Time Value of Money

Readings: Chapter 6

At the end of this unit students should be able to:  

1. Convert time value of money (TVM) problems from words to time lines.

2. Explain the relationship between compounding and discounting, between future and present value.

3. Solve for time or interest rate, given the other three variables in the TVM equation.

4. Distinguish among the following interest rates:  Nominal (or Quoted) rate, Periodic rate, and Effective (or Equivalent) Annual Rate; and properly choose between securities with different compounding periods.

5. Compute the future value of some beginning amount, and find the present value of a single payment to be received in the future.

6. Find the future value of a series of equal, periodic payments (an annuity) as well as the present value of such an annuity.

7. Explain the difference between an ordinary annuity and an annuity due, and calculate the difference in their values.

8. Calculate the value of a perpetuity.

9. Demonstrate how to find the present and future value of an uneven series of cash flows.

10. Solve time value of money problems that involve fractional time periods.

11. Construct a loan amortization schedule

TVM is one of the most fundamental aspects of finance. It is important because it recognizes that with the presence of positive interest rates a dollar today is worth more than a dollar any time in the future. The primary reason of this is the existence of investment opportunities i.e. money today can be invested to earn further wealth in the future.

TVM concepts are important to making decisions which compare future cash flows to be obtained at different points in time. It is necessary to compare the value at some common point usually at Time (t) = 0 (now) or at some arbitrary time (t) into the future.

Future Value (Compounding)

The terminal value of a lump sum (a single cash flow) when invested, and the intervening interest payments when invested over a given period  n  at a specified rate of interest  k is called the future value of a lump sum.

Assumptions:

1.     The interest rate remains constant

2.     The interest earned is reinvested along with the original principal

Illustration 1: If we have $1000 to be invested at 20% over the next three years, what are the future values at the end of each year?

                                                          Year

1000 + ( 1000 x.20) = 1200                 1

1200 + ( 1200 x.20) = 1440                 2

1440 + ( 1440 x .20) = 1728                3

* The lump sum or principal is $1,000, the number of periods is 3 and the interest rate is 20%

However suppose we wanted to find the value at year 50? - "We wouldn't want to be repeating the process 47 more times. There is a shorter method which is now described:

We will call the principal, PV or PMT, terminal value, FV for Future Value, the interest rate k and the number of years n, then :

After year 1

Similarly after year 2:  

If we continue for year 3, we would have:  

Generally:

Therefore the previous result of $1,728 could have been found by:

Also, the value at year 50,could be found by :

Using tables

Future value interest factors can be found from Table 1 (See below) which is called Future Value Interest Factors, FVIF, $1 compounded at k percent for n Periods: FVIF_k%,n = (1+k)ⁿ. By matching appropriate interest rates and the corresponding periods we obtain the future value factors.

 Table 1:-FVIF

Growth Rates

TVM concepts can be use to estimate growth rates. We may wish to find the average growth in GDP, inflation or in an index etc. over some given time period.

Example  

If the Jamaican Stock Exchange market index grew from 240.38 at the end of the year in 1983 to 25,745.88 at year-end in 1992 what was the average growth rate per year? 

Note

1.     We are interested in the geometric average which accounts for compounding. The geometric average accounts for compounding over the period whereas the arithmetic average does not do that

2.     The number of yearly digits between 83 and 92 inclusive is 10, but the number of compounded years is n-1, which is 9.

Example

How long will it take to triple your money given the existing interest rates of 10%

Frequent Compounding - Lump sum

In the future value formulae we have used so far we assumed that compounding took place annually i.e. once a year. If compounding takes place more than once per year e.g. semiannually, we have to modify our formulae as follows:    

Using tables form	Using algebraic form
Where m is the number of times interest is compounded per year

Note - where compounding takes place more than once per year interest is earned on interest at a faster rate hence the future value (FV) is larger. (Ceteris Paribus)

Example

What is the future value of $500 given an interest rate of 15% p.a. invested over 2 years and interest is compounded as follows:

a)     Annually  

b)    Semiannually

c)     Continuously  

  Effective Annual Interest Rates (EAIR)

As can be seen the number of compounding periods increase the FV increase all thing being equal. This gives rise to the effective annual interest rate (EAIR) or the annual percentage rate (APR) which is that rate of interest, which equates to the nominal rate subject to multiple compounding.

  Example

What is the EAIR of 15% p.a. compounded:

 a)              Semi-annually?

b)      Continuously?

