Present value Essay

The concept of present value is one of the key concepts in financial decision-making, since it takes into account the opportunity costs associated with future payments; in other words, “a dollar today is better than a dollar tomorrow”. Use of the notions of present and future value allows to compare cash flows on a rational basis. Moreover, this idea is widely used in accounting: according to FASB standards, the concept of present value is used to identify the fair value of liabilities and assets when it is not possible to determine the fair market price (Khan, 2004). Valuation of assets and calculation of the real value of deferred payments is also done using the idea of present value (Baker & Powell, 2005). Using the time value of money, economics can take into account such issues as risk, interest rate and inflation in the process of evaluating investments. The idea of the present value is the background for all long-term financial decisions, and this is why the concept of present value is one of the first topics taught in finance courses.
1. Calculation of future values of investments
Since it is not specified in the formulation of the task whether the interest is simple or compound, it is supposed that the investments are performed using the compound interest approach. It is possible to calculate the future value of this investment in two ways: using step-by-step calculation and applying formulae for the future value directly. The calculations in this paper are performed in detail and the exact mathematical formulae for the future values are given for each investment.
a. $14,000 if invested for five years at a 9% interest rate
Step-by-step approach: each year the sum will be increased by 9%, i.e. the value of the investment a year later will be equal to value of the investment at the previous year the multiplied by the factor of 1.09. Table 1 shows factors and future values for each year.

Year
Factor
Value

0
1.0000
$14,000.00

1
1.0900
$15,260.00

2
1.1881
$16,633.40

3
1.2950
$18,130.41

4
1.4116
$19,762.14

5
1.5386
$21,540.74

Table 1. Future values of the investment
Formula for future value of the investment with compound interest: where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods (Needles & Powers & Crosson, 2010). In our case, PV = $14,000; i = 0.09 (9%), and N = 5. Thus, . The values obtained by two methods are equal, which proves the correctness of the calculations. For further similar calculations, only formulas will be applied.
b. $29,500 if invested for three years at a 5% interest rate
Formula for future value of the investment with compound interest: where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. In our case, PV = $29,500; i = 0.05 (5%), and N = 3. Thus, .
c. $29,900 if invested for seven years at an 3% interest rate
Formula for future value of the investment with compound interest: where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. In our case, PV = $29,900; i = 0.03 (3%), and N = 7. Thus, .
d. $18,500 if invested for ten years with a 1.9% interest rate
Formula for future value of the investment with compound interest: where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. In our case, PV = $18,500; i = 0.019 (1.9%), and N = 10. Thus, .
2. Calculation of present values of investments
a. $27,500 to be received three years from now with a 1% interest rate
Present value is a concept reverse to the future value, and can be calculated using the following formula: , where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods (Parrino & Kidwell, 2009). It is again suggested that compound interest is being used (as it is the most common for investment tasks and bank deposits). In our case, FV = $27,500, N = 3, i = 0.01 (1%). Thus, It is also possible to determine the present value of future payment using step-by-step approach, it is similar to Table 1-4, but the calculations will descend from the future value (the last year) to the current year. Table 5 shows the example of such calculation for current investment. For year 2, the present value of the investment is calculated as the value of the initial investment in year 3 ($27,500) divided by the factor of interest rate (1.01), or multiplied by 0.9901. The relations are similar for the next years. As a result, present value of the investment is $26,691.23, which is equal to the result obtained using the previous method. This proves the correctness of the calculations.
Year
Factor
Value

0
0.9706
$26,691.23

1
0.9803
$26,958.14

2
0.9901
$27,227.72

3
1.0000
$27,500.00

Table 5. Present values of the investment
b. $43,000 to be received five years from now with a 7% interest rate
The formula for the present value is: , where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. It is again suggested that compound interest is being used (as it is the most common for investment tasks and bank deposits). In our case, FV = $43,000, N = 5, i = 0.07 (7%). Thus,
c. $125,000 to received two years from now with a 11% interest rate
The formula for the present value is: , where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. It is again suggested that compound interest is being used (as it is the most common for investment tasks and bank deposits). In our case, FV = $125,000, N = 2, i = 0.11 (11%). Thus,
d. $550,000 to be received eight years from now with a 2% interest rate.
The formula for the present value is: , where FV – future value of the investment, PV – present value of the investment, i – interest rate, and N – number of periods. It is again suggested that compound interest is being used (as it is the most common for investment tasks and bank deposits). In our case, FV = $550,000, N = 8, i = 0.02 (2%). Thus,
3. Annuity problem
Suppose you are to receive a stream of annual payments (also called an “annuity”) of $200,000 every year for three years starting this year. The interest rate is 5%. What is the present value of these three payments?
For annuity, it is also possible to use both step-by-step approach and calculations using predetermined formulas. In this paper, both approaches will be implemented. For a step-by-step approach, it is necessary to determine the present value of each annual cash flow. It is supposed that this is an ordinary annuity, e.g. that payments are done in the end of the year (e.g. the interest rates will be the same as in the beginning of the next year). For year 0, present value of the annuity will be equal to: For year 1, present value of the annuity will be equal to: For year 2, present value of the annuity will be equal to: The summary present value of these three annuity payments is:
The present value of an ordinary annuity can be determined using the following formula: , where PMT is the value of annual payment, i – interest rate, and N – number of periods (Vance, 2002). In our case, PMT = $200,000; i = 0.05 (5%), and N=3. Thus, Present values of the annuity received by both methods coincide, and this fact proves that the calculations are correct.
4. Bank deposit problem
Suppose you are to receive a payment of $220,000 every year for three years. You are depositing these payments in a bank account that pays 3% interest. Given these three payments and this interest rate, how much will be in your bank account in three years?
The question of the task basically means that it is necessary to calculate future value of an annuity. This can be done using the formula for the future value of the annuity or by using step-by-step approach. Let us perform both types of calculations. Again, it is supposed that the annuity is ordinary (e.g. that the payments are received and deposited in the end of the year), and that compound interest is considered, since these are the most common business situations.
For step-by-step approach, it is necessary to calculate the future values of each annual payment. Future value of the payment done in the first year: .
Future value of the payment done in the second year: .
Future value of the payment done in the third year will be the same as the value of the investment (since the deposit will be done in the end of the 3-year period): .
Total future value of the annuity will be:
The formula for the future value of the annuity is the following: , where PMT is the value of the annual payment, i – interest rate, N – number of periods (Brechner, 2009). In this case, PMT = $220,000; N = 3, i = 0.03 (3%).
Both calculations yield the same result, which proves that they are correct.
Conclusion
There are various mathematical apparatus for calculating future and present value of annuities and perpetuities using simple and compound interest. The techniques applied in this paper illustrate the methods used for determining present and future values of different cash flows. The applications of the concept of the time value of money are diverse: from determining the real value of annual payments and deposit cash flow for the bank account to evaluation of investment opportunities and financial decision-making.