# Net Present Value

### Overview of Future Value of Money

The future value of current money, a factor in deciding whether to invest in a new capital project, can be calculated using a formula; the rate of return will be linked to the risks associated with the endeavor.
According to the concept of time value of money, money that is held now is worth more than the same amount of money in the future. For an investment to be worthwhile, the future value of the money must be greater than the present value. The investment will earn returns, generally calculated as interest. The future value of money can be calculated using a formula. It’s important to note that time value of money formulas can be presented in a number of ways. For example, in some formulas n x t will be simplified into just n for total number of periods rather than number of periods per year times number of years.
${\rm{FV}}={\rm{PV}}\;\times\;\left[1+\left(\frac{i}{n}\right)\right]^{(n\times t)}$
Where:
\begin{aligned}\rm{FV}&=\text{Future Value}\\\rm {PV}&=\text{Present Value}\\i&=\text{Rate of Interest}\\n&=\text{Number of Periods Per Year}\\t&=\text{Number of Years}\end{aligned}
Therefore, $1,000 today will have a value of$3,207.14 in 10 years at 12 percent annual interest compounded semiannually.
\begin{aligned}\rm{FV}&=\1{,}000\times\left[1+\left(\frac{0.12}2\right)\right]^{(2\times10)}\\&=\3{,}207.14\end{aligned}
When a business makes decisions about investing in new projects or expanded product lines, analysis of present and future values is beneficial. The interest rate or rate of return will be linked to the risks associated with the endeavor. For investors, there are two types of risk: market and business risk. Market risk is the systematic danger of loss because of the movement of greater economic influences. Business risk is the possibility that the venture will not generate the expected profits. For businesses using the principles for capital budgeting, the latter type is often referred to as a standalone risk, or the danger of loss from a single capital structure, division, or unit decision. Investors and businesses will make decisions about the risk and return trade-off for the venture using the interest rate as the defining factor. Risk and return trade-off is the capital structure profile for the balance between the danger of loss and the income. Capital budgeting decisions compare money earned in the future to money spent today, so the future value of money is discounted to the present value in order to compare the two appropriately.

### Net Present Value Calculation

The net present value of an investment can be calculated by analyzing the projected future cash inflows, rolled back to the present value, and cash outflows of the investment.
Defining the capital structure is an important step in the planning process. Capital structure is the manner in which a company funds the investments used to further its business goals, using debt, stocks, and retained earnings (net profits minus dividend payments). When a business is trying to create the optimal capital structure through capital budgeting, it will analyze the projected cash inflow, rolled back to the present value, and cash outflow of the investment. This analysis will give the business the net present value (NPV) of the project.
${\rm{NPV}}=\sum_{t=1}^T\frac{C_t}{(1+r)^t}-C_i$
Where:
\begin{aligned}t&=\text{Number of Periods of Cash Flow}\\T&=\text{Final Time Period of the Investment}\\C_t&=\text{Cash Inflow}\\C_i&=\text{Cash Outflow (or the Initial Investment})\\r&=\text{Interest Rate}\end{aligned}
If Vision Inc. is planning to buy a piece of machinery for $100,000 that allows the company to generate$75,000 per year for three years, the net present value of the investment at 7 percent can be calculated.
\begin{aligned}{\rm{NPV}}&=\frac{\75{,}000}{(1+0.07)^1}+\frac{\75{,}000}{(1+0.07)^2}+\frac{\75{,}000}{(1+0.07)^3}-\100{,}000\\\\&=\96{,}823\end{aligned}
Each term is adjusted from the future value of money to the present value. Without this adjustment, it appears that the income from this investment is $125,000. The interest rate, also called the discount rate in this case, represents the required rate of return by the investor. It is equivalent to the interest rate that could be earned if the business invested in securities or other financial markets. Vision Inc. only has$100,000, and there is another machine that will generate 60,000 per year for four years with the same interest rate. The NPV for this project can be calculated. \begin{aligned}{\rm{NPV}}&=\frac{\60{,}000}{(1+0.07)^1}+\frac{\60{,}000}{(1+0.07)^2}+\frac{\60{,}000}{(1+0.07)^3}+\frac{\60{,}000}{(1+0.07)^4}-\100{,}000\\\\&=\103{,}233\end{aligned} NPV analysis shows that the second option has the higher present value of money, so Vision Inc. should opt for that choice. In general, investments that generate an NPV equal to or greater than 0 will be considered, and the second step is to identify the highest NPV. A further analysis of the competing investments involves applying an internal rate of return (IRR) analysis to compare all possible investments. Many companies use both an NPV and IRR approach to selecting investments. ### Net Present Value as Applied to Decision-Making Capital decisions are made based on net present value. Decisions using net present value require money invested and revenue earned. The investment for a capital budget decision generally comes from operating income or retained earnings in the form of cash. The net present value decision affects the degree of combined leverage (DCL); it shows the effect of a change in sales on the earnings in the form of earnings per share (EPS), or the profit allocated to a single share of common stock. The degree of combined leverage (DCL) is the ratio between a measure of a company's fixed and variable costs and is a measure of a company's use of borrowed funds. $\rm{DCL}=\frac{\text{Percent Change in EPS}}{\text{Percent Change in Sales}}$ It can also be written as the product of the degree of operating leverage (DOL) and the degree of financial leverage (DFL). The degree of operating leverage (DOL) is a measure of the change of a company's operating income with respect to change in sales. The degree of financial leverage (DFL) is a measure of the effect of capital structure changes on the operating income of a company. The DCL formula can also be expressed using DOL and DFL. $\rm{DCL}=\rm{DOL}\times\rm{DFL}$ In NPV decisions, the cash inflow represents a change in sales revenue. If current net sales are250,000, the percent change in sales is 24 percent.
$\frac{\60{,}000}{\250{,}000}=0.24\;\text{or}\;24\%$
If Vision Inc. reported an EPS of $3 and estimates an EPS of$3.25 after the purchase, the change in EPS would be 8 percent.
$\text{Change in EPS}=\frac{(\3.25-\3)}{\3}= 0.08\;\text{or}\;8\%$
If the degree of operating leverage is 1, then the DCL is 0.33. This means that for every dollar made by increasing sales, the earnings per share will increase by \$0.33.
${\rm{DCL}}=\frac{0.08}{0.24}=0.33$
The NPV decision then flows into decisions based on the increase in sales from the new investment. It is part of a larger capital structure plan that gives the optimum debt/equity mix, or the best possible blend of long-term liabilities and equity in the capital structure to produce optimal earnings.