Finance — ISEN 302

The internal rate of return for a series of cash flows is defined as the discount rate with makes the net present value go to zero. Thus, a higher internal rate of return is desirable. Alternative definitions include: (1) the discount rate such that EUAW goes to zero (2) For a loan, the interest paid on balance such that the balance is zero after the final payment (3) For an investment, the interest earned on the unrecovered investment such that the unrecovered investment is zero after the last cash flow.

IRR may be found by number of techniques

• Utilize equations $[\bold{1-3}]$ repeatedly to find a $r$ such that $NPV=0$
• Utilize factors analogous to equations $[\bold{1-3}]$ repeatedly to find a $r$ such that $NPV=0$
  • Interpolation may be conducted using rate guess $R_1,R_2$ and associated net present values $P_1,P_2$, according to $R_I=R_1+(R_2-R_1)(P_1)/(P_1-P_2)$
• Utilize IRR(cashflow_amounts, [rate_guess]) to find the rate directly from a series of cash flows, starting at year 0. Sheets Docs: IRR

Incremental Analysis

When evaluating internal rate of return, we only want to select alternatives that exceed our minimum attractive rate of return (MARR). But decisions are made at the margins, so when comparing high cost alternative (A) against low cost alternative (B), we need to evaluate if the increment in investment from alternative (A) to (B) exceeds our MARR, not merely if the increment from doing nothing to (B) exceeds MARR.

Procedure Summary

In summary, we will need to (1) arrange alternatives in order of increasing cost (2) Consider each possible increase in investment. If the IRR of the increment in each cash flow exceeds MARR, then that increment in investment is desirable, and the more costly investment is selected for future comparisons (3) Proceed to make these comparisons until all alternatives have been evaluated and selection is made.

Procedure Algorithm

1. Arrange alternatives $(1:n)$ in increasing order of initial cost.
2. Evaluate $R_I$ of alternative $1$.
   1. If $R_I<MARR$, evaluate of alternatives $(2:n)$ from step 2.
   2. If $R_I\ge MARR$, proceed to step 3. Alternative $1$ is our current choice
3. For the next pair of alternatives, $i$ and $i+1$, made up of cash flows $C_{i1},...,C_{ij}$ and $C_{(i+1)1},...,C_{(i+1)j}$, evaluate $R_I$ of $(C_{(i+1)1}-C_{i1}\>,\>...\>,\>C_{(i+1)j}-C_{ij})$.
   1. If $R_I<MARR$, evaluate alternatives $(1,3:n)$ from step 3. Alternative $i$ is still our current choice
   2. If $R_I\ge MARR$, evaluate alternatives $(2:n)$ from step 3. Alternative $i+1$ is now our current choice
4. Select the current choice as the best alternative for our MARR.