Risk and Reward: Evaluating the Probability of Investment Decisions

[Kinetica]

Homework Statement

You are a risk-neutral individual whose net worth is $10,000 and you are thinking of opening a Donut Franchise. To open the franchise you must invest $5,000. If you buy the franchise, the probability is 1/3 that you will make $500,000, 1/3 that you will break even, and 1/3 that you will lose the entire investment. What would be your decision? In general, at what probabilities would you change your decision?

The Attempt at a Solution

Without investing, my expected wealth is E(L1)=10,000*1=10,000.

With investing, my expected wealth besides 10,000-5,000=5,000 of my initial wealth is

E(L_2)=(500,000-5,000)*(1/3)+(5,000-5,000)*(1/3)+(0-5,000)*(1/3)=163,333.30.

Therefore, the expected wealth from the franchise investment is higher than the initial wealth without gamble. But the decision of which decision is better depends entirely on the risk-taking propensities of the individual and which decision this person chooses.


There is also another possible way of solving it, and I don't know which one is correct:

With investing, my expected wealth is
E(L_2)=(510,000-5,000)*(1/3)+(10,000-0)*(1/3)+(10,000-5,000)*(1/3)=173,333.30


Thanks a lot.

 

[clamtrox]

Seems to me like neither one is correct. I don't know anything about finance, but I'd say making $500000 means your networth is $510000, breaking even means it's $10000 and losing all means it's $0.

 

[Ray Vickson]

Seems to me like neither one is correct. I don't know anything about finance, but I'd say making $500000 means your networth is $510000, breaking even means it's $10000 and losing all means it's $0.

The question is really not about finance at all; it is just about gamble evaluation.

In this case, if you choose to invest, then your final wealth will be: (i) 5000 + 0 (if you lose); (ii) 5000 + 5000 (if you break even); and (iii) 5000 + 500,000 if you win. Here the common term 5000 is what you have left of your initial 10,000 after investing 5000 of it.

Actually, in the "win" case (iii) it is not absolutely clear whether you end up with 5000 + 500,000 or 5000 + 505,000, because it is not clear from the wording whether you add what you "make" (= 500,000) to the original 5000 investment, or whether that 5000 investment is absorbed in the 500,000. In such a case you can always include both possibilities and give two clearly-distinguished answers.

RGV

 

[clamtrox]

The question is really not about finance at all; it is just about gamble evaluation.

I mean that the calculation itself is trivial: the problem is in the interpretation of the question. I'd have done it differently than you (different from either way of counting case iii). Though as a disclaimer, I'm not a native english speaker so I'm probably wrong.

 

[Ray Vickson]

I mean that the calculation itself is trivial: the problem is in the interpretation of the question. I'd have done it differently than you (different from either way of counting case iii). Though as a disclaimer, I'm not a native english speaker so I'm probably wrong.

Doing it differently is OK if you show your work and explain your reasoning; it is not OK if you just write down numbers without explanation.

RGV

 

[Kinetica]

Hi! Can you please tell me your logic behind these calculations? This is what professor suggested but for me the net worth of 510,000 does not make sense because you invest 5,000 anyway. This leaves us with 505,000 at most.

This is the actual hint:

"The wording is somewhat ambiguous. However, take the problem to mean that in the three cases the net worth of the individual at the end is $510,000, $10,000, and $5,000 respectively".

Seems to me like neither one is correct. I don't know anything about finance, but I'd say making $500000 means your networth is $510,000, breaking even means it's $10000 and losing all means it's $0.

 

[Ray Vickson]

Hi! Can you please tell me your logic behind these calculations? This is what professor suggested but for me the net worth of 510,000 does not make sense because you invest 5,000 anyway. This leaves us with 505,000 at most.

This is the actual hint:

"The wording is somewhat ambiguous. However, take the problem to mean that in the three cases the net worth of the individual at the end is $510,000, $10,000, and $5,000 respectively".

I have taught similar courses in the past and would never have given the "hint" your instructor gave you. I don't understand where the figures come from. (Actually, I do understand, but do not agree.) They seem to not follow the worded description of the problem, at least if one uses the normal definition of terms like "break even", etc. However, you must try to use the instructor's interpretation rather than your own---unless you can really prove your case--- because for now the instructor has control over your life.

RGV

 

[Kinetica]

You know what - the instructor, who gave this hint is not my own instructor, I just happened to find her probability class online where she assigned this homework poblem. My instructor gives no hints.

The final net worth that you suggested, I still have to calculate its expected value instead of a linear utility function (which is not given but by common knowledge is just =aC)?

 

[Ray Vickson]

You know what - the instructor, who gave this hint is not my own instructor, I just happened to find her probability class online where she assigned this homework poblem. My instructor gives no hints.

The final net worth that you suggested, I still have to calculate its expected value instead of a linear utility function (which is not given but by common knowledge is just =aC)?

The expected value IS the EV of a linear utility function, because utility functions are equivalent up to scale and location; that is, you get the same optimal decisions using the functions u(x) and v(x) = a*u(x) + b for constants a, b with a > 0. Therefore, you might as well take a = 1 and b = 0.

RGV

 

[Kinetica]

OK, the only thing that bugs me in this problem is that I don't know

if I have to subtract $5000 of the "anyway" investment from each of the options (and multiply by 1/3?

OR I have to add $5000 of the amount that is left of 10 000 to each option, and then, multiply by 1/3?

 

[Ray Vickson]

OK, the only thing that bugs me in this problem is that I don't know

if I have to subtract $5000 of the "anyway" investment from each of the options (and multiply by 1/3?

OR I have to add $5000 of the amount that is left of 10 000 to each option, and then, multiply by 1/3?

Try to think about it systematically. In this problem the two "time scales" are very different (that is, if we don't invest, we have $10,000 now, but if we do invest, we might have about $500,000, but only several years from now). However, in the "invest" case, let's *pretend* that the outcome is determined right away, so to compare decisions we just look at the expected amount of money in our jeans when we go home today.

Option 1: no investment. We go home with $10,000.

Option 2: invest. We must immediately spend $5,000, so before any returns we are left with $5,000 in our jeans. With probability 1/3 we reap $500,000 (or maybe $505,000---depending on interpretation) and would go home with $505,000 or $510,000 at the end of the day. With probability 1/3 the investment breaks even; in the normal interpretation of this terminology, that means that we just get back our initial $5,000 investment and so we would go home with $10,000 in our pocket. With probability 1/3 we lose all our investment, so would get back nothing from the franchise; we would go home with $5,000. Since the $5,000 is common to all three outcomes, we could drop it when computing the expected value (then add it back at the end). Or, if you prefer, you could keep that extra $5,000 throughout; you would end up with the same numbers either way.

RGV

 

[Kinetica]

Thank you for the detailed explanation.

I did not know what to do with the part where you earn 5,000,000, either your investment of 5,000 disappears or you add it to the new earnings. Now, I see your point.

Thanks!