Annual worth (AW) is an analysis technique based on the concept of equivalencies [1]. Annual worth is normalizes assets, which allows comparison of assets with drastically different characteristics (such as lifespan, initial cost, and servicing cost). AW converts rising and falling costs of an investment over a given time period into a constant cost-per-year value. For example, if the elephant you purchased in year 0 was expected to live 50 years, and the cost of caring for it fluctuated over time, you might want to know how much that elephant was ordinarily going to cost you each year (in today's dollars). Note that because AW takes the time-value of money into account, AW analysis will not result in the same value as a simple average of all the annual costs. Because AW is directly based on the net present worth (PW) calculation, you must be able to accurately calculate PW.

AW calculations will give the same decision result as PW calculations (unless N = 1, in which case AW calculations will give an erroneous result), but is usually preferred over PW as a reporting tool (in most cases), as it is easier to understand. AW 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 using discounted payback
- Find the annual worth (AW), a constant series cost, using (A|P,i,N) where
*P*is the present worth calculated in step 1,*i*is the given MARR, and*N*is the number of years you will be using that asset (the "planning horizon"). This gives the annual worth.

It is important to be aware of planning horizons when evaluating annual worth. Planning horizons give an idea of how long a particular asset or investment will be serviceable. 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

If cash flow is constant after the period zero, you do not need to calculate present worth. Something that is worth $10 a year for 5 years has an annual worth of $10.

# The Effect of Choice Environment on Annual Worth

## 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.