The Invest and Earn Problem

• Problems of this type have one or more initial negative cash flows - followed by positive cash flows to the end of the project

• Standard Solution– Sweep cash flow back into pot at time of decision– Basic solution

• use NPV and required rate of return directly

• all other measurement devices are variations on the NPV

Evaluating Invest and Earn Problems

• NPV directly incorporates your required rate of return and allows you to decide based on whether 0 or greater

• PVR can be used but it requires more work and not much better an indicator (this time you decide whether its 1 or greater)

• IRR is just playing with the NPV problem i value until the NPV is zero– a lot of work if you already know your critical rate of

return.

Variations on the Invest and Earn Problem

• The problem goes on for an indefinite length of time– Response - the first 20 or 30 years pretty much

establish the NPV / just truncate the problem

• The problem starts with a negative cash flow - goes positive - then goes negative at the end.– Example - mine development problems are

usually like this because the land must be restored when mining ceases

Negative Begin and End Variations

• This case challenges what people mean by a project rate of return because the project obligates money that will never be used in the profitable part of the project– Raises havoc with the IRR because part of the

investment will never be invested in the project to obtain a return from within the project

• How can a rate be internal?

Responses to the Negative End Challenge

• Use NPV– many companies set a required rate of return to

invest• They have many opportunities to invest and only take

those that meet the required rate

– if you have all these investment opportunities you just put your negative end money in another project and earn your required rate until you need it.

– If you have all these opportunities NPV answer is simple and hassle free

The IRR Bombshell

• Cash flows times magic numbers have the form of a polynomial– we know from math that polynomials have as many

roots as axis crossings– that means there is more than 1 interest rate that makes

the NPV zero!• From a physical standpoint because some of the negative cash flow

money grows inside the project, and some of it grows outside its just telling you about rates of growth inside and outside of the project that make NPV zero– Of course that means the whole concept of internal growth is crap

Multiple Roots and NPV• As long as you have plenty of opportunities to

invest at your required rate of return NPV is not effected– Who really cares whether the money grew in the

project as long as your getting your return

• Problem comes up when money or opportunities to invest at the rate are not a dime a dozen– To make answer make sense the money outside the

project must be growing at required rate

What if it Doesn’t?

• Situation can occur often– Many businesses have developed specialties in one

type of business or another - they are good at handling that type of risk

• They may not be able to handle risk well in other lines of business

– Money outside of project may not be able to be locked into long term commitments

– Many businesses can make 2 or 3 times more in their field than in the general market

The External Rate of Return• Plan to divert some money to lower rate investments

just as you would in real life– This may allow you to have sinking funds etc.

• Because the money you grew outside the project grows at a different rate - where and how much you put out when does impact the answer– in general minimize the amount of cash you run outside the

investment

• You remember Herby and Hanna Housings cash flow had the foul characteristic– 40% was the right answer if they could invest some of their

savings from renting at 40%

ERR Example

• Herby and Hanna Housing Cash Flow for buying instead of renting

Start

Year 1 Year 2 Year 3 Year 4 Year 5Year 6

The ERR Method

• Take all initial negative cash flows and discount them back to time zero– Herby and Hanna Housing Example

• -$3560 at time zero

• Drops directly into the pot without any discount factor

• If you have multiple negative cash flows you will have several P/F factors to discount these back

– Since the interest rate is unknown you have to just put in the formula for now

More ERR Method

• Look at your opportunities for short term fluid investments and select a rate of return for the growth of money outside the project– Lets say Herby and Hanna Housing decide to

put their savings in a Money Market at 4%

• All positive cash flows are discounted forward to the end of the project at 4% rate of return.