• The EAIR is always greater than the nominal or stated rate, k.

Annuities

An annuity is a series of unbroken constant payments of cash flows made at equal time intervals. 

a) Ordinary Annuity

An ordinary annuity, by definition, is one where the payments or cash flows take place at the end of some given time period for e.g. most mortgages and loan payments. "This is also called a deferred annuity". E.g.:

Future Value of ordinary annuity

We define an annuity by counting the number of payments that are made. The future value of an ordinary annuity is found at the point at which the last payment is made. So for an  n  period ordinary annuity the future value is found at the end of the nth year.

Looking at the above cash flow pattern, let's see what the future value would be at the end of five years, using the lump sum approach, given an interest rate of 20%.  

  * It should be noted that in the case of an ordinary annuity the first payment is made at the end of the first period and therefore for an  n  period annuity the first payment is compounded over   n-1   periods.

If we were to use the annuity tables (See Table 2 - FVIFA below), we would apply the following formula:          

  Table 2:-FVIFA

b) Annuity Due

An annuity due, by definition, occurs where payments or cash flows take place at the beginning of some time period (e.g. rent and university tuition fees)   E.g.:

Future Value of an Annuity Due

The future value of an annuity due is found at a point which is one period after the last payment is made. So for an  n  period annuity-due the future value is found at the beginning of the n+1 ^th year.  

Using the lump sum approach, with a rate of 20%, the value at the end of five years would be:  

This difference here is that the cash payments are received one period earlier and therefore earn interest for an extra period.  

Before we use the tables to solve this example we need to understand the relationship between an Ordinary Annuity and an Annuity Due:

    Therefore, the solution to this example is:

c) Perpetuity

A perpetuity is similar to an annuity, however it extends into infinity. E.g.    

Examples of perpetuities include The British Consol. (this is a perpetual bond issued by the British Government in 1815 where the proceeds had been used to pay off many smaller issues that had been floated in prior years to pay for the Napoleon Wars. They are called consol because they were used to consolidate the past debts. These perpetual bonds just keep on paying interest each year with no maturity date set.

A more common example would be the pension that one receives (assuming that one lives forever one will continue to receive their periodic pension amounts.)

Growing perpetuities refer to the situation where an amount of money increases at a fixed rate indefinitely .

There is no future value of a perpetuity. We can however find the present value ( later).

  d) Embedded Annuity

This exists in a series of cash flows some of which do not form part of an annuity. 

Note - the entire cash flow stream is also known as an "Uneven cash flow stream"

Future value of an embedded annuity (uneven cash flow stream)

This again can be found by finding the sum of the future value of the individual cash flows. Using the FVIF tables - Table 1 to find the future value at the end of year 6 of the above pattern, we would have the following (assuming k=10%):  

The payment of $1,500 can be treated as an annuity. Whether it is an ordinary annuity or an annuity due is dependent on where one recognises the starting point of the payments to be. If the $1,500 is treated as an ordinary annuity the starting point is time period 1 and the ending point is time period 4. If it is treated as an annuity due the starting point is time period 2 and the ending point is time period 5.

  (a) If we treat as ordinary annuity, the formula for which is:

             FVA_O  =  PMT  x FVIFA _k%,n

  then the future value of the series of $1,500 at time period 4 is found by:  

Consequently, the future value of the series of $1,500 at time period 6, can now be found by applying the lump sum formula:

        FV =  PV x FVIF_k%,n 

Hence, at time period 6:       

As a short cut, the future value of the series of $1,500 at time period 6 could be found by

                             FV = 1,500 (FVIFA _10%,3 )( FVIF_10%,2)

                                   = 1,500 (3.31)(1.21)

                                   = 6007.65

An alternate way to treat it as an ordinary annuity is to find the future value at time period 6 as follows:

·        Include imaginary additional payments at time periods 5 and 6, then

·        Subtract the future value of these additional payment from the future value of the '5' payments.

Eg:

(b) If we treat it as an annuity due, the formula for which is

FVA_D =[ PMT(FVIFA _k%,n)] ( 1+k),

then the starting point is time period 2 and the ending period is time period 5. Therefore at time period 5:

Consequently, the future value of the series of $1,500 at time period 6, can now be found by applying the lump sum formula:

        FV =  PV x FVIF_k%,n 

Hence, at time period 6:   

  Reverting to the original question, the final figure for the future value at the end of year 6, would be :

  Note - the $2,000 was put in at the end of year six, thus no interest payment was made.

