22 Internal Rate of Return and Net Present Value
22.1 Internal Rate of Return
Suppose that a transaction has net cashflows of amounts \(C_0, C_1,.., C_n\), at times \(t_0,t_1,.,t_n\). The internal rate of return for the transaction is any rate of interest satisfying the equation \[\sum_{k=0}^n C_kv^{t_k} = 0\]
Example: Susan deposits \(\$8,000\) into an investment account. After 4 years, she withdraws \(\$5,000\), and at the end of 8 years she withdraws the remaining balance of \(\$5,414.21\). Calculate the annual effective internal rate of return.
Solution: The internal rate of return is the rate that results in the present value of the cash outflow being equal to the present value of the cash inflows: \[\begin{align}8,000 &= 5,000v^4 +5,414.21v^8 \\ \implies v^4 &=0.83856 \implies i = 4.50\%\end{align}\]
If there are more than 2 cash flows, it can be difficult ot solve for the IRR exactly, but we can often use the BA II to find the IRR.
Example: An industrialist is considering the purchase of a new machine for \(\$50,000\). If purchased, the machine can be used to convert \(\$1,000\) of raw materials at the beginning of each year into widgets than can be sold for \(\$9,000\) at the end of each year. The machine will last for 10 years, and it is worthless thereafter.
Calculate the annual effective internal rate of return resulting from the purchase of the machine and operating it for 10 years.
Solution: The net cash flows for times 0 through 10 are:
- \(CF_0 = -50,000 - 1,000 = -51,000\)
- \(CF_1 = CF_2 = ... = CF_9 = -1,000 + 9,000 = 8,000\)
- \(CF_{10} = 9,000\)
With the BA II Cash Flow Worksheet, we can enter the cashflow with the following:
[CF] CFo = -51,000 [ENTER] ↓C01 = 8,000 [ENTER] ↓ F01 = 9 [ENTER] ↓C02 = 9,000 [ENTER] ↓ F02 = 1[IRR] [CPT] -> IRR = 9.3361
Thus the internal rate of return is \(9.3361\%\)
22.2 Net Present Value
The net present value of a project is the present value of a project’s cash flows, discounted at the required rate of return:
\[NPV = \sum_{k=0}^n C_kv^{t_k}\]
Example: An investor pays \(100,000\) for a 5-year investment that produces cash flows of \(70,000\) at the end of 3 years and at the end of 4 years. The cash flows are reinvested at an annual effective rate of interest of \(3\%\).
Using an annual effective interest rate of \(7\%\), calculate the net present value of the 5-year investment.
Solution: With the BA II, enter the following cashflows:
CF0 = -100,000C01 = 0 F01 = 4CO4 = 70,000 * 1.03 + 70,000 * (1.03)^2[NPV] I = 7 ↓ [CPT] -> NPV = 4,354.7962