Discount some forward - some back

Initial Negative ValuesBucket

Subsequent ValuesForward Bucket

Discounting Positive Flows Forward

Comparative Cash Flow if Herby External Rate InfoBuys Instead of Rents Annual % 0 4

Buying Cash "i" 0 NPV 16925.48 0.003333333Over Rent Position n P/F n forward F/P

-3560 -3560 0 1 -3560 7322.33 -3537.67 1 1 22.33 72 1.270742 28.3756661622.33 -3515.34 2 1 22.33 71 1.26652 28.2813948422.33 -3493.01 3 1 22.33 70 1.262312 28.1874367222.33 -3470.68 4 1 22.33 69 1.258119 28.0937907522.33 -3448.35 5 1 22.33 68 1.253939 28.000455922.33 -3426.02 6 1 22.33 67 1.249773 27.9074311322.33 -3403.69 7 1 22.33 66 1.245621 27.8147154122.33 -3381.36 8 1 22.33 65 1.241483 27.7223077222.33 -3359.03 9 1 22.33 64 1.237358 27.6302070322.33 -3336.7 10 1 22.33 63 1.233247 27.5384123222.33 -3314.37 11 1 22.33 62 1.22915 27.4469225842.33 -3272.04 12 1 42.33 61 1.225067 51.85706844

391.14 -2880.9 13 1 391.14 60 1.220997 477.5806078

Later Negative Flows Also Discount Forward

Initial Negative ValuesBucket

Subsequent ValuesForward BucketInitial Investment

Savings DiscountedForward at 4%

230.56 4283.77 59 1 230.56 14 1.047691 241.5557254-419.44 3864.33 60 1 -419.44 13 1.044211 -437.9837282-25.81 3838.52 61 1 -25.81 12 1.040742 -26.86153922

-404.84 3433.68 62 1 -404.84 11 1.037284 -419.9340261-404.84 3028.84 63 1 -404.84 10 1.033838 -418.5388965-404.84 2624 64 1 -404.84 9 1.030403 -417.1484018-404.84 2219.16 65 1 -404.84 8 1.02698 -415.7625267-404.84 1814.32 66 1 -404.84 7 1.023568 -414.3812559-404.84 1409.48 67 1 -404.84 6 1.020167 -413.004574-404.84 1004.64 68 1 -404.84 5 1.016778 -411.6324657-404.84 599.8 69 1 -404.84 4 1.0134 -410.26491615995.5 16595.3 70 1 15995.5 3 1.010033 16155.98878

ERR Problem Set Up

Initial Negative ValuesBucket

Subsequent ValuesForward BucketInitial Investment

Future Savings andCosts Discountedforward at 4%

-404.84 2219.16 65 1 -404.84 8 1.02698 -415.7625267-404.84 1814.32 66 1 -404.84 7 1.023568 -414.3812559-404.84 1409.48 67 1 -404.84 6 1.020167 -413.004574-404.84 1004.64 68 1 -404.84 5 1.016778 -411.6324657-404.84 599.8 69 1 -404.84 4 1.0134 -410.26491615995.5 16595.3 70 1 15995.5 3 1.010033 16155.98878

0 16595.3 71 1 0 2 1.006678 00 16595.3 72 1 0 1 1.003333 0

330.18 16925.48 73 1 330.18 0 1 330.18

Sum 21482.23536

-$3560

$21,482

Solving the ERR

Initial Negative ValuesBucket

Subsequent ValuesForward BucketInitial Investment

Future Savings andCosts Discountedforward at 4%

-$3560

$21,482

Now discount the future pot back into the big pot at time 0.Note this makes for a very easy IRR problem.

$3560 = 1/((1+i)73 ) * $21,482

Solve for i

(1+i)73 = $21,482/$3560

(1+ i) = 6.034^(1/73)

(1 + i) = 1.0249

i = 0.0249/monthAdapt to 1 year (1+.0249)12 = 1.3437 or 34.37%

Teachers Attitude Problem• I’m a Mining and Mineral Resources Engineer

– Every project I do will have negative cash flows to build and negative at the end to reclaim

• I never see IRR work smoothly - so I don’t like it

• IRR assumes money outside the project grows at the same rate as money in the project– I sell mining projects because they are better

More Bad Attitude

• I don’t like ERR either– I took all my money from the project and put it

into CDs and never invested in another project– If the company is for real it invests in certain

types of projects repeatedly - projects are not one and onlys

– ERR invested all earnings outside the project - how stupid

The Tweeked IRR

• Problem is that committing to the project means committing not only money now but money in the future– I will have to cover that future commitment by

taking some of my earnings and setting them aside to meet that expense (probably in lower earnings - low risk short term fluid investments)

• Its kind of like a sinking fund

– I’m certainly not going to set all my earnings in

How to Tweek

Take Just Enough of YourFuture Earnings to offsetthe future negative cash flow

How do I do that?

