Investment Criteria: Present Worth

Present worth (PW) analysis is an equivalence method of analysis of investment decisions in which a project's cash flows are presented as a single present value. PW is one of the most basic and efficient methods available for determining the acceptability of a project on an economic basis.

PW analysis takes two forms: simple payback, which ignores the time-value of money, and discounted payback, which accounts for the time-value of money. PW analysis results in a single criterion. For simple payback, that criterion is the investment payback period. For discounted payback, that criterion is whether the PW is greater than zero. If the investment passes the criterion, the investment is good.

Net Present Worth

The present worth of all incoming cash flows is compared with all the outgoing cash flows and the difference in the two flows is the net present worth. This net present worth is a basic tool for determining whether an investment is a good one.

When Present Worth is not the Best Choice

When your investments have unequal lives (one lasts 5 years and another lasts 15) use Annual Worth (AW) to solve.

Present Worth Analysis Methods

Simple Payback: Present Worth Analysis Without the Time-Value of Money

Simple Payback Method - the amount of time that it takes to recover a project's cost.

Example: You invest $10,000 in weatherizing your home sauna, which is in your previously drafty garage. The newly insulated sauna saves $1,000 per year in gas. Since it takes 10 years to recover the cost of the project, the simple payback period is ten years.

Video Transcript Available here

Just for entertainment, there is an interesting take on simple payback here using energy and water.

Strength of Simple Payback Method

It's Simple

If all you want to know is the amount of time it will take you to recover the cost of a new pump, refrigerator, or welder, and you do not care about how time affects the value of that money, then go no further than the Simple Payback Method.

Weaknesses of Simple Payback Method

Quick and Limited view of Financial Performance of the Investment

One of the drawbacks to the Simple Payback Method is that it does not account for future savings or costs beyond the payback period. For example, a piece of equipment may allow you to continue to save $1,000 per month in gas bills after the payback period, but these savings are not expressed in the simple payback period. In addition, the equipment may require enormously expensive maintenance at some time after the simple payback period, and this cost is also not reflected in the analysis.

Time Value of Money is not Considered

Example:
What if you were selling something old that you don't need anymore. This old thing is worth $2,000 even though it doesn't work. You don't need the money now, but you want to take a trip to Peru in two years and will put the money towards that trip. Both of your neighbors, Linda Walker and Cat Daddy, really like this old thing and offer you $2,000 for it.

Rick "Cat Daddy" lives in a sky-blue house with bars on the windows. He owns two cats named "Power" and "Control" and enjoys yelling into his cell phones early in the morning, watching Fox News and hosting sketchy friends in a backyard tent city. Rick offers to pay you $1,000 now and $1,000 in two years for the old thing; the simple payback is two years.

Linda Walker lives in a tiny beige house with the nicest fern garden on the block. She is 64 years old and has red hair. Her two favorite past times are dumpster-diving and free-piling. Every Sunday night she puts on her headlamp and sorts through the neighborhood's recycling bins looking for the NY Times. On Mondays, she likes to sit in bed with the fan on, wearing nothing but her deceased mother's best jewelry, while she reads the entire week's worth of NY Times. Linda Walker offers to pay you $2,000 in two years for the old thing; the simple payback is therefore two years.

By just looking at the Simple Payback, the offers from Linda Walker and Cat Daddy are equal. However, if you consider the time value of money, Rick's offer is better. If you consider the likelihood of Rick going to prison before he pays the second $1,000, Linda Walker's offer is far superior.

How to Calculate Simple Payback

To calculate simple payback, add up the costs or benefits in each time period through the payback period as though the interest rate were zero. Do not discount costs or benefits. Costs or benefits after the payback period are not considered: it is as if the interest rate (and therefore, the cost) jumps to infinity after the payback period.

When to use Simple Payback

Simple payback is often a good criteria to use when deciding whether a more detailed investment criteria analysis should be made, but it is only useful in certain cases. It should only be used when the payback period is short. It is a poor criteria to use when evaluating whether to make long-term investments. When benefits are not in the near future, the results will deviate from those calculated using discounting (present worth) and a minimum acceptable rate of return. Simple payback should not be used for assets that have cyclical costs and benefits over time.

