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Capital Budgeting Practice

Assume you have just retired as the CEO of a successful company. A major publisher has offered you a book deal. The publisher will pay you $1 million upfront if you agree to write a book about your experiences. You estimate that it will take three years to write the book. The time you spend writing will cause you to give up speaking engagements amounting to $500,000 per year. You estimate your opportunity cost to be 10%.
Should you accept this deal? Plot a diagram that measures NPV (on the y-axis) vs. discount rate (on the x-axis) to solve this problem. (Hint: Have your scale on the x-axis go to 50% (discount rate)).
Determine the IRR for this deal. (Hint: IRR is the point at which NPV = 0)
Suppose you inform the publisher that it needs to sweeten the deal before you will accept it. The publisher offers $550,000 advance and $1,000,000 in four years when the book is published.
Should you accept or reject the new offer? Again, plot a diagram that measures NVP (on the y-axis) vs. discount rate (on the x-axis) to solve this problem. (Hint: Have your scale on the x-axis go to 50% (discount rate)).
Determine the IRRs for this deal (Hint: There are two IRRs for this problem).
Discuss if the IRR rule for making budgetary decisions can be used in this case.
Finally, you are able to get the publisher to increase your advance to $750,000, in addition to the $1 million when the book is published in four years.
Should you accept or reject this new offer? Again, plot a diagram that measures NVP (on the y-axis) vs. discount rate (on the x-axis) to solve this problem. (Hint: Have your scale on the x-axis go to 50% (discount rate)).
Determine the IRR for this deal.
State three conclusions regarding the use of IRR vs. NPV that you can make from questions 2-4. Which is the stronger method to use (IRR or NPV), and why?

Full Answer Section

$NP V = C F_{0} + ( 1 + r ) ^{1} C F ^{1} + ( 1 + r ) ^{2} C F ^{2} + ( 1 + r ) ^{3} C F ^{3}$ $NP V = 1, 000, 000 + ( 1 + 0.10 ) ^{1} -500 , 000 + ( 1 + 0.10 ) ^{2} -500 , 000 + ( 1 + 0.10 ) ^{3} -500 , 000$ $NP V = 1, 000, 000 - 1.10 500 , 000 - 1.21 500 , 000 - 1.331 500 , 000$ $NP V = 1, 000, 000 - 454, 545.45 - 413, 223.14 - 375, 657.40$ $NP V = 1, 000, 000 - 1, 243, 425.99$ $NP V = - $243, 425.99$

Since the NPV is negative $(- $243, 425.99)$ , you should reject this deal at a 10% opportunity cost.

NPV vs. Discount Rate Diagram (Offer 1)

To plot the diagram, we'll calculate NPV at various discount rates (0%, 10%, 20%, 30%, 40%, 50%).

Discount Rate ( $r$ )	$C F_{0}$	$C F_{1} / (1 + r)^{1}$	$C F_{2} / (1 + r)^{2}$	$C F_{3} / (1 + r)^{3}$	NPV
0%	1,000,000	-500,000	-500,000	-500,000	-500,000
10%	1,000,000	-454,545	-413,223	-375,657	-243,426
20%	1,000,000	-416,667	-347,222	-289,352	-53,241
25%	1,000,000	-400,000	-320,000	-256,000	24,000
30%	1,000,000	-384,615	-295,858	-227,583	92,000
40%	1,000,000	-357,143	-255,102	-182,216	205,539
50%	1,000,000	-333,333	-222,222	-148,148	296,297

Diagram Plot (Conceptual):

X-axis: Discount Rate (%) from 0% to 50%
Y-axis: NPV ($)
Plot the points from the table above.
You will see the NPV curve starting negative, crossing the X-axis between 20% and 25%, and then becoming positive.

Determine the IRR for this deal.

The IRR is the discount rate at which NPV = 0. From the table above, we can see that the NPV becomes positive between 20% and 25%. We can approximate it through interpolation or use a financial calculator/software.

Using a financial calculator or software (e.g., Excel's IRR function): $C F_{0} = 1, 000, 000$ $C F_{1} = - 500, 000$ $C F_{2} = - 500, 000$ $C F_{3} = - 500, 000$

The IRR for Offer 1 is approximately 22.95%.

New Offer (Sweetened Deal)

Offer 2: $550,000 advance and $1,000,000 in four years when the book is published, while still giving up $500,000/year for 3 years.

Initial Cash Flow ( $C F_{0}$ ): +$550,000 (upfront advance)
Opportunity Cost (Cash Outflow) for Year 1 ( $C F_{1}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 2 ( $C F_{2}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 3 ( $C F_{3}$ ): -$500,000
Cash Flow Year 4 ( $C F_{4}$ ): +$1,000,000 (publication payment)
Opportunity Cost ( $r$ ): 10%

Should you accept or reject the new offer?

Let's calculate the NPV at a 10% opportunity cost.