• Determine what rate you can get on secure, fluid, short-term investments– For Hanna and Herby it was 4%

• Create a temporary pot at the beginning of the negative cash flow

Tweeking Procedures

• Discount the future negative cash flows back into the temporary pot at the chosen rate.

Discounting Back

137.83137.83137.83137.83157.83604.95230.56230.56230.56230.56230.56230.56230.56230.56230.56 Temp230.56 n P/F Pot

-419.44 0 1 -419.44-25.81 1 0.996678 -25.7243

-404.84 2 0.993367 -402.155-404.84 3 0.990066 -400.818-404.84 4 0.986777 -399.487-404.84 5 0.983499 -398.16-404.84 6 0.980231 -396.837-404.84 7 0.976975 -395.518-404.84 8 0.973729 -394.204-404.84 9 0.970494 -392.895

Total -3625.24

-$3625.24

Next Step

-$3625.24

Now discount part of your positive cash flow forward into the pot at 4%to balance the red ink.

How much and when?

• Part of the project earnings are to be discounted forward to cover the red ink– Because the rate of return on this sort of “sinking

fund” money is less than the project itself we want to use as little as late as possible

• The simple approach is to use the last so many savings flows to cover the red ink– In practice one may not want to turn in zero earnings

quarters– They spend more for a good investor smoke screen

I’ll Use the Simple Approachn F/P Cash Flow Cumulative Amount

137.83 16 1.054688 145.3676 3718.639115 3625.24 Critical Amount137.83 15 1.051184 144.8846 3573.271518137.83 14 1.047691 144.4033 3428.386871137.83 13 1.044211 143.9236 3283.983567157.83 12 1.040742 164.2602 3140.060009604.95 11 1.037284 627.5049 2975.799771230.56 10 1.033838 238.3616 2348.294858230.56 9 1.030403 237.5697 2109.933214230.56 8 1.02698 236.7805 1872.363469230.56 7 1.023568 235.9938 1635.582992230.56 6 1.020167 235.2098 1399.589162230.56 5 1.016778 234.4284 1164.379364230.56 4 1.0134 233.6495 929.9509938230.56 3 1.010033 232.8733 696.3014557230.56 Temp 2 1.006678 232.0996 463.4281618230.56 n P/F Pot 1 1.003333 231.3285 231.3285333

-419.44 0 1 -419.44

I will sweep enough future earnings forward at an available external rate to coverthe future red ink commitment.

My New Cash Flow

Future Negative Cash Flowis cancelled by positive cashflows swept forward.

Now sweep the rest of the cash back into the main pot.

The Sanitized IRRNote that I now have a classic cash flow problem - invest and make moneythere after.

I can get a regular trick free IRR on this thing.

IRR on this cash flow is 39.01%

Three Answers - 1 Problem

• Regular IRR was over 40%– but to get it - Herby had to invest his savings at

over 40% (I don’t think so)

• ERR was 34.37%

• If Herby diverts funds into a money market to cover the payments after he graduates he will get 39.01%

Summary of the Invest and Earn Problem

• Problem has a negative cash flow of initial investments and then positive cash flows in the future

• Can determine if investment is worth it using– IRR– NPV– PVR (PVR is really designed for something else)

Sabotaged Invest and Earn Problems

• These problems have a negative cash flow at the end (it costs both to get in and to get out at the end)– IRR becomes meaningless because some of the

money will not grow in the project

– The numbers from the IRR may do funny things (with no guarantee it will be obvious)

• The IRR will be Sabotaged

• The NPV may or may not be.

How do I know what rates are Sabotaged?

• If doing an IRR - your sabotaged so forget it

• If doing an NPV– Look at your required rate of return (it’s the i

value you use in your magic numbers)– How many opportunities does your company

have to get this rate?• Very few - Your sabotaged

• A lot - Maybe your ok

Continuing Exercise

• Your doing NPV and your company has many opportunities to get the rate of return i– Do these opportunities provide regular and

reliable means for your company to come up with the negative cash flow amounts in their individual projects internally?

• No - your NPV is sabatoged

• Yes - your ok - do a regular NPV problem solution and forget it.

Reasons Why You Need to Know this Stuff

• Financial planning and structure of your projects is always one of the biggest factors in determining their earnings potential