Annual worth is the second analysis technique based on the concept of equivalencies you will learn, with present worth being the first [1]. Annual worth is a way of taking two assets which have drastically different conditions (lifespans, initial costs, and servicing costs) and allows us to determine which is a better choice. Because annual worth is directly based on the net present worth calculation, it is incredibly important to fully understand and have a great working knowledge of present worth to correctly answer questions regarding annual worth.

Annual worth will give the same decision result as present worth calculations, but is usually preferred in most cases as it is easier to grasp for most clientele. A great example of this idea is that the present worth for a car could be $250,000 over the average life span of ownership, while the annual worth would be reported as $5,500 [2]. In a case such as this, it is much easier to understand why annual worth can be a more user friendly reporting method. Annual worth calculations are helpful decision making tools in the real world as they allow [1]:

- an asset to be accurately added into an annual cash flow statement
- the unit costs calculation that can be used to determine reasonable pricing for items on sale
- make or buy decisions to be made
- minimum cost analysis to be easily computed

The calculation of the annual worth is determined by converting all the costs and/or benefits of the project into a series of constant annual benefits/costs for the life of the project [3]. A great example of this principle is the butter over bread model. To find the present worth we scrape all the butter on the bread to one edge (i.e. move all the costs/benefits into time period 0, present worth calculation) and then we smooth the butter out evenly over the bread (i.e. divide the costs/benefits evenly over the lifespan of the project) [2].

# How to calculate Annual Worth

- Calculate the present worth of the asset
- Multiply present worth by factor notation (A|P,i,N) where
*i*is the given MARR and*N*is the number of years you will be using that asset. This gives the annual worth.

AW=PW(A|P,i,N)

It is important to be aware of planning horizons when evaluating annual worth. Planning horizons give an idea of how long a particular asset, service investment, etc… will be used for. There are several places to get a planning horizon:

- Ask the people you are working for
- The planning horizon is specified in a contract
- The planning horizon is an infinite time period
- If evaluating several service investments with different service lives, choose the least common multiple between them as the length of the planning horizon

## Annual Worth Directly — A Special Case

There is a case that should be considered special. Normally, when you calculate annual worth you have to first calculate present worth and then transform it into annual worth. If you have a problem where the cash flow is constant after the current period, there is no need to make a calculation. Something that is worth $10 a year for 5 years has an annual worth of $10.

# The Criteria

## Annual Worth in Unconstrained Choice

- Choose all assets with an annual worth greater than zero.

## Annual Worth in Exclusive Choice

- If all investments have equal service lives, choose the asset with the largest annual worth.

# Annual Worth Examples

## Example 1

MARR = 10 %Year | A | B | C |
---|---|---|---|

0 | -10 | -20 | -30 |

1 | 8 | 0 | 0 |

2 | 0 | 0 | 0 |

3 | 0 | 15 | 0 |

4 | 0 | 0 | 25 |

A) With no requirement to provide service, which asset would you choose?

B) With the requirement to provide 12 years of service which asset would you choose?

First Calculate Present Worth

PW(A) = -10 + (8/1.1) = -2.73

PW(B) = -20 + (15/1.1^3) = -8.73

PW(C) = -30 +(25/1.1^4) = -12.9247

Calculate Annual Worth

PW(A|P, i = 10%, n)

A: n = 1 B: n = 3 C: n = 4

AW(A): -2.73(A|P,i=10%, 1)=-2.73(1.1) = -3.003

AW(B): -8.73(A|P,i=10%, 3) = -3.51033

AW(C): -12.92(A|P,i=10%, 4) =-12.92(.3155) = -4.08

Answer A) If there is no requirement to provide service, choose none of the assets since they all have negative present worth and thus negative annual worths.

Answer B) If there is a requirement to provide 12 years of service, choose asset A because it has the most positive (least negative) annual worth.

## Example 2

You are required to invest in a particular widget for your company. You have a choice between two widgets. Widget A has an initial cost of 75 dollars. It has a two year service life after which you can sell it for 30 dollars. Widget B has an initial cost of 90 dollars. It has a 3 year service life after which it can be sold for 30 dollars. If you are going to have to replace your widget an infinite amount of times after it's service life is over, which widget should you choose? The MARR is 10%.

*The first thing we need to do is to calculate the present worth of a single service life of each investment.*

Widget A:

PW=-75+30(1+.1)^{-2}

=-50.21

Widget B:

PW=-90+30(1+.1)^{-3}

=-67.46

*Widget A has a service life of 2 years, so N1=2. Widget B has a service life of 3 years, so N2=3.*

Widget A:

AW=-50.21(A|P,i=.1,N1=2)

=-28.9

Widget B:

AW=-67.46(A|P,i=.1,N2=3)

=-27.1

*Widget B has the greater annual worth and is the better choice.*

## Example 3 (Midterm 2 Annual Worth)

Use the following table to answer the questions

A | B | C | D | |
---|---|---|---|---|

Asset Life | 2 | 3 | 4 | 6 |

PW (10%) | 1.51 | .96 | .15 | 1.64 |

AW (10%) | .61 | .55 | .05 | .38 |

IRR | 22% | 12% | 11% | 16% |

**1) Given a choice of the four investments and no requirements to provide a service for a specific amount of time, which asset would choose?**

A) A

B) B

C) C

D) D

E) Can't tell from the data

**Answer:** Since we have no obligation to a particular amount of time, we can ignore the lives of each of the assets. We do not need to look at the IRR, either. Since the lives of the assets do not matter, following the criteria, we would chose the asset with the highest present worth, which is 1.64, or asset D.

**2) Suppose you are required to provide 12 years of service with the style of asset you choose. Which style of asset would you choose?**

A) A

B) B

C) C

D) D

E) Can't tell from this data

**Answer:** This problem is the exact opposite of the one before. Again, since we are assuming we will be involved for 12 years no matter which asset we choose, we can ignore the life of the assets. Since the life of the assets are all equal, following annual worth criteria, we would chose the asset with the greatest annual worth, which is asset A, with AW of .61.