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Investment - Present Value and Future Value
Two basic time value of money challenges are finding the value of a collection of cash flows now (present value) and the value as of a point in time in the future (future value).
Question
If you are offered GBP1 today or GBP1 in a year’s time, which would you choose?
Most individuals say GBP1 today because it provides them the choice of whether to spend or invest the money today and avoid the risk of never getting it at all. However, the GBP1 to be received in the future is worth less than GBP1 received now. The GBP1 to be received in the future is today worth GBP1 minus the opportunity cost and the risk of living without it for one year. The present value is derived by discounting the future cash flow by the interest rate. The rate of interest in this context can be called the discount rate.
Time influences the value of money because delay causes opportunity costs and risk. If you obtain a return of r% for waiting one year, GBP1 × (1 + r%) is the future value after one year of GBP1 invested now. Put another way, GBP1 is the present value of GBP1 × (1 + r%) received in a year’s time.
A saver may want to know how much money is needed today to create a certain quantity in the future given the rate of interest, r.
With compound interest, today’s worth is GBP100 and the interest rate is 10%, hence the projected value after two years is £100 × (1 + 0.10)2 = £121. The present value – the comparable value today — of GBP121 in two years, given that the annual interest rate is 10%, is GBP100.
Two basic time value of money challenges are finding the value of a collection of cash flows now (present value) and the value as of a point in time in the future (future value).
Question
If you are offered GBP1 today or GBP1 in a year’s time, which would you choose?
Most individuals say GBP1 today because it provides them the choice of whether to spend or invest the money today and avoid the risk of never getting it at all. However, the GBP1 to be received in the future is worth less than GBP1 received now. The GBP1 to be received in the future is today worth GBP1 minus the opportunity cost and the risk of living without it for one year. The present value is derived by discounting the future cash flow by the interest rate. The rate of interest in this context can be called the discount rate.
Time influences the value of money because delay causes opportunity costs and risk. If you obtain a return of r% for waiting one year, GBP1 × (1 + r%) is the future value after one year of GBP1 invested now. Put another way, GBP1 is the present value of GBP1 × (1 + r%) received in a year’s time.
A saver may want to know how much money is needed today to create a certain quantity in the future given the rate of interest, r.
With compound interest, today’s worth is GBP100 and the interest rate is 10%, hence the projected value after two years is £100 × (1 + 0.10)2 = £121. The present value – the comparable value today — of GBP121 in two years, given that the annual interest rate is 10%, is GBP100.
Before you can determine present or future values, you must know the relevant interest or discount rates to employ as well as the number of time periods under consideration. The rate will usually depend on the overall level of interest rates in the economy, the opportunity cost, and the riskiness of the assets under consideration.
Note that the interest and discount rates are the same percentage rates, but the language varies according on context. Calculating present values helps investors and analysts to translate cash flows of different amounts and at different points in the future into sums in the present that can be compared with each other. Likewise, the cash flows can be translated into the values they would be comparable to at a common future point.
Example: Comparing Investments
You are choosing between two investments of equal risk. You believe that given the risk, the proper discount rate to apply is 9%. Your starting investment (outflow) for each is GBP500. One investment is predicted to pay out GBP1,000 three years from now; the second investment is expected to pay out GBP1,350 five years from now. To choose between the two investments, you must compare the value of each investment at the same moment in time.
Note that the interest and discount rates are the same percentage rates, but the language varies according on context. Calculating present values helps investors and analysts to translate cash flows of different amounts and at different points in the future into sums in the present that can be compared with each other. Likewise, the cash flows can be translated into the values they would be comparable to at a common future point.
Example: Comparing Investments
You are choosing between two investments of equal risk. You believe that given the risk, the proper discount rate to apply is 9%. Your starting investment (outflow) for each is GBP500. One investment is predicted to pay out GBP1,000 three years from now; the second investment is expected to pay out GBP1,350 five years from now. To choose between the two investments, you must compare the value of each investment at the same moment in time.
As you can see, the investment with a payoff of GBP1,350 five years from now is worth more in present value terms, hence it is the better investment.
You are deciding between the same two assets, but you have reassessed their dangers. You now regard the five-year investment to be riskier than the first and estimate that a 15% return is required to justify making this investment.