Discounted Payback: Present Worth Analysis Including the Time-Value of Money

In this class, we use the generic term "present worth" to mean "discounted payback." PW is often referred to as lifecycle cost. PW is the lumped-together time-period-zero value of future amounts in a cash flow.

How to Calculate Present Worth

You will be given an interest rate and a series of cash flows. For example, find the PW of the following series when the interest rate is 5%:

Year Asset
0 -$200
1 -$300
2 -$400
3 -$500
4 -$100

The first step is to determine whether the nominal interest rate needs to be converted to an effective interest rate. You must convert when any of the three periods (nominal, compounding, or effective) are unalike. For this example, assume all periods are annual because no periods are stated. Therefore, no conversion is required.

The second step is to look for the four common patterns in the cash flow and (if necessary) break the flow apart into discrete, easily-calculable chunks. There are always many ways to break down any cash flow. It can be easier to see the patterns if you draw the cash flow out, like this:

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Figure 1. Given cash flow.

The above example can be broken down into two summed flows: a constant-series flow $(A = -100)$ plus a linear gradient flow $(G = -100)$. Draw out the two flows separately, like this:

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Figure 2. Constant-series portion of given cash flow.


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Figure 3. Linear gradient portion of given cash flow.


NOTE: BOTH FLOWS ARE TIME-SHIFTED. A constant-series flow should have zero value in time period 0, and a linear gradient flow should have zero value in both time periods 0 and 1. The constant-series has been shifted to the left one time period, and the linear gradient has been shifted left two time periods.

Third, use factor notation and time-shifting to convert the pictures into numbers.

  • Constant-series (reference Figure 2)
    1. Find A (the annual payment, or "annuity") using Figure 2. $A = -100$ because there is $100 flowing out of the business (or your pocket) every period.
    2. To find N, count the number of non-zero annuities in the series (there are 5).
    3. In the front cover of your book (or on the note sheet provided with all exams) you will find this: $(P|A, i, N)$. This is factor notation meaning "find P (the present worth) given A (the annuity), i (the effective interest rate), and N (the number of effective periods the annuity occurs)." To the right of the factor notation is a formula. Calculate P using the formula, given $A=-100$, $i=.05$, $N=5$. Simultaneously time-shift by dividing by $(1 + i)^n$. For this example, $n=-1$ because the series has been shifted one period to the left. Note that $N$ in the numerator and $n$ in the denominator sum to the last time period of the series (time period 4). The combined calculation looks like this:
(1)
\begin{align} \text{in factor notation: }P_A=\frac{-100(P|A, i=.05, 5)}{(1.05^{-1})} \end{align}
(2)
\begin{align} P_A=-100\left[\frac{1.05^5-1}{.05(1.05^5)(1.05^{-1})}\right] \end{align}
  • Linear gradient series (reference Figure 3)
    1. Find G (the gradient, or more simply, the slope: dy/dx) using Figure 3. $G = -100$ because the change in y, the amount, is -100 per one change in x (time period). -100/1 = -100.
    2. To find N, count the number of non-zero annuities in the series and add one. This is a special rule that holds true only for the linear gradient. To remember this rule, think of a standard linear gradient that has a line connecting the tops of the amounts (Figure 4). That line extends down to the red line. Every place the line crosses either the timeline or an amount is counted as a part of the series. Therefore, because no series ever has any amount in time period zero, we must have two zero-amount entries at the beginning of a linear gradient series. For this example, $N=5$.
    3. In the front cover of your book (or on the note sheet provided with all exams) you will find this: $(P|G, i, N)$. This is factor notation meaning "find P (the present worth) given G (the linear gradient), i (the effective interest rate), and N (the number of effective periods the linear gradient occurs)." To the right of the factor notation is a formula. Calculate P using the formula, given $G=-100$, $i=.05$, $N=5$. Simultaneously time-shift by dividing by $(1 + i)^n$. For this example, $n=-2$ because the series has been shifted two periods to the left. If you have trouble seeing the shift, compare Figure 4 to Figure 3. Note that $N$ in the numerator and $n$ in the denominator sum to the last time period of the series (time period 3). The combined calculation looks like this:
(3)
\begin{align} \text{in factor notation: }P_G=\frac{-100(P|G, i=.05, 5)}{(1.05^{-2})} \end{align}
(4)
\begin{align} P_G=-100\left[\frac{1.05^5-.05(5)-1}{.05^2(1.05^5)(1.05^{-2})}\right] \end{align}

Finally, add $P_A$ and $P_G$. The sum is the total present worth of the given cash flow.

