C2 May 2000 #50

[sunshine]

How about another finance question?!

In the beginning of 1999, a company invested 10 mil in a project expected to earn 2 mil at the beginning of each year beginning in 2001. Immediately after the investment is made, the company learns that 100 of its most experienced workers plan to retire by the end of 2000. W/out these workers' skills, the project can earn nothing until 2005.
Assume that the risk-free rate is zero and the company's beta is greater than zero.
Based on the discounted payback rule, which of the following actions would be recommended?

A. Abandon the project
B. Continue the project and do nothing, forgoing the 4 years of profits
C. Invest 2 mil immediately to hire and train new employees, preserving the earnings in 2003 and 2004
D. Give w to each of the potential retirees at the beginning of 2004 if they delay retirement until then.
i) sqrt(w/10) employees are expected to accept the offer
ii) The cash flows in years 2001-2004 would be 2 mil times the percent of the potential retirees who accept the offer.
E. No recommendations can be made w/out knowing the discount rate.

The answer is D.

[Gandalf]

Yuck! That was a real exam question, too, I see. It was not obvious that you get to choose the value of w. I thought you were supposed to determine that D was better than C (or vice versa) even without knowing the value of w, just like (for D to be correct) you're supposed to conclude D is better without knowing the discount rate. And it seemed to me D was better for some w's; C for others.

So I peeked at the SOA's solution to see what's going on.

Oh, we get to choose w. Well, as long as I'm peeking, let's see if I like the rest of what they did.

w=26,667; 51.64 workers; 2,754,000 undiscounted gain all OK.

Wrong formula for revenue. If they took the derivative of that formula to get MB, they never would end up with 26,667.

Wrong expression for number of workers. Plugging 26,667 into their expression gives .5164, not 51.64

[sunshine]

Yeah, their solution didn't make much sense to me at all. I think I'll just write off this question as a faulty qustion.
Thanks Gandalf!

[retaker]

Forget that question! :crazy:

[Mathlete]

Yuck is right!

Those turdburglers (Thank you, B&B). :lol:

[GuyInWestGrove]

Two or so years later, this thread was helpful.

I came up with (after getting the question wrong and looking at the SOA solution) a bonus of $60,000 with 77% of the workers taking it.

This would give ~ $1.5 million / year for 4 years in cash flow.

A couple of questions --

1. Are my figures above right?

2. Why does the question say to use the "discounted payback rule"? (specifically, Would the answer be different if the question said to use NPV?)

3. What does the risk-free rate of zero have to do with anything?

[Gandalf]

1. Yuck, yuck, yuck, yuck and more. This question is even dumber than I thought, as the optimal value of w should depend on the discount rate.

However, the following statements are true:
For i = 0, a bonus of 80,000/3 (as the SOA solution has) is optimal.
It produces 51.64% acceptance. Total revenues with bonus = .5164*2,000,000*4 = 4,131,182. Total cost of bonus = 51.64*26,667 = 1,377,061. Net improvement = 2,754,121.

I'm not sure where you got your 60,000, but it is not optimal. Total revenues = 6.2 million, but you now pay out 4.6 million in bonuses, so the net improvement is only 1.6 million.

And the SOA is correct that no other alternative at i = 0 has net improvement > 2,754,121, and C (which has positive net improvement at i=0) has costs earlier than this, so it is true that D with bonus = 26,667 beats all the other choices.

The SOA solution implies bonus =26,667 is optimal, since marginal revenues = marginal cost. With a discounted payback rule, that was the wrong test for optimal. Test should be marginal PV of revenues = marginal PV of cost.

2. I don't know what the text says. Does it describe a "discounted payback rule"? I have a hard time believing it could be anything other than NPV. (Unless maybe one is structured as "chose the better IRR" where the other is structured as "chose the higher NPV at a specified desired discount rate). Anyway, on this problem you should certainly choose the same answer either way, if you interpret D as saying "if you can choose the best value of w"

3. I think it's there to establish that the discount rate you should use for the cash flows is positive. The SOA solution says "positive discount rate (implied by the problem wording)"; I suppose risk-free rate = 0 is what implies that; or maybe beta > 0; I have no idea why beta > 0 matters. If you were discounting at a negative interest rate D would not necessarily have the highest NPV.

[GuyInWestGrove]

I'm not sure where you got your 60,000, but it is not optimal.


I tried to set Marginal cost (derivative of (w dollars per worker) * (sqrt(w/10) ) equal to Marginal revenue (derivative of 4 years * ( (sqrt(w/10) / 100 ) * 2 million.


2. I don't know what the text says. Does it describe a "discounted payback rule"?


The text does talk about it, but I had ignored it till this question. Brealey: "The discounted payback rule asks, "How many periods does the project have to last in order to make sense in terms of net present value?".

[Gandalf]

I'm not sure where you got your 60,000, but it is not optimal.


I tried to set Marginal cost (derivative of (w dollars per worker) * (sqrt(w/10) ) equal to Marginal revenue (derivative of 4 years * ( (sqrt(w/10) / 100 ) * 2 million.

Differentiation error?
Marginal cost = d/dw [(w^(3/2))/sqrt(10)] = [(3/2) w^(1/2)]/sqrt(10)

Marginal revenue = d/dw [w^(1/2) * 80,000 / sqrt(10)] = (1/2) w^(-1/2) 80,000 / sqrt(10)

[(3/2) w^(1/2)]/sqrt(10) = (1/2) w^(-1/2) 80,000 / sqrt(10)

Multiply both sides by 2 w^(1/2) sqrt(10)

3 w = 80,000
w = 80,000 / 3

[quote=Gandalf]
2. I don't know what the text says. Does it describe a "discounted payback rule"?


The text does talk about it, but I had ignored it till this question. Brealey: "The discounted payback rule asks, "How many periods does the project have to last in order to make sense in terms of net present value?".
OK, then it becomes equivalent to NPV in this test, since you won't be paid back before 2004, and cash flows are equal thereafter. So whichever has the greatest NPV by 2004 will pay back first.

The tests would produce different "which to do" if stream A had NPV = +1 through 1999 thru 2005 and +10 1999 thru infinity, while stream B had NPV = -1 1999 thru 2005 and +20 1999 thru infinity. Do A by "discounted payback rule" and B by "highest NPV rule". I suspect few decisions are made exclusively by the discounted payback rule, but IRL at my company one thing that must be documented in pricing proposals is the year at which cumulative NPV becomes positive.