Present value of GBP1,350 in five years discounted at 15% =
You are deciding between the same two assets, but you have reassessed their dangers. You now regard the five-year investment to be riskier than the first and estimate that a 15% return is required to justify making this investment.
Present value of GBP1,350 in five years discounted at 15% =
The investment paying GBP1,000 in three years (discounted at 9%) is, in this situation, preferable than the investment paying GBP1,350 in five years (discounted at 15%) in present value terms. Its present value of GBP772.18 is larger than the present value of GBP671.19 on the five-year investment.
The example illustrated the following three elements that must be addressed while comparing investments:
The cash flows each investment will generate in the future
The timing of these financial flows
The risk involved with each investment, which is reflected in the discount rate
Present value evaluates the joint influence of these three components and gives an effective means of evaluating investments with varying risks that have different future cash flows at different points in time.
Net Present Value
Present value is useful for comparing investments when the beginning outlay for each investment is the same, as was the case in the previous example. But investments may not have the same initial cash outflow, and withdrawals may occur at times other than time zero (the time of the initial outflow). The net present value (NPV) of an investment is the present value of future cash flows or returns minus the present value of the cost of the investment (which typically, but not always, occurs solely in the initial period). Using NPV rather than present value to evaluate investments is especially relevant when the investments have differing initial costs. The example below shows this concept.
Example: Comparing Investments Using Net Present Value
The NPV of the investment in the above example that is paying GBP1,350 in five years (discounted at 15%) if it initially cost GBP500 is
£671.19 – £500.00 = £171.19
The NPV of the investment paying GBP1,000 in three years discounted at 9% if it initially cost GBP700 is
£772.18 – £700 = £72.18
This amount is less than GBP171.19, making the investment paying GBP1,350 in five years discounted at 15% worth more in present value terms.
If expenses were to occur at times distinct from time zero, then they would likewise be discounted back to time zero for the purposes of comparison and computation of the NPV. If the NPV is zero or larger, the investment is yielding at least the discount rate. An NPV of less than zero implies that the investment is generating less than the discount rate, hence should not be made.
Application of the Time Value of Money
The time worth of money notion can help address many typical financial challenges. If you save in a deposit account, it can inform you by how much your money will grow over a specific number of years. Time value of money problems can involve both positive cash flows (inflows or saves) and negative cash flows (outflows or withdrawals).
The example illustrated the following three elements that must be addressed while comparing investments:
The cash flows each investment will generate in the future
The timing of these financial flows
The risk involved with each investment, which is reflected in the discount rate
Present value evaluates the joint influence of these three components and gives an effective means of evaluating investments with varying risks that have different future cash flows at different points in time.
Net Present Value
Present value is useful for comparing investments when the beginning outlay for each investment is the same, as was the case in the previous example. But investments may not have the same initial cash outflow, and withdrawals may occur at times other than time zero (the time of the initial outflow). The net present value (NPV) of an investment is the present value of future cash flows or returns minus the present value of the cost of the investment (which typically, but not always, occurs solely in the initial period). Using NPV rather than present value to evaluate investments is especially relevant when the investments have differing initial costs. The example below shows this concept.
Example: Comparing Investments Using Net Present Value
The NPV of the investment in the above example that is paying GBP1,350 in five years (discounted at 15%) if it initially cost GBP500 is
£671.19 – £500.00 = £171.19
The NPV of the investment paying GBP1,000 in three years discounted at 9% if it initially cost GBP700 is
£772.18 – £700 = £72.18
This amount is less than GBP171.19, making the investment paying GBP1,350 in five years discounted at 15% worth more in present value terms.
If expenses were to occur at times distinct from time zero, then they would likewise be discounted back to time zero for the purposes of comparison and computation of the NPV. If the NPV is zero or larger, the investment is yielding at least the discount rate. An NPV of less than zero implies that the investment is generating less than the discount rate, hence should not be made.
Application of the Time Value of Money
The time worth of money notion can help address many typical financial challenges. If you save in a deposit account, it can inform you by how much your money will grow over a specific number of years. Time value of money problems can involve both positive cash flows (inflows or saves) and negative cash flows (outflows or withdrawals).
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