Present Value
Present value is the opposite of future value, as it reveals how much a dollar to be received in the future is worth today. The math is simply the reciprocal of future value calculations:
Where "i" is the interest rate per period and "n" is the number of periods
For example, $1,000 to be received in 5 years, when the interest rate is 7%, is presently worth $712.99 ($1,000 X (1/(1.07)5). Stated differently, if $712.99 is invested today, it will grow to $1,000 in 5 years. Present Value Tables are available in the appendix. Use the table to find the present value of $50,000 to be received in 8 years at 8%; it is $27,013.50 ($50,000 X .54027).
Annuities
Streams of level (i.e., the same amount each period) payments occurring on regular intervals are termed "annuities." For example, if you were to invest $1 at the beginning of each year at 5% per annum, after 5 years you would have $5.80. This amount can be painstakingly calculated by summing the future value amount associated with each individual payment, as shown at below.
But, it is much easier to use to an Annuity Future Value Table. The annuity table is simply the summation of individual factors. You will find the "5.80191 " factor in the 5% column, 5 year row. These calculations are useful in financial planning. For example, you may wish to have a target amount accumulated by a certain age, such as with a retirement contribution account. These tables will help you calculate the amount you need to set aside each period to reach your goal. See the book Appendix for this table.
Conversely, you may be interested in an Annuity Present Value Table. This table (which is simply the summation of amounts from the lump sum present value table - with occasional rounding) shows factors that can be used to calculate the present worth of a level stream of payments to be received at the end of each period. This table is found in the Appendix to the book. Can you use the table to find the present value of $1,000 to be received at the end of each year for 5 years, if the interest rate is 8% per year, is $3,992.71? Look at the 5 year row, 8% column and you will see the 3.99271 factor.
Returning to the Original Question
How do you compute the payment on a typical loan that involves even periodic payments, with the final payment extinguishing the remaining balance due? The answer to this question is found in the present value of annuity calculations. Remember that an annuity involves a stream of level payments, just like many loans. Now, think of the payments on a loan as a series of level payments that covers both the principal and interest. The present value of those payments is the amount you borrowed, in essence removing ("discounting") out the interest component. This may still be a bit abstract, and can be further clarified with some equations. You know the following to be true for an annuity:
Present Value of Annuity = Payments X Annuity Present Value Factor
A loan that is paid off with a series of equal payments is also an annuity, therefore:
Loan Amount = Payments X Annuity Present Value Factor
Thus, to determine the annual payment to satisfy a $100,000, 5-year loan at 6% per annum:
$100,000 = Payment X 4.21236 (from table) Payment = $100,000/4.21236
Payment = $23,739.64
You can safely conclude that 5 payments of $23,739.64 will exactly pay off the $100,000 loan and all interest. Simply stated, the payments on a loan are just the loan amount divided by the appropriate present value factor. To fully and finally prove this point, let's look at a typical loan amortization table. This table will show how each payment goes to pay the accumulated interest for the period, and reduce the principal, such that the final payment will pay the remaining interest and principal. You should study this table carefully:
The journal entries associated with the above loan would flow as follows:
1-1-x1 |
Cash |
Pennies 100,000 |
||
Note Payable |
100,000.00 |
|||
To record note payable |
||||
12-31-x1 |
Interest Expense |
6,000.00 |
||
Note Payable |
17,739.64 |
|||
Cash |
23,739.64 |
|||
To record Interest payment |
||||
12-31-x2 |
Interest Expense |
4,935.62 |
||
Note Payable |
18,804.02 |
|||
Cash |
23,739.64 |
|||
To record Interest payment |
||||
12-31-x3 |
Interest Expense |
3,807.38 |
||
Note Payable |
19,932.26 |
|||
Cash |
23,739.64 |
|||
To record Interest payment |
||||
12-31-x4 |
Interest Expense |
2,611.44 |
||
Note Payable |
21,128.20 |
|||
Cash |
23,739.64 |
|||
To record Interest payment |
||||
12-31-x5 |
Interest Expense |
1,343.75 |
||
Note Payable |
22,395.89 |
|||
Cash |
23,739.64 |
|||
To record Interest payment |
A Few Final Comments on Future and Present Value
• Be very careful in performing annuity related calculations, as some scenarios may involve payments at the beginning of each period (as with the future value illustration above, and the accompanying future value tables), while other scenarios will entail end-of-period payments (as with the note illustration, and the accompanying present value table). In later chapters of this book, you will be exposed to additional future and present value tables and calculations for alternatively timed payment streams (e.g., present value of an annuity with payments at the beginning of each period).
• Payments may occur on other than an annual basis. For example, a $10,000, 8% per annum loan, may involve quarterly payments over two years. The quarterly payment would be $1,365.10 ($10,000/7.32548). The 7.32548 present value factor is reflective of 8 periods (four quarters per year for two years) and 2% interest per period (8% per annum divided by four quarters per year). This type of modification does not only pertain to annuities, but also to lump sums. For example, the present value of $1 invested for five years at 10% compounded semiannually can be determined by referring to the 5% column, ten-period row.
• Numerous calculators include future and present value functions. If you have such a machine, you should become familiar with the specifics of its operation. Likewise, spreadsheet software normally includes embedded functions to help with fundamental present value, future value, and payment calculations. Following is a screen shot of one such routine: