–       Net present value is the present value of all of the future cash flows less the initial cost.

u       IRR

–       IRR is that interest rate that exactly equates the present value of all of the future cash flows with the initial cost.  In other words, the NPV=0

Calculation of NPV and IRR

The basic steps in calculating the Net Present Value and Internal Rate of Return are as follows:

u       First be sure to clear the calculator by hitting the cream colored     key and then the clear all             key.

 

u        Enter the proper payments per year. You first must type in the appropriate number (eg. monthly would be 12, quarterly would be 4, etc.), then push the cream colored     key, and then the P/YR         key . Usually, for IRR and NPV problems, you deal with annual cash flows. Consequently, the number of payments per year is usually set to 1.

u        

u        

 

 

 

 

NPV and IRR (Continued)

u        

u        

u        

u       Next, if you wish to compute a Net Present Value, you must enter the investor's cost of capital and then press the I/YR        key. If you are only interested in calculating an internal rate of return, this step is omitted.

NPV and IRR (Continued)

u      Now enter the cash flow for the first year and then press the cash flow     key. If this cash flow remains the same for any number of consecutive years, you enter the number of consecutive years and then press the cream colored      key and then the repeat Nj       key.

u      Repeat the above step for any different consecutive cash flow.

 

 

NPV and IRR (Continued)

u     After all of the cash flows are entered, press the cream colored      key and then the NPV     key.  If you wish to compute the Internal Rate of Return, simply press the cream colored       key and then the IRR      key.

Impact of Interest Rates

The monthly payment for a $100,000 loan at various interest rates:

u      8% = $733.76

u      9% = $804.62

u      10% = $877.57

u      11% = $952.57

u      12% = 1,028.61

Impact of Amortization Term

The monthly payment for a $100,000 loan at 12% annual interest at various maturities:
Number                              Total Interest
of Years     Payment        Paid During Loan     

u 30 years = $1,028.61       $370,300.53

u 15 years = $1,200.17       $216,030.25

u Most lenders charge about ½ point interest less for a 15 year loan so that would be 11.5%
15 years = $1,168.19        $210,274.17
Thus, increasing your payment by $139.58 per month would save $160,026.36 of interest over the life of the loan.