Frequent Compounding - Annuities

In the future value formulae above, we assumed that compounding took place annually i.e. once a year. If compounding takes place more than once per year e.g. semiannually, we have to modify our formulae as follows:    

Using tables form	Using algebraic form
Where m is the number of times interest is compounded per year

Present Value (Discounting)

The current value of lump sum to be received in the future is called the present value. In most cases it is the reverse of compounding and the process of finding the present value of a future sum of money is called discounting. The appropriately determined interest rate that is used to find the present value of the future sum is called the discount rate  .

  Class question 2

What is the present value of $150,000 to be received:

1.  in six (6) years at 7% per annum?

2.  In four (4) years at 7% per annum?

3. In four (4) years at 10% per annum?

• Given a fixed sum of money to be invested some time in the future the present value changes depending on the interest rate and the time period. The higher the interest rate the lower the present value. The greater the time period the lower the present value.

Class question 3

Use a present value formula to answer the following question: Given that, interest rates are at 30%, would you prefer a scholarship of $20,000 payable now or a refund of your tuition of approximately $51,000 at the end of 3 years?   

Present value of an annuity

 The present value of an annuity is an amount of money now, which is equivalent to a series of payments over some future annuity period.

Alternatively, it is the deposit which is required now to allow equal withdrawals at the end (or beginning) of each period in the annuity.

a) Ordinary Annuity

The present value of an ordinary annuity is determined by the following formula:

Note - when this formula is applied, the result is at a point which is one period before the first payment is made. 

Let us find the present value at Time 0 of the following ordinary annuity cash flow pattern given an interest rate of 10%: 

Note - here again we could have found the answer by discounting the cash flows individually and summing the results.     

b) Annuity Due

The present value of an annuity due is determined in a manner similar to the future value of an annuity due (discussed earlier), therefore:

Note - when this formula is applied, the result is at the point where the first payment is made.

Let us find the present value at Time 0 of the following annuity due cash flow pattern given an interest rate of 10%: 

  c) Present value of an embedded annuity (uneven cash flow stream), assuming k = 10%  

d) Present Value of a Perpetuity  

  This is an approximation which recognizes the fact that cash flows very far into the future do not contribute much to present value, that is, despite the infinite cash flows the more distant ones do not contribute much or is of negligible value

Example

What is the PV today of the cash flow stream of $1000 which starts at year1 with a discount rate of 10%?  

•   (Note - always treat perpetuity as ordinary annuity, that is, the starting point is one period before the first payment began)

What is the PV today of the cash flow stream of $1000 which starts at year 3 with a discount rate of 10%?  

What is the PV today of the cash flow stream of $1000 which starts at year 0 with a discount rate of 10%?  

  Loan Amortisation

Some loans are repaid in equal periodic installments each comprising of interest and principal so that the next period's interest charge is reduced; as a result, interest charges are larger in the beginning of the repayment period. The total interest paid over the life of the loan is sometimes called the finance charges. This method of repayment recognises the reducing balance of the principal and takes into account the fact that you are paying the lender k% per period on his outstanding money and charging the borrower k% per period on his outstanding balance. The loans are called amortised loans and the following formula is applicable (assuming period-end payments i.e. an ordinary annuity):

Example

What is the monthly payment on a loan of $20,000 at 36% per annum repayable over 2 years with the first payment being made at the end of the month. (ordinary annuity) 

  Amortization Schedule:

• Note - this can easily be done using a spreadsheet package e.g. Excel, Lotus

At the end of period 23, i.e. start of period 24 we know that only one annuity is outstanding, therefore the outstanding balance at this time equals: 

Likewise, the closing balance at the end of period 3 is:  

The example above stated that payments were made at month end. Let us now look at what happens when the payments are made at the beginning of the month, with everything else remaining constant. 

This is effectively now an annuity due. To determine the monthly payment, we now have:

The amortisation schedule for the first 3 months will now look as follows: (Note - the first payment consists only of principal i.e. no interest)

Note - the ordinary annuity formula would still be applicable to find the outstanding balance at any point in time.

E.g. The outstanding balance at the end of month 3 is:

Note: any actions that reduce the principal outstanding also reduce the interest payment:  

E.g.

·        Increase monthly payments

·        Make adhoc payments to principal

·        Make an initial lump sum payment to principal