# Two assets with identical expected returns and STD's

Assume that Assets A and B have expected returns of 8% and standard deviations of 16%. Which of the following statements about the standard deviation of returns for a portfolio comprised of Assets A and B is most accurate? a. The weight of each asset as a percentage of the total portfolio is needed to derive the portfolio standard deviation b. The covariance between the correlation of the two assets is needed to derive the portfolio standard deviation. c. The standard deviation of the portfolio will be less than the sum of the asset-weighted standard deviations of each of the two assets. d. The standard deviation of the portfolio will be more than the sum of the asset-weighted standard deviations of each of the two assets. This completely stumped me! If the two assets have the same risk wouldn’t the protfolio have the same risk, regardless of the weight of each? Could you explain?

What if they had correlation = -1? E.g., A is long Yen and B is short Yen - same vol…

You need both (a) and (b) to calculate SD of the portfolio, so none pf these answers independently is sufficient. Correlation cannot be greater than 1 in absolute value, so consider this: (a+b)^2= a^2+b^2+2ab a=w1*SD1 = asset weighted SD asset 1 b=w2*SD2 = asset weighted SD asset 2 2ab=2*w1*SD1*w2*SD2, multiply this with a factor smaller than 1 (that being correlation) Add them all and you obtain a value lower than (a+b)^2. Square root it, you would obtain the same value is the correlation between the assets is 1, or a smaller value if the correlation is lower than 1. Is the answer C?

Joey: I’m still not there. The two assets have same expected return and same expected sigma. If r=-1.00, that does not change sigma. If you put 100% of the portfolio into Asset A, what difference would it make? The return is the same, and sigma is the same, and vice versa for asset B. Putting half here and half there, still return and sigma are same. I must be missing something big Map1: The answer is not C.

Disclaimer: I am not stating anything below that any one else already does not know. I am trying to solve my way to an answer below. var port = wa^2*sa^2 + wb^2 * sb^2 + 2*wa*wb*cov(ab) cov (ab) = cor (ab) * sa * sb Now I am not sure of the term --> covariance between the correlation of the two assets. So given (b) contains this – I am inclined to say B is wrong. If cor(ab) = 1 --> sigma port = wa*sa + wb * sb because the other term becomes 2*wa*wb*sa*sb If cor(ab) = -1 sigma port = wa*sa - wb * sb for any cor(ab) between -1 and +1 --> c) is correct. But this is not always correct. So © is not always right. The sigma port can never be > wa * sa + wb * sb. So (d) is wrong. Given this – I would think (a) is the right choice. Is that the answer, Dreary?

cpk123, yes the answer is c, and I think you are right, although it is a bit unintuitive. Lets think about it: Since sa = sb, this means: 1) sigma portf = (wa + wb) s (if r = 1.00, where s=sa=sb). 2) sigma portf = (wa - wb) s (if r = -1.00, where s=sa=sb). So, if wa = wb, i.e., 50% in A and 50% in B, the portfolio’s sigma becomes 0! That is, if you combine two assets with equal weighting in a portfolio, where they are perfectly negatively correlated, and their sigma’s are equal, you will get rid of risk completely. Makes sense. What if r=0? I could work it out, but I think sigma of portfolio will be teh weighted average of the two sigma’s.

Is it (a) or ©. You responded to map1 saying © is wrong. But in response to mine, you have said © is right. CP

Sorry, I meant a is right.

