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The following table is from page 171 of Fundamentals of Investing (11th edition) by Gitman, Joehnk, Smart. Please consider only the X, Y and XY columns (second, third, fifth).

Portfolio XY comprises assets X and Y in the proportion $2:1$. As you can see, while the average expected (here, "expected" is not used in the statistical sense, but to mean the forecast value) returns of assets X and Y have a standard deviation of $3.16$ and $6.32$ respectively, portfolio XY's expected return has a standard deviation of $0$!

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Some further context: The authors are trying to illustrate the power of diversification: by replacing $\frac13$ of the original quantity of X with Y, the expected return of the portfolio is increased, while its risk (the standard deviation of the expected return) is decreased.

But how can the portfolio's risk possibly become nil !? Can someone pinpoint what is amiss?

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There is no paradox. The textbook's conclusion is correct within its own framework. Zero standard derivation simply doesn't mean risk-free. In real market, $5-\sigma$ events occur much frequently than expected. The risks associated with these events are usually unhedgeable and cannot be measured properly using "standard derivation". People sometimes use another concept "value at risk" to compensate "standard derivation". The "value at risk" is simply an estimate of the maximum potential lose within 95% level of confidence. – achille hui Feb 1 at 22:51
@achillehui Thank you! Could you post this as an Answer? Also, I was wondering if my comment below at 22:56:44 makes sense to you? – Ryan Feb 1 at 23:18
Can't see the time stamp. No idea which comment you referred to. What you said below does make sense (to a certain extent). – achille hui Feb 2 at 3:56

3 Answers

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There is no paradox. The textbook's conclusion is correct within its own framework. Zero standard derivation simply doesn't mean risk-free. In real market, 5-$\sigma$ events occur much more frequently than expected. The risks associated with these events are usually unhedgeable and cannot be measured properly using "standard derivation". People sometimes use other tools like "value at risk" to compensate "standard derivation". The "value at risk" is simply an estimate of the maximum potential lose within 95% level of confidence.

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It's because $X$ and $Y$ are perfectly negatively correlated: $Y = 40 - 2X$. So a mix of $\frac{2X + Y}{3}$ is a constant, which is why its standard deviation is zero.

Edit:

As you mentioned, it seems paradoxical that the risk of a combination of two risky assets can be zero. This is because of the intuitive notion that $\mathrm{risk}(X+Y) = \mathrm{risk}(X)+\mathrm{risk}(Y)$. However, this is only true if $X$ and $Y$ are uncorrelated. The correct way to calculate the variance (more or less the same thing as risk) of the sum of two variables is shown here.

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Yes according to the textbook, X and Y have a correlation coefficient of -1. The problem though, is, while the computation seems straightforward, it doesn't seem correct that a portfolio comprising two risky assets is risk-free. Don't you smell a paradox? – Ryan Feb 1 at 21:55
Thanks. I wasn't expecting the combined risk to be higher; I was thrown off because zero is, well, zero, not $0.00001$. But I've since figured out that (a) all the data values are approximatations, and so is the $0$ std dev, (b) the forecast is merely an extrapolation of historical data, so even if we predict a $0.000001$ std dev and a return of 13.3%, this does not preclude the possibility that I can still suffer a 90% loss. Meaning, I was forgetting that we are dealing with data here, as opposed to inevitable combinatorial-type probability logic (am I making sense?). – Ryan Feb 1 at 22:56
(a) It's true that you'll never find perfectly negatively correlated stocks in real life; and (b) with a stddev that low, a 90% loss is next to impossible, but still has probability greater than 0, which I guess is your point – valtron Feb 1 at 23:06
BTW is there some term to distinguish data-based statistical inference and data-based probability distributions from theoretical probability distributions? – Ryan Feb 1 at 23:15
I guess the correct term for my point (b) is confidence interval. I was completely neglecting that even if the std dev is predicted to be $0$, it is still only so within a certain confidence interval. – Ryan Feb 1 at 23:26
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Hint:

Let's play a game, where I flip a coin and you can bet any amount on Heads or on Tails.
For every \$1 you bet on Heads, you get \$0.90 if it lands Heads, and lose the \$1 otherwise.
For every \$1 you bet on Tails, you get \$0.90 if it lands Tails, and lose the \$1 otherwise.

Assume you bet \$1 on Heads and \$1 on Tails:

  1. What is the probability of landing on Heads? What is the payoff when it lands on Heads?
  2. What is the probability of landing on Tails? What is the payoff when it lands on Tails?
  3. What is the payoff structure look like?
  4. What is the variance?
  5. What is the standard deviation?

No matter the outcome of the coin flip, you will get exactly \$0.90 each time. This is completely risk-free! You should play this game with me many many times.

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@Ryan the parentheses have been removed and elaborated. Work through the example. – Calvin Lin Feb 1 at 21:51
So the variance and standard deviation are both $0$, since the loss is constantly $20$cents. What is your point though? How can a portfolio comprised of two risky stocks have zero risk ?? – Ryan Feb 1 at 22:05
@Ryan The loss is constantly \$1.10 actually. Consider the risky stocks as betting \$1 on Heads, and betting \$1 on Tails. The combined portfolio has 0 risk. – Calvin Lin Feb 1 at 22:06
Oh yes sorry, it should be $\$1.1$. I know that the two stocks have a perfect un-correlation, hence their risk negating each other (and in this case, apparently cancelling out entirely). But is this reasonable though? I mean, you have two risky stocks. Even if the risk becomes 0.0000000000001%, there's still some risk. But to have zero risk? Oh ok, since the forecast are estimates, then the 0 std dev is consequently also an estimate; I shall accept my own such explanation for the time being. – Ryan Feb 1 at 22:10
@Ryan Think of this simple example. You place \$1000 with fund manager A, and \$1000 with fund manager B. Both of them charge you a fee of \$100 per year. A buys (and sells) some stocks. B does the exact same opposite of A at the exact same time (think of B as the counterparty to all of A's trades). What is your outcome given any set of stock prices in a year? Do you have any risk? Does it matter how risky the stocks are? – Calvin Lin Feb 1 at 22:16
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