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Basic Time Value of Money Principles

The fundamental idea supporting the concept of time value of money is actually very simple.  Simply put, a dollar received today is worth more than a dollar received a year from now.  It is worth more because the dollar can be invested and allowed to earn interest for that year.  At the end of the year, the dollar received today is worth the original dollar plus the interest it has earned during the year.

Solving time value of money problems involve either compounding or discounting either lump sums (or single payments) or annuities (a series of equal payments).

There are two types of compounding problems.
1) the future value of a lump sum and
2) the future value of an annuity.

There are two types of discounting problems.
1) the present value of a lump sum and
2) the present value of an annuity.

Special forms of these four types include:

  1.   sinking fund problems,

  2.   mortgage payment problems, 

  3.  and payments in advance.

return to Using the HP10B and HP10BII calculator.

 

COMPOUNDING

In financial matters, the word compounding means the deposit of a dollar sum into a bank account or other interest-bearing investment, the drawing of interest periodically on the invested amount, and the further drawing of interest on the interest received in prior periods.  The compounding period is that period of time when the interest is paid.  That interest paid the first period now earns interest in subsequent periods.  The shorter the compounding period, the greater the accumulation over time.  

For example, if $10,000 is deposited into a bank account that pays 6% interest compounded annualy will accumulate to  $10, 600 by the end of the year ($10,000 + $600 interest).  If instead, the bank pays that same interest rate compounded monthly, the account will accumulate $10, 616.78.  Compounding monthly earned $16.78 more interest.  This is because the $50 interest paid from the very first month (6%/12*$10,000=$50) then earns interest the remaining 11 months.  The second month the account earns $50.25 ($10,050*6%/12).  Each month the interest earned is added to the account and earns interest in the subsequent month.

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THE FUTURE VALUE OF A LUMP SUM (SINGLE PAYMENTS)

If you deposit $1,000 in an investment yielding 8 percent, during the first year you will make $80 in interest. Thus, the total value of your investment after the passage of a year will be $1,080.

If you leave the $1,080 in the investment, how much will the value of your investment be at the end of two years? You will, of course, have your $1,080 from the first year, plus 8 percent on the $1,080 during the second year. For the second year you would receive $86.40 in interest to add to your $1,080. The total value of your investment at the end of two years would therefore be $1,166.40.

Similarly, for the third year, your interest would be $93.31, and the total value of your investment at the end of the third year would be $1,000 plus $86.40 plus $93.31 for a total of $1,259.71.

As can be seen, the process of compounding involves investment of a given sum, drawing of interest on that sum for a period, and then drawing of interest on the original sum plus the interest accumulated in prior periods. The general formula for compounding is:

 

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THE FUTURE VALUE OF AN ANNUITY (SERIES OF EQUAL PAYMENTS)

Suppose that, instead of depositing $1,000 at the beginning of a period, you deposit $1,000 at the end of each year for three years. What would be the total value of your investment at the end of the three years, drawing interest at 8 percent per year? The process of compounding in this case is similar to that of the example above, except that the payments are usually made at the end of each year, and the values thereof are added together. Thus, a $1,000 payment would be made at the end of year one, another $1,000 payment would be made at the end of year two, and another payment would be made at the end of year three.

The payment at the end of year one would be worth $1,000; it would draw interest of $80 during the second year, and $1,000 would be added to it, making a total value at the end of year two of $2,080. $166.40 would accumulate in interest during the third year, and $1,000 would be added at the end of the third year, making a total value of $3,246.40. This amount, then, is the future value of a series of three payments of $1,000 each, accumulating interest at 8 percent annually. Such a series of payments is called a level annuity

The general formula for the accumulation of a given amount each period is easily derived using Equation 1. The derivation of the general formula may be seen in Equation 2.

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SINKING FUND ACCUMULATION

A sinking fund is an account into which is deposited money every period so that the account will grow to the desired amount in the future.  The future value of a series of equal payments was the unknown that had to be calculated in the previous compounding situation. Now suppose that you wish to invest a certain amount which over a three-year period will accumulate to $1,000. How much would you have to invest each year for three years at 8 percent interest to reach the desired figure?

This question is literally and mathematically the reciprocal of the prior question that asks, "How much is the future value of a series of known payments?"  To find how much must be invested or put into a sinking fund drawing interest at 8 percent, one need only take the reciprocal of the future value. Thus, if $1000 were deposited at the end of the year for three years into an account paying 8% per year, the account would accumulate to $3,246.40.  The reciprocal in this example would be $1,000 divided by $3,246.40, or one divided by 3.2464 which would yield .30803.  The quotient must then be multiplied by 1,000, since it is the sum which we wish to accumulate.  The resulting answer is approximately $308.03, which means that if this amount were invested at the end of each three years, at the end of the three year period the value of the accumulated total would be $1,000. The formula for the Sinking fund (SF) is easily found by solving Equation 2 for the payment (PMT) as seen in Equation 3.

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DISCOUNTING

Discounting may be regarded as the reciprocal process of compounding. In compounding we seek to find the future value of a payment or series of payments, or the payments necessary to accumulate to a desired future value. In discounting we know the future payments, or the total amount to be paid off, and we must calculate the present value of those future payments or of the individual payments necessary to amortize (pay off) a present value.

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THE PRESENT VALUE OF A LUMP SUM (SINGLE PAYMENT)

The present value of a lump sum payment due at some time in the future is the reciprocal of the future value of a lump sum payment invested today. Thus, its formula is shown in Equation 4 below.

The present value of $1,000 due at the end of the fifth year is $680.58. Or to state it slightly differently, if an investor pays $680.58 today for the right to obtain $1,000 in five years, (s)he will have made an 8 percent return on the investment.

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THE PRESENT VALUE OF AN ANNUITY (SERIES OF EQUAL PAYMENTS)

A series of equal payments due in the future is also known as a level annuity. The formula for the present value of an annuity is shown in Equation 5. While not illustrated, this formula is also easily derived using the same algebraic technique used in Equation 2. The algebraic solution to Equation 5 is left as a student exercise.

This calculation of an annuity's present value is extremely important because it demonstrates that the value of any income stream, whether the payments are equal or whether they vary, can be obtained by summing the present values of the individual payments. Thus, the present value of an annuity of $1.00 per year for 5 years is the sum of the present values of annual payments. These amounts are shown below and summed:

If the amount to be invested annually is $1,000 the present value of the annuity is therefore 3.992710 x 1,000 = $3,992.71.

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INSTALLMENT TO AMORTIZE A PRINCIPAL AMOUNT

The periodic payments necessary to repay a present value, such as an amount of borrowed money, constitute the amortization or repayment of the loan. The principal amount borrowed is nothing more than the present value of the series of periodic payments discounted at the interest rate. The amortization installment formula shown in Equation 6 is the reciprocal of the present value of an annuity formula.

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PAYMENTS IN ADVANCE

Some income streams provide payments at the beginning of the periods. For example, apartment rentals are often paid at the beginning of the rental period. Such streams can be compounded or discounted easily by recognizing that the beginning of any period is the end of the prior period. For example, an ordinary annuity of $1,000.00 for five years, whose value discounted at 8 percent is $3,992.71 can be converted to payments in advance by shifting all payments backward one year. The resulting annuity requires a payment at time zero (now), which is worth 100 percent of the amount paid, plus payments at the end of each year (beginning of the next year) for four years. The value of the annuity with payments in advance is therefore $4,312.13.

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