# What is internal rate of return

The **internal rate of return** of an asset (or loan) is the interest rate(s) at which the present worth of the asset is equal to zero. It can be thought of as the break-even rate of return. It is the interest rate that would make you just indifferent between buying and not buying the asset.

# How do you calculate internal rate of return

To compute the IRR of an asset, first we want to set the present worth of the asset equal to zero, since we're looking for the "break-even" rate. In other words, we want to find the interest rate $i$ where the present worth of the asset is equal to zero. We are essentially solving the following equation for the variable $i$.

(1)where $A_0$ is the cashflow in period 0, $A_1$ is the cashflow in period 1, and so on for each of the $N$ cashflows. The variable $i$, once found, will be the internal rate of return.

## Easy case of a period zero cost and a period N benefit

One of the simplest scenarios is an asset (or loan) with a single cashflow in period zero and a single cash flow in one other period, period $N$. In this case, the expression consists of only two terms:

- $A_0$, the present worth of the cashflow in period zero.
- $A_N\left(1+i\right)^{-N}$, the present worth of the cashflow in period $N$.

For this simple case, we can actually solve for $i$ algebraically. For example,

(2)However, once you have more than two cash flows, solving algebraically quickly becomes a nightmare. When you're dealing with a more complicated series of cashflows, you'll want to use a numerical solver like the one on your calculator to solve for $i$. *Note: you want to use the nSolve function on your calculator, not the regular solve function. You may have to try different starting points to get all the roots if there is more than one.*

## Root finding

# Unconstrained choice criteria

If the asset is an investment, and the IRR is greater than or equal to the MARR, then buy it. However, the inverse is true when the asset is a loan. In other words, buy an asset on a loan when the MARR is greater than the IRR.

## Is it an investment or a loan?

The relationships between three criteria must be examined:

Present Worth (PW) tells you how your cash flow would change if you were to take on the project in question, relative to the no-action base level of zero cash flow. Qualitatively, a positive value means your cash flow would increase. Negative means decrease.

Minimum Acceptable Rate of Return (MARR) is the lowest rate at which you will offer up your money. It is based on complicated risk analysis and in this class it is always given if needed.

Internal rate of return (IRR) is the interest rate for which your asset would have a present worth of zero, i.e. the break-even rate, at which you are exactly indifferent between taking on the project and not. If you can invest in a project where your MARR is lower than the IRR, you will have done better than broken even, and you should do it. On the other hand, if you are looking at borrowing money, you want the IRR to be lower than the MARR (if your MARR is lower, you would be better off using your own retained earnings than borrowing).

Four possible PW/IRR/MARR combinations can arise:

1 | 2 | 3 | 4 | |

PW | + | + | - | - |

IRR vs MARR | IRR>MARR |
IRR<MARR |
IRR>MARR |
IRR<MARR |

Good Investment | Good Loan | Bad Loan | Bad Investment |

Analysis:

First, consider PW. If it's positive, buy it. If it's negative, don't. If it's zero you're indifferent, but in this class we say go for it.

Consider the assets you've decided to buy:

1. IRR>MARR: Money being used will bring you a greater return than money being sat on. Putting your money into this project has positive present worth to you, so it must be a good **investment**.

2. IRR<MARR: Money being sat on will bring you a greater return than money being used. Keep your money. Since this asset has positive present worth to you, it must be cheap money from outside, i.e. it must be a good **loan**.

Now, the assets you didn't buy:

3. IRR>MARR: If this were an investment, you would have chosen to take it because money being used would bring you a greater return than money being sat on. Yet you didn't take it. Why not? It must be a bad **loan**.

4. IRR<MARR: If this were a good loan, you would have taken it. Yet you didn't take this asset. Why not? It must be a bad **investment**.

# Exclusive choice procedure

- Order assets from lowest initial investment to highest.
- Eliminate all assets with IRR < MARR.
- Declare the least expensive (lowest initial cost, the absolute value of $A_0$) asset as the candidate.
- Of the assets which remain, choose the next-lowest-investment asset as the challenger.
- Find the incremental IRR of the candidate and the challenger.
- If incremental IRR > MARR, the challenger is a better choice. Eliminate the candidate, set the challenger as candidate, and repeat from step 4.
- If incremental IRR < MARR, the candidate is a better choice. Eliminate the challenger and repeat from step 4.

- Once all challengers are exhausted, keep the candidate as the best asset.

You can also find a lot of worked examples of the procedure in answers to essential question three. Focus on the last question. Another resource are the video answers to question three in youtube.