Expected return for n = ∞, normally dist. assets, portfolio theory

[slakedlime]

Homework Statement

My economics exam is in a few days. My professor posted solutions to a sample final, and I'm confused by one of the answers. I won't have access to him before the exam, so I can't ask him to clarify. I'm hoping that someone here can help.

____

QUESTION:
You have $100 at hand to invest. You can invest in many assets with independent returns over the next month following a normal distribution N ~ (\bar{R}, σ^{2}_{R}).

1) If you choose a portfolio with equal weight on n such assets. What will be the distribution of your wealth at the end of the month?

2) Use this calculation to illustrate the benefits of diversification. In particular, if you are risk averse, what value of n you would choose.2. Homework Equations & attempt at a solution

Please see the attachment for my professor's solution. What I don't understand is highlighted in green. For n = ∞, why is the expected return at least 100(1+\bar{R}) and not simply 100\bar{R}, which is the case for n in general (as my professor has shown)?

I understand why the standard deviation of the portfolio approaches zero as n approaches infinity, and why a portfolio of n assets is (theoretically) riskless.

____

An explanation, or guidance would be much appreciated.

 

[BruceW]

I think you are right that the return is $100\bar{R}$. And what your professor actually says is: "I would get $100(1+\bar{R})$ for sure at the end of the period". So, if he is not talking about the return, what is he talking about here? Think what does $100(1+\bar{R})$ correspond to? (I'm guessing that essentially this is just a problem of slightly vague wording).

 

[slakedlime]

Thank you for your reply Bruce. If we break down 100(1+\bar{R}) into its components, we have 100 + 100\bar{R}. Is my professor trying to say that this is the final accumulation that I end up with? That is, i) I will have my original $100. ii) on top of that, I will have returns that should theoretically equal 100\bar{R}? That is, I can expect to always keep my original $100 in this scenario because my portfolio is "riskless".

If I don't have a riskless portfolio, i.e. I invest n times, then I won't be able to keep my original $100 for certain. Is that the case? If yes, how would I calculate the final accumulated amount? Would that simply be my returns, i.e. 100\bar{R}, or would I be keeping some part of my original investment on top of that?

Please let me know if I am understanding you correctly, or if I have misunderstood something. Thank you.

 

[Ray Vickson]

Homework Statement

My economics exam is in a few days. My professor posted solutions to a sample final, and I'm confused by one of the answers. I won't have access to him before the exam, so I can't ask him to clarify. I'm hoping that someone here can help.

____

QUESTION:
You have $100 at hand to invest. You can invest in many assets with independent returns over the next month following a normal distribution N ~ (\bar{R}, σ^{2}_{R}).

1) If you choose a portfolio with equal weight on n such assets. What will be the distribution of your wealth at the end of the month?

2) Use this calculation to illustrate the benefits of diversification. In particular, if you are risk averse, what value of n you would choose.2. Homework Equations & attempt at a solution

Please see the attachment for my professor's solution. What I don't understand is highlighted in green. For n = ∞, why is the expected return at least 100(1+\bar{R}) and not simply 100\bar{R}, which is the case for n in general (as my professor has shown)?

I understand why the standard deviation of the portfolio approaches zero as n approaches infinity, and why a portfolio of n assets is (theoretically) riskless.

____

An explanation, or guidance would be much appreciated.

I think there is an error in the document. Look at the last sentence of question 1, where it says "...your wealth at the end is $100 N(\bar{R}, \sigma_R^2/n)$. So, it implies that you start with $100 and end up with $100X, where $X \sim N(\bar{R}, \sigma_R^2/n)$. For $n \to \infty$ this clearly implies that your wealth will be $100 \bar{R}$. Question 2 contradicts this; it implies that your wealth will be $100 (1+\bar{R})$.

Added note: for the situation describe in question 1, the investment is riskless when $n = \infty$; you end up with all of your original $100, plus an earned amount of $100(\bar{R}-1)$. That means that if $\bar{R} > 1$ you make a profit for sure.

 

[BruceW]

Thank you for your reply Bruce. If we break down 100(1+\bar{R}) into its components, we have 100 + 100\bar{R}. Is my professor trying to say that this is the final accumulation that I end up with? That is, i) I will have my original $100. ii) on top of that, I will have returns that should theoretically equal 100\bar{R}? That is, I can expect to always keep my original $100 in this scenario because my portfolio is "riskless".

yes. That's what I think he is saying. Your return is (almost) certain to be $100\bar{R}$ and your original investment is $100$, so at the end, the money you will have is (almost) certain to be $100+100\bar{R}$.

If I don't have a riskless portfolio, i.e. I invest n times, then I won't be able to keep my original $100 for certain. Is that the case? If yes, how would I calculate the final accumulated amount? Would that simply be my returns, i.e. 100R¯, or would I be keeping some part of my original investment on top of that?

well, if the portfolio has some risk, then (as the professor's solution says), the return will be some random amount $100R^p$ this could be 10, or a million, or negative a million, or anything. So, true, in some cases, you will end up with less than the original $100$, and in some cases, you will even end up with a negative amount! You know that the average return is $100\bar{R}$, so the average money you end up with is $100+100\bar{R}$, i.e. the same as for the riskless portfolio.

 

[BruceW]

I think there is an error in the document. Look at the last sentence of question 1, where it says "...your wealth at the end is $100 N(\bar{R}, \sigma_R^2/n)$. So, it implies that you start with $100 and end up with $100X, where $X \sim N(\bar{R}, \sigma_R^2/n)$. For $n \to \infty$ this clearly implies that your wealth will be $100 \bar{R}$. Question 2 contradicts this; it implies that your wealth will be $100 (1+\bar{R})$.

I think it is just bad wording. When it says "your wealth is" I think it means "the return will be". And when it says "I would get $100(1+\bar{R})$ for sure at the end of the period" I think it means "the total money I will have at the end". (since return is how much you gain, not the total money).

 

[Ray Vickson]

I think it is just bad wording. When it says "your wealth is" I think it means "the return will be". And when it says "I would get $100(1+\bar{R})$ for sure at the end of the period" I think it means "the total money I will have at the end". (since return is how much you gain, not the total money).

Yes, I do understand the differences, but the OP is confused by it. I maintain that the document--exactly as written--has errors in it.

 

[slakedlime]

Thank you both for your answers. I was really confused by the poor wording of the document, but I think I understand what my professor is trying to say. :)