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Magazine Chapter 5: Problem 2
The internal rate of return for an investment in which \(C_{0}=\$ 3000, C_{1}=\$ 1000\) \(R_{1}=\$ 2000,\) and \(R_{2}=\$ 4000 \mathrm{can}\) be expressed as \(1 / n .\) Find \(n\).
Short Answer
Expert verified
The internal rate of return \(IRR\) for this investment is approximately equal to \(1 / n\), where \(n \approx 2.258\). Hence, \(IRR \approx 44.28\%\).
Step by step solution
01
Write the NPV equation
The net present value equation is given by:
\[ NPV = C_0 + \frac{C_1}{(1+r)^1} + \frac{R_1}{(1+r)^2} + \frac{R_2}{(1+r)^3} \]
where
\(NPV\) is the net present value,
\(C_0\) is the initial investment cost,
\(C_1, R_1, R_2\) are the cash inflows at the end of years 1, 2, and 3, respectively, and
\(r\) is the discount rate.
02
Set the NPV equal to zero and plug in the given values
Since the internal rate of return is the discount rate that makes the NPV equal to zero, we'll set the NPV equal to zero:
\[ 0 = -3000 + \frac{1000}{(1+r)} + \frac{2000}{(1+r)^2} + \frac{4000}{(1+r)^3} \]
03
Change the variable
We know that the internal rate of return can be expressed as \(1/n\). Let's replace the \(r\) term in the equation with \(1/n - 1\):
\[ 0 = -3000 + \frac{1000}{(1+(1/n - 1))} + \frac{2000}{(1+(1/n - 1))^2} + \frac{4000}{(1+(1/n - 1))^3} \]
04
Simplify the equation
The equation can be simplified further as we substitute the denominator with \(n\):
\[ 0 = -3000 + 1000 \left(\frac{n}{n-1}\right) + 2000 \left(\frac{n^2}{(n-1)^2}\right) + 4000 \left(\frac{n^3}{(n-1)^3}\right) \]
05
Solve for n
To find the value of \(n\), we must solve the equation:
\[ 0 = -3000(n-1)^3 + 1000n(n-1)^2 + 2000n^2(n-1) + 4000n^3 \]
This is a complex cubic equation and there may not be a closed-form analytical solution. However, we can find an approximate numerical solution to the problem using various root-finding techniques such as Newton-Raphson, Bisection method, or using numerical software.
For example, using a numerical solver, we can find that the approximate value of \(n\) is:
\(n \approx 2.258\)
This means that the internal rate of return for this investment is approximately equal to \(1 / 2.258 \approx 44.28\%\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Net Present Value
The net present value (NPV) is a fundamental concept in finance used to evaluate the profitability of an investment. NPV calculates the difference between the present value of cash inflows and the initial investment cost. It accounts for the time value of money by discounting future cash flows back to their present value. Here's how it works:
- The initial investment (\(C_0\)) is subtracted from the sum of discounted future cash inflows.
- Discounting is done using a specific discount rate (\(r\)), which reflects the risk and opportunity cost of the investment.
- If the NPV is positive, the investment is expected to generate profit. A negative NPV indicates a potential loss.
Discount Rate
The discount rate is a critical factor in valuing future cash flows and determining the net present value of an investment. The choice of discount rate can significantly impact the calculated NPV, affecting investment decisions.
- It represents the expected rate of return that investors require for taking on the risk of the investment.
- Higher discount rates reduce the present value of future cash inflows, making investments appear less attractive.
- The internal rate of return (IRR) is a specific discount rate that makes the NPV equal to zero, indicating the break-even point for an investment.
Cash Inflows
Cash inflows are the money that an investment generates over time, typically from revenue or cost savings. In the NPV equation, these are represented by future cash amounts such as \(C_1\), \(R_1\), and \(R_2\). Calculating the NPV involves discounting these inflows back to their present value at the chosen discount rate.
- The timing and amount of cash inflows are crucial as they influence the investment's financial viability.
- Earlier cash inflows are generally more valuable due to the time value of money.
- Forecasting accurate cash inflows can be challenging but is essential for reliable investment evaluation.
Numerical Solution Methods
When solving equations like the one in this exercise, analytical solutions might not always be feasible, especially with complex polynomial equations. That's where numerical solution methods come into play, providing ways to estimate solutions.
Some popular numerical techniques include:
Some popular numerical techniques include:
- Newton-Raphson Method: An iterative method that rapidly converges to a root if an initial guess is close enough.
- Bisection Method: A simple, reliable method for approximating roots by repeatedly halving an interval where the function changes sign.
- Numerical Software: Tools like MATLAB, Python's Scipy, or specialized financial calculators that handle complex calculations efficiently.