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Figure 4. Standard linear gradient series flow, $G = -100, N = 5$.

The Effect of Choice Environment on Present Worth

Present Worth Criteria in an Unconstrained Choice Environment

You may be presented with a variety of investment choices, and on top of this, each choice may each have different options available. In this case we should adhere to the following rule: Choose ALL investments that have a present worth greater than or equal to 0.

Since this is unconstrained choice we are free to choose as many investment choices as we deem worthy by present worth analysis, but we can only have one option per investment.

Why Present Worth Criteria in an Unconstrained Choice Environment Works

Download audio here.

At one point in the audio the criteria is misspoken as PW>1, it should be PW>0

This rule works under the assumption that the money is not doing anything else if not invested. Unconstrained choice allows us to see all of our options and invest in all that are greater than or equal to zero. In a close run one single choice may not be sufficient so unconstrained choice takes care of this problem and displays our options.

Example: Present Worth Analysis in an Unconstrained Choice Environment

For this example assume MARR = 10%.

Year Option A Option B
0 -10 -10
1 15 11
PW(10%) 3.63 0

Sample Calculations for PW:
A: -10 + (15 / (1 + 0.1)^1 ) = 3.63
B: -10 + (11 / (1 + 0.1)^1 ) = 0

Analysis results: Both options should be purchased since the PW of both investments is greater than or equal to 0.

Present Worth Criteria in an Exclusive Choice Environment

Exclusive choice differs from unconstrained choice in that we only want to choose one investment option. An example would be choosing an investment for a business, such as building a house. The house can be built more durable but it takes longer to build or less durable but built much more quickly. You can't choose both options, you must choose only one.

In the case of exclusive choice, we should adhere to the following rule:
We choose the single option that has the largest present worth.

Why Present Worth Criteria in an Exclusive Choice Environment Works

Video Transcript Available here

This rule also works because money is not doing anything if it's not being invested. Unlike unconstrained choice it looks at a list of different things that are needed and finds the one with the greatest benefit.

The Phone Lines Example

Video Transcript Available here

Continuing the example with some calculations

Video Transcript Available here

Why the answer is the answer

Video Transcript Available here

Example 2

This is a pretty cool and quick analysis. It goes pretty fast.

Example 3

Assume that MARR = 10%

Year A B C D
0 -10 -20 -200 -1
1 15 30 215 4
PW(10%) 3.63 7.27 -4.34 2.63

Sample Calculations for PW:
A: -10 + (15 / (1 + 0.1)^1 ) = 3.63
B: -20 + (30 / (1 + 0.1)^1 ) = 7.27
C: -200 + (215 / (1 + 0.1)^1 ) = -4.34
D: -1 + (4 / (1 + 0.1)^1 ) = 2.63

Investment option B is the winner for exclusive choice because it has the highest present worth.

Example 4

MARR = 10%

YEAR A B C
0 10 10 10
1 0 0 -3
2 0 0 -3
3 0 -15 -3
4 0 0 -3
5 -15 0 0
PW 0.686 -1.269 0.49

PW(A)= 10+(-15/1.15) = 0.686
PW(B)= 10+(-15/1.13) = -1.269
PW(C)=10-3(P\A, i=0.10, n=4)
PW(C)=10-3((1.14-1)/(0.10(1.1)4))= 0.49

Remember:
Assets - as many as you want with one choice from each category, choose non-negative values.
Asset - choose one highest positive value.

Q1: Which assets would you choose?
A: Assets A and C

Q2: Which asset would you choose?
A: Asset A

More Examples
  1. Prior Pop Quiz
  2. Pop Quiz Oct. 31
  3. Sample Question
  4. Mid Term 2 Question
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