# Applications of Money-Time Relationships

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Applications of Money-Time Relationships
And Methods For Evaluating Investments

Overview Minimum Acceptable Rate of Return (MARR)
Two kinds of choice problems: Choose ALL the profitable projects? Choose ONE most profitable project from many alternatives (mutually exclusive alternatives)? Methods for comparing mutually exclusive alternatives You actually already know these methods. They are just renamed slightly. PW Present Worth uses PV present value (also called NPV or Net Present Value) AW Annual Worth uses AV annuity value FW Future Worth uses FV future value In each method, the MARR should be used as i% Rates of Return Calculating IRR (Internal Rate of Return) The IRR is the i% where PV=0. Calculating ERR (External Rate of Return) A bit more complicated. Covered after IRR. Direct use of IRR and ERR do not allow comparison of mutually exclusive alternatives The  method for comparing mutually exclusive alternatives

Tunnel Construction Example (\$ figures in million hk\$)

Tunnel Construction Example
We are going to consider two problems A. Choose all profitable projects. B. Choose most profitable project. We will assume MARR = 6%

Choose all the profitable projects
Several equivalent ways to identify all the profitable projects. Given a MARR% requirement, Find all projects where IRR > MARR% OR Find all projects where PW>0 at i%=MARR% Find all projects where AW>0 at i%=MARR% (assumes repeatability)

Choose most profitable project
Constraints besides money (management time, difficult to obtain resources or equipment, land) may mean that not all projects can be funded. Most profitable means the highest \$ amount. Find project with highest PW (or AW) given i%=MARR%. Important: Do not compare IRR%. IRR% can be misleading. Seek the highest \$ amount, not the highest % return.

Tunnel life cycle The company builds the tunnel. We will treat this as a year 0 cost. The company operates the tunnel for L years, and receives an annual revenue from vehicle tolls. There is also an annual cost for salaries, repair, etc. After a useful life L years, the company sells the tunnel for a “salvage value” \$S

Let’s look at the raw data again
We have the annual cost but not the annual revenue. We are going to need to know the annual revenue. Annual revenue = Fare x Yearly Traffic. Is yearly traffic = Daily Traffic x 365? (no! the planners tells us it is less, they guess x325)

Now we have revenue, but need a way to evaluate the projects
In the next slide, we’ll choose the PW (present worth) method and see how to apply it to the problem.

Present worth method Although the tunnel project is simple, drawing a cash flow diagram cam help to avoid mistakes. Initial Cost C0 Year 1 Year L Salvage Value \$S Cash flow of \$A/year for L years, Where \$A = Annual Revenue – Annual O&M Cost Find \$PW?

Present worth calculation
Initial Cost C0 Year 1 Year L Salvage Value \$S Cash flow of \$A/year for L years, Where \$A = Annual Revenue – Annual O&M Cost Find \$PW? PW = -C0 + A * (P/A,i%,L) + S * (P/F,i%,L)

Organizing the data Arrange items according to life cycle (initial cost, annual O&M, salvage) Calculate PW of each class of items Add the PWs to get the total PW for each project

Organizing the data: 1 Arrange
Life cycle: Initial costs, then annual costs, then salvage

Organizing the data: 2 calculate each PW
Each PW is calculated with the standard formula. The initial cost is Year 0 and is thus a PW. The other costs need a conversion factor.

Conclusions from PW analysis
The Sha Tin, and 2 Lantau projects are profitable because PW > 0 at MARR=6%. Of the profitable projects, the Sha Tin project is the MOST profitable. It produces the highest Present Worth. The Tsuen Wan project is not profitable at MARR=6%

Rates of Return The IRR is the most common rate of return
The IRR is the i% where Present Worth = 0 It is usually algebraically difficult to solve for IRR%, so Use a spreadsheet and improved guesses to solve Or use a graph

Double click the spreadsheet, then play with i% until PW=0 for each project. The results are the Rates of Return IRR%.

IRR% results All except Tsuen Wan are above the 6% MARR.
A project is profitable when IRR>MARR But the highest IRR% may not give the highest PW.

Some problems with IRR Does not allow comparison of projects
Why not? Compare Can be ‘patched up’ with delta method. Easy to define, but Difficult to calculate Trial and error, graphing, Newton’s method To solve f(I)=0, use I**=I*-(f(I)/f’(I)) Assumes project revenues are reinvested at IRR%, which can be unrealistic if IRR% is very high.

We need a volunteer The volunteer needs to have \$100 in notes and \$1 in coins. There is no chance of losing money.

We need a volunteer Investment A costs \$1 and earns a 100% rate of return during this class. Investment B costs \$100 and earns a 20% rate of return during this class. Choose which investment you prefer

Of course \$1 x 100% = \$1 profit \$100 x 20% = \$20 profit
The lower IRR% may give a higher profit, if the required dollar investment is higher.

Delta method Choose the highest IRR investment, in this case A.
Then look at cash flow Investment A -\$1, then +\$2 Investment B -\$100, then +\$120 Delta = (B-A) has cash flows of -\$99, then + \$118 The IRR of Delta is about 20%. If 20% is over your MARR, then you should spend the extra money (the \$99) to go from investment A to investment B. You will then earn \$19 more profit.

Other IRR complaints Too hard to calculate Unrealistic when high
ERR, or external rate of return, tries to solve these problems. It does NOT solve the project comparison problem.

ERR External Rate of Return
Procedure - Project life is L years. Determine which periods (years, months) have net revenue and which periods have net costs Find FV (future value) of all net revenues, at MARR% Find PV (present value) of all net costs, at MARR% Find the E% that solves (FV of Revenues) = (PV of costs) * (P/F,E%,L) Or E% ={ [(FV of Rev)/(PV of Costs)](1/L)-1}

Odds and Ends MARR – where does it come from? Annual worth method
often worthwhile when projects repeat

MARR The MARR is usually given to you in textbook problems. In real life, where does it come from? Usually upper management decision, involving: Rates of interest on borrowed money Supply and demand for money in the company Expectations of investors Return on companies stock Return on stock index for the industry Project Risk (though adjusting MARR upwards for risk can be a bad move – since it weights present results over future)

Next Week: Annual Worth method
Useful for projects that repeat or when an annual value is desired for comparison to other projects with annual values. [Our tunnel comparisons did not repeat – the company planned to sell the tunnel] Main technical step is the annual capital recovery (CR) amount. Annual CR = (Initial Cost) (A/P,i%,L) – (Salvage Value) (A/F,i%,L)

Summary To find the most profitable project or investment:
compare PW (or sometimes AW) using i%=MARR% Do not compare IRR of projects; misleading To find all the profitable projects or investments: All of these methods produce equivalent answers Compare IRR to MARR Compare ERR to MARR Look for positive PW, using i%=MARR% Look for positive AW, using i%=MARR%

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