7.4 Net Present Value
An important method to evaluate large investments in the public and private sector is called Net Present Value (NPV) analysis. Before presenting the method, we need to introduce compounding and discounting interest.
The concept of compounding interest illustrates how an initial deposit grows over time due to interest being calculated on both the initial principal and the accumulated interest. Consider an example with an initial deposit of $100 and a 5% interest rate compounded annually over three years. At the start of year 1, the initial balance is $100. With an interest rate of 5%, the interest earned during the first year is $5 (\(100 \cdot 0.05\)), resulting in an ending balance of $105. In year 2, interest is calculated on the new balance of $105, earning $5.25 (\(105 \cdot 0.05\)), bringing the ending balance to $110.25. By year 3, the balance of $110.25 earns $5.51 in interest (\(110.25 \cdot 0.05\)), resulting in a final balance of $115.76. The general formula for compounding interest can be written as follows: \[FV = PV \cdot (1+r)^t\] where \(r\) is the interest rate, \(t\) is the number of time periods (years), and \(PV\) and \(FV\) are present and future value, respectively.
Discounting interest is the opposite operation in which we determine the present value (PV) given a flow of money to be received in the future. The interest rate used to determine the present value is called the discount rate \(r\). This concept reflects the idea that money received in the future is worth less than money received today due to factors like inflation or opportunity cost (e.g., investments). For example, if you receive $100 every year for the next three years and the discount rate is 6%, the present value of each $100 payment decreases over time. In the first year, the present value of $100 is calculated as follows: \[\frac{100}{(1+0.06)}= 94.34\] In the second year, the present value of $100 is calculated as: \[\frac{100}{(1+0.06)^2}=89.00\] In the third year, it is \[\frac{100}{(1+0.06)^3}=83.96\] By discounting the future payments, the total present value of receiving $100 each year for three years is the sum of these discounted amounts, which is less than the nominal $300 but represents the equivalent value in today’s terms, i.e., $267.30. Put differently, if you put $267.30 in a bank account today at 6% interest rate then you can withdraw exactly $100 at the end of each of the following three years. The general formula is written as follows: \[PV_t=\frac{FV_t}{(1+r)^t}\] Consider the following setup for a Net Present Value calculation. Suppose you receive a cash flow of $100,000 in each of the next ten years at an interest rate of 6%. The NPV formula is generally written as follows: \[NPV=\sum_{t=1}^T \frac{X_t}{(1+r)^t}\] In the example given, this translates into the following equation: \[NPV=\sum_{t=1}^T \frac{100,000}{(1.06)^t}=736,008.71\] Next, consider the example of renovating a football field with two options for the turf. The first option is artificial turf with initial cost is $500,000 but does not require any maintenance for 10 years. The second option is natural turf, which requires spending $200,000 initially but annual maintenance is required. Those maintenance cost vary from year to year due to re-seeding and fertilization requirements. Assuming a discount rate of 5%, the cost, discount factors (DF), and present values (PV) are summarized for the natural turf below. The NPV of the natural turf is $506,986.05 making it more expensive than the artificial turf. Note though that the NPV is very sensitive to the discount rate chosen. For example, in the case of the football field, a discount rate of 6% makes the artificial turf the cheaper option.
| Year | Cost | DF | PV |
|---|---|---|---|
| 0 | 200 | 1.000000 | 200.00000 |
| 1 | 30 | 1.060000 | 28.30189 |
| 2 | 50 | 1.123600 | 44.49982 |
| 3 | 30 | 1.191016 | 25.18858 |
| 4 | 50 | 1.262477 | 39.60468 |
| 5 | 30 | 1.338226 | 22.41775 |
| 6 | 50 | 1.418519 | 35.24803 |
| 7 | 30 | 1.503630 | 19.95171 |
| 8 | 50 | 1.593848 | 31.37062 |
| 9 | 30 | 1.689479 | 17.75695 |
| 10 | 50 | 1.790848 | 27.91974 |
Due to the sensitivity of the NPV with regard to the discount rate, the internal rate of return (IRR) may calculated as an alternative. The IRR calculates the discount rate that would cause the present value of benefits (returns) to equal the present value of costs. In the case of the NPV, the IRR is calculated such that \(NPV=0\). In the case of the football field, the IRR calculation yields to a discount rate which makes the natural turf identical to the artificial turf. The IRR (numerically) calculates \(r\) such that:
\[C_0=\frac{N_1}{1+r}+\frac{N_2}{(1+r)^2}+\cdots+\frac{N_t}{(1+r)^t}\]
where \(C_0\) is the initial cost and \(N_t\) is the return in time period \(t\). Note that you need a computer to calculate the IRR. In Excel, it is the function IRR. In the case of the football field, the discount rate of 5.47% makes the two investment options equal.
To annualize capital cost, i.e., spreading the capital cost over the project life, we much calculate and use the annualization factor. \[AF=\frac{r}{(1+r)^n-1}+r\] The annualization factor allows to distribute an initial lump sum over \(n\) years of life. The amount per period is such that when discounted, you get the initial capital costs