$NP V = C F_{0} + ( 1 + r ) ^{1} C F ^{1} + ( 1 + r ) ^{2} C F ^{2} + ( 1 + r ) ^{3} C F ^{3} + ( 1 + r ) ^{4} C F ^{4}$ $NP V = 550, 000 + ( 1.10 ) ^{1} -500 , 000 + ( 1.10 ) ^{2} -500 , 000 + ( 1.10 ) ^{3} -500 , 000 + ( 1.10 ) ^{4} 1 , 000 , 000$ $NP V = 550, 000 - 454, 545.45 - 413, 223.14 - 375, 657.40 + 1.4641 1 , 000 , 000$ $NP V = 550, 000 - 1, 243, 425.99 + 682, 904.17$ $NP V = 1, 232, 904.17 - 1, 243, 425.99$ $NP V = - $10, 521.82$

Since the NPV is still negative $(- $10, 521.82)$ , you should reject this new offer at a 10% opportunity cost.

NPV vs. Discount Rate Diagram (Offer 2)

We need to calculate NPV at various discount rates to plot the diagram and find IRRs.

Discount Rate ( $r$ )	$C F_{0}$	$C F_{1} / (1 + r)^{1}$	$C F_{2} / (1 + r)^{2}$	$C F_{3} / (1 + r)^{3}$	$C F_{4} / (1 + r)^{4}$	NPV
0%	550,000	-500,000	-500,000	-500,000	1,000,000	50,000
5%	550,000	-476,190	-453,515	-431,919	822,702	111,078
10%	550,000	-454,545	-413,223	-375,657	682,904	-10,522
15%	550,000	-434,783	-378,072	-328,758	571,753	-19,860
20%	550,000	-416,667	-347,222	-289,352	482,253	-20,988
25%	550,000	-400,000	-320,000	-256,000	409,600	-16,400
30%	550,000	-384,615	-295,858	-227,583	349,603	-8,453
35%	550,000	-370,370	-274,348	-203,221	299,634	1,695
40%	550,000	-357,143	-255,102	-182,216	258,074	13,613
50%	550,000	-333,333	-222,222	-148,148	197,531	43,828

Diagram Plot (Conceptual):

X-axis: Discount Rate (%) from 0% to 50%
Y-axis: NPV ($)
Plot the points from the table above.
You will observe that the NPV curve starts positive, crosses the X-axis (becomes negative) somewhere between 0% and 10%, then turns around, crosses the X-axis again (becomes positive) somewhere between 30% and 35%, and then remains positive. This pattern indicates multiple IRRs.

Determine the IRRs for this deal.

Because the cash flow stream changes sign more than once (from positive to negative, then back to positive), this is a non-normal cash flow stream, leading to multiple IRRs.

From the table:

The first IRR (where NPV goes from positive to negative) is between 5% and 10%.
The second IRR (where NPV goes from negative to positive) is between 30% and 35%.

Using a financial calculator or software (e.g., Excel's IRR function or iterative method): $C F_{0} = 550, 000$ $C F_{1} = - 500, 000$ $C F_{2} = - 500, 000$ $C F_{3} = - 500, 000$ $C F_{4} = 1, 000, 000$

The two IRRs for Offer 2 are approximately 7.17% and 31.93%.

Discuss if the IRR rule for making budgetary decisions can be used in this case.

No, the IRR rule for making budgetary decisions cannot be reliably used in this case (Offer 2).

This situation exemplifies the problem of multiple IRRs, which occurs when the cash flow stream is "non-normal," meaning it has more than one sign change (e.g., positive, then negative, then positive again).

Problems with Multiple IRRs:

Ambiguity: With two IRRs, which one should be used for comparison against the cost of capital? Both indicate an NPV of zero.
Misleading Decision: If your cost of capital (e.g., 10%) falls between the two IRRs (7.17% and 31.93%), the IRR rule would suggest acceptance if $I RR > r$ . However, the NPV at 10% is negative, indicating rejection. This creates a contradiction.
Reinvestment Rate Assumption: The IRR method implicitly assumes that cash flows are reinvested at the IRR itself. When there are multiple IRRs, this assumption becomes highly problematic and unrealistic, especially if one IRR is very high.

Conclusion: Due to the multiple IRRs arising from the non-normal cash flow pattern, the IRR rule is unreliable and should not be used as the sole or primary decision criterion for Offer 2. In such cases, the Net Present Value (NPV) is the superior and unambiguous decision-making tool. If NPV is positive at the cost of capital, accept; if negative, reject.

Final Offer (Increased Advance)

Offer 3: $750,000 advance and $1,000,000 in four years when the book is published, while still giving up $500,000/year for 3 years.

Initial Cash Flow ( $C F_{0}$ ): +$750,000 (upfront advance)
Opportunity Cost (Cash Outflow) for Year 1 ( $C F_{1}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 2 ( $C F_{2}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 3 ( $C F_{3}$ ): -$500,000

Sample Answer

Initial Book Deal Offer

Offer 1: $1 million upfront, give up $500,000/year for 3 years.

Initial Cash Flow ( $C F_{0}$ ): +$1,000,000 (upfront payment)
Opportunity Cost (Cash Outflow) for Year 1 ( $C F_{1}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 2 ( $C F_{2}$ ): -$500,000
Opportunity Cost (Cash Outflow) for Year 3 ( $C F_{3}$ ): -$500,000
Opportunity Cost ( $r$ ): 10%

Should you accept this deal?