Introduction
In the previous post, we covered the time value of money (TVM) concept. In this post, we will look at how the TVM concept can be used to make financial decisions.
Discount Rate
When we discount a future sum or cash flow stream, we must choose a rate which compensates us for the riskiness of the future cash flows. This rate is known as the discount rate.
The least risky assets are short-term government securities, such as U.S. Treasury Bills. The interest rate on these securities is known as the risk-free rate. Because the risk-free rate is the rate earned on the least-risky securities, it follows that the rate on all other assets must be at a premium to this rate.
Assets can vary in riskiness; therefore, discount rates will vary. However, we should note two things. First, discounting as a financial decision tool only makes sense when we are confident in the cash flow forecast. When future cash flows are highly uncertain, other valuation methods might be better. Second, it makes sense to generate several different cash flow forecasts and attach probabilities to each of them. You can then either calculate the present value of each forecast or the present value of a single-probability-weighted forecast. The point is that the discount rate should not be the only tool for managing financial uncertainty.
The discount rate can be thought of as having two sides. On one side, the discount rate represents the capital provider’s minimum required return. On the other side, the discount rate represents the company’s cost of capital.
The discount rate should represent a weighted average of the debt and equity capital the firm uses to finance its operations. When the discount rate accounts for both forms of capital, it is referred to as the weighted average cost of capital (WACC). We will discuss the WACC further in a subsequent post.
Net Present Value
The net present value (NPV) is a tool which allows an investor or firm to estimate the amount of economic value an investment creates.
To calculate an investment’s NPV, we calculate the present value of an asset’s future cash flows and subtract from that value the asset’s cost:
NPV = PV – Cost
When the NPV is positive, the investment creates economic value. When the NPV is negative, the investment destroys economic value. Thus, the decision rule for using NPV is:
Accept an investment when NPV > Cost; Reject an investment when NPV < Cost
Because the NPV brings all value to the present, it allows decision-makers to compare investment alternatives.
Let’s look at two examples of the NPV concept.
Example:
Joe is looking to purchase a 50% interest in a plumbing supply business. He believes that the company’s cash flows can grow by 10% per year for three years at which time cash flows will grow by 2% in perpetuity. Currently, the company generates $200,000 per year in discretionary cash flow. Joe believes 16% is an appropriate return. Is this a good investment if it is being offered for $500,000?
Solution:
To see if this investment generates positive economic value, we must subtract the $500,000 price from the present value of the cash flows.
The present value of the cash flows is the sum of the present values of the cash flows in the first three years and the present value of the perpetuity in year 4.
First, we calculate the present value of the cash flows in the first three years with the following steps:
- Calculate the cash flows for each of the years by multiplying each subsequent year’s cash flow by the growth rate: 110,000, 121,000, 133,100
- Calculate the discount factor for each cash flow
- Sum 1 and the discount rate
- Raise the result to the corresponding year in which the cash flow occurs
- Divide 1 by the result of the above step: 0.86, 0.74, 0.64
- Multiply each cash flow by the corresponding discount factor: 94,600, 89,540, 85,184
- Sum the results: 269,324
Next, we calculate the present value of the perpetuity value in year 4 and beyond
- Calculate the cash flow in the terminal year by multiplying the last cash flow in the high-growth period by 1 plus the steady growth rate: (133,100 x 1.02) = 135,762
- Divide the result by the difference of the discount rate and the steady growth rate: 969,729
- Divide the result of the above step by the sum of 1 and the discount rate, all raised to the year corresponding to the steady growth phase: 535,762
Now, we sum the present value of the high growth period and the present value of the perpetuity:
269,324 + 535,762 = 805,086
Subtract the cost of the investment from the investment’s present value:
805,086 – 500,000 = 305,086
This investment generates $305,086 in positive economic value. Therefore, Joe should accept the investment.
Example:
A restaurant sees that a fully outfitted food truck is for sale for $100,000. The owners of the restaurant estimate that operating the food truck will add $25,000 in net cash flow to the company. They also estimate that the truck’s useful life is 10 years. The company’s cost of capital is 14%. Should the restaurant purchase the food truck?
Solution:
To calculate the NPV of this investment we subtract the price of the truck from the present value of the ten $25,000 annual cash flows. Using the formula for the present value of an ordinary annuity factor:
25,000 x 5.21 = 130,250
Subtracting the cost of the investment from the value of the investment:
130,250 – 100,000 = 30,250
The investment creates $30,250 in economic value.
Internal Rate of Return
Another decision-making tool which is often used in real estate and private equity is the internal rate of return (IRR). The IRR is the rate which equates the present value of the cash flows with the investment’s cost. In other words, it is the rate which produces an NPV of zero.
Because of the iterations needed to calculate the IRR, it is best to use a financial calculator or computer spreadsheet to aid in the calculations.
The decision rule for using the IRR is:
Accept an investment when IRR > cost of capital; Reject an investment when IRR < cost of capital
Let’s return to the food truck example. Plugging the cash flows from this project into a financial calculator, we get an IRR of 21.41%. Because the company’s cost of capital is 14%, the company should pursue the project.
Modifying the IRR
There are two criticisms of the IRR. First, when an investment’s cash flows oscillate between positive and negative values, the IRR calculations can yield multiple values.
The second problem is that the IRR assumes the cash flows will be reinvested at the IRR, which in many cases will be highly unrealistic.
In the food truck example, we calculated a project IRR of 21.41%. This calculation assumes that the $25,000 in cash flow per year will be reinvested at this rate, equaling a future value of roughly $696,000 at the end of year 10. To achieve this, the owners would have to find investments generating compound returns at the IRR for each cash flow. The probability of doing just that is extremely low. In this sense, the IRR is a fiction.
In other words, the IRR assumes that the $25,000 per year for 10 years is the equivalent of $696,000 received at the end of year 10. In both cases, the IRR is the same.
There is a method of calculating the IRR which allows us to deal with these two issues. This method is known as the modified internal rate of return (MIRR). The MIRR assumes reinvestment at the cost of capital or some other reasonable rate.
Once we have determined the rate we want to use, we can calculate the MIRR using the following steps:
- Calculate the present value of any negative cash flows, including the initial outlay (which occurs at time 0 and is already discounted)
- Calculate the future value of the positive cash flows
- Find the rate which equate the present and future values
Returning to the food truck example, suppose the owners want to know the MIRR if the cash flows compound at the company’s cost of capital of 14%?
Because there are no negative cash flows after the initial cost, the PV of the negative cash flows is the $100,000 initial investment.
The future value of ten $25,000 payments at 14% is $483,432. The rate which equates this value with the $100,000 initial investment is 17.07%.
Because the MIRR is greater than the company’s hurdle rate of 14%, the investment is worth pursuing.
Conclusion
The net present value and internal rate of return concepts are important tools for anyone making investment decisions. These tools can be used in conjunction with other metrics, but by measuring the investment against the investor’s or firm’s required return, these measures show the economic value which the investment is expected to create. These measures are also useful in helping to compare competing investments, which allows investors and firms to find the best use of their capital.
