CEO McIntosh is considering an investment option:

His firm only likes investments with a MARR of $10\%$ or more, to decide whether this option is

acceptable, the CEO first decides to calculate the IRR of the cash flow stream. He does this by

calculating the $NPV(Y)$ of the cash flow stream given a rate of return $(x)$ and plots the result

$($generally the point where this plot crosses the $x$ axis is the IRR$).$ The equation he plots is $(Y=)$:

His plot looks like this:

Plots:

$($this graph was made using wolfram alpha$)$

Unfortunately, this plot crosses the $x$ axis at $3$ places! So$,$ it has multiple possible rates of

return. These can be seen in the graph, though the exact values show up in the link. The zeroes

are approximately: $x=(-1\mathrm{.}49,0\mathrm{.}24,1\mathrm{.}12).$ The CEO wants to understand what each of these

numbers mean. He knows that since $x$ is $(1+r)$ if he were to invest $$1$ and get an interest rate of

$\text{'}x\text{'},$ his money after a single period would change by $x.$ So$,$ for the first rate $($i$.$e$.-1\mathrm{.}49),$ this means

his $$1$ of money $($assets$),$ would become $$1\mathrm{.}49$ of debt.

Similarly, if he waits another period, it will turn into $$2\mathrm{.}22$ of assets since $FV2=1\mathrm{.}49(1+i{)}^2.$

Notice in the even years our negative rate becomes positive. The CEO knows this is impossible,

otherwise he could get rid of all his company's debt by investing it here for $1$ period. Real interest

rates either shrink or grow the investment, they cannot negate it$,($i$.$e Thus, he knows the first rate can be

ignored because of the mathematical model not lining up with reality.

continue shrinking on and on$.$ Although this is not a good investment rate, he knows he must take it seriously because

the number has a realistic interpretation. For the last rate $(1\mathrm{.}12),$ if he invests $$1$ at time $0,$ he will have $$1\mathrm{.}12$ after

$1$ period, and it would continue growing on and on$.$ This is a good rate for an investment, he hopes this is what the

'true' interest rate is but isn't sure.

He knows that the multiple rates occur due to there being multiple negative cash flows and that they are

generally hard to deal with, so he has hired you as an analyst to figure out what happens if he simplifies things.

a$.$ The CEO first notices that he could always turn multiple negative cash flows into a single negative cash flow

by adding all the negative cash flows together today, getting that much money at time $0,$ and just holding

onto it until he needs to make the payments. Calculate the IRR of the above cash flow stream if all the

negative cash flows are moved to year $0.$

Hint: Consider that you are moving all the negative cash flow amounts to a dresser drawer, so at time

$0$ you put $$8,000$ into the drawer. You then can retrieve each payment from the cash in the drawer

and make the payments today $(0)$ year $1$ and year $5.$ Therefore, the years with current negative cash

flows will go to $0$ and year $0$ will show the full $-$$$8,000.$ Then you can use the IRR function in Excel to

solve the IRR of the cash flow.

b$.$ The CEO also knows that he doesn't have to just 'hold on$\text{'}$ to the money at time $0,$ he can find a temporary

fund to invest it and then withdraw money to pay for the negative cash flows. However, he's not sure how

well his investment will have to do to make the whole plan worthwile. If the minimum attractive interest

rate for the company is $10\%,$ calculate what the temporary fund's interest rate $($annual effective$)$ will have

to be$.($i$.$e$.$ what fund interest rate makes the overall project have an IRR of $10\%)$

Hint: Now consider that you are putting funds into an account that earns interest. Now instead of having

to put all $$8000$ away you can put away less and let it earn interest to still be able to make the payments

in year $0,1,$&$5.$ Therefore, in year $0$ you have a PV function with an unknown rate. You can reference

an empty cell for the rate and use the IRR function to solve for the IRR of the cash flow. When you

change the value in the empty rate cell notice the PV in year O goes down and the IRR goes up$.$ You need

to find the correct rate!

c$.$ With the assumptions below, think about what happens to $$1$ after multiple periods of investing have

passed.

IRR

$-100\%$IIRR

$0\%$