Nasty:)

I think it’s none of the above. a) Yep you need this b) Well you need the covariance or the correlation not “The covariance between the correlation” c) I’d like this with a “less than or equal” d) just way out

Dreary Wrote: ------------------------------------------------------- > Assume that Assets A and B have expected returns > of 8% and standard deviations of 16%. Which of > the following statements about the standard > deviation of returns for a portfolio comprised of > Assets A and B is most accurate? > > This completely stumped me! If the two assets > have the same risk wouldn’t the protfolio have the > same risk, regardless of the weight of each? > Could you explain? My best bet is (a). Here’s my thinking… > a. The weight of each asset as a percentage of the > total portfolio is needed to derive the portfolio > standard deviation Remember the formula of the std deviation of a portfolio of assets (let’s assume two assets): without writing the equation, its sum of the the square of the individual weights times their respective variances plus 2 times the product of the weights times the covariance of both assets, ALL taken to the one-half power (did I get that right? oh well, you get the drift…). It’s enticing to think that weighting doesn’t matter when both assets provide similar measures. But that’s only true when calculating the expected return of the portfolio. With the standard deviation, more weight is given to larger weights due to the magnitude from squaring each variable. Try a few examples on your calculator and you’ll see what I mean. > b. The covariance between the correlation of the > two assets is needed to derive the portfolio > standard deviation. This is confusing at the least and misrepresented at best. You need the covariance between returns of the two assets, not the between correlation. Correlation is standardized form of covariance. The correlation of an asset with itself is 1. So are we trying to find the covariance of all the individual correlations? It doesn’t make sense. > c. The standard deviation of the portfolio will be > less than the sum of the asset-weighted standard > deviations of each of the two assets We don’t know for sure. It all depends on correlation. If correlation is 1, then we’re dealing a perfect positive relationship that’s equivalent to the asset weighted standard deviations. If it’s less than 1, then the the portfolio standard deviation will be lower. Remember the relationship: the lower you go in correlation, the lower the overall risk in the portfolio. > d. The standard deviation of the portfolio will be > more than the sum of the asset-weighted standard > deviations of each of the two assets. Not true. The asset weighted standard deviations occur at a correlation of 1. Correlation, by definition, cannot be greater than 1. And since portfolio standard deviation declines when correlation declines, it can’t be true that the standard deviation of the portfolio is greater than the asset weighted deviation.

By the way, not to drag out this topic, but here’s a simple proof as to why the portfolio standard deviation of two assets with a correlation of 1 simplifies down to the weighted standard deviation of each asset. Let’s start off with the variance of a portfolio: Var(Portfolio) = (w1^2)(s1^2) + (w2^2)(s2^2) + 2(w1)(w2)Cov[1,2] Remember that covariance = r(s1)(s2) We know in this case that r = 1, so let’s leave out r Now substitute… Var(Portfolio) = (w1^2)(s1^2) + (w2^2)(s2^2) + 2(w1)(w2)(s1)(s2) Let’s rearrange this: Var(Portfolio) = (w1^2)(s1^2) + 2(w1)(w2)(s1)(s2) + (w2^2)(s2^2) Notice the pattern? That’s right, it’s a perfect square trinomial, a^2 + 2ab + b^2. Factoring reduces the expression (using the pattern (a+b)^2)… Var(Portfolio) = [(w1)(s1) + (w2)(s2)]^2 Now take the square root to find the standard deviation… StdDev(Portfolio) = (w1)(s1) + (w2)(s2) There you have it. With a correlation between two assets of 1, our expression of portfolio standard deviation simplifies to the sum each asset’s weight times its standard deviation.

similarly when r=-1 w1s1 - w2s2 would be your answer. bcos the covariance term would become -2 w1 w2 s1 s2.

yes, that looks good, but What if r=0? Var(Portfolio) = (w1^2)(s1^2) + (w2^2)(s2^2) + 2(w1)(w2)Cov[1,2] since Cov=0, then Var(Portfolio) = (w1^2)(s1^2) + (w2^2)(s2^2) sigma = ? , how do you simplify that other than to: (w1s1+w2s2) * (w1s1-w2s2)?

that would not be right. w1^2*s1^2 + w2^2*s2^2 --> is a^2 + b^2 which is not (a+b) (a-b) (a+b) ( a-b) = a^2 - b^2 In the above case bcos s1 = s2 it becomes var portfolio = (w1^2+w2^2) * s^2 I do not think any further simplification can be done. CP

Factor using i. Imaginary portfolio weights. Imaginary P/L - very new millenium.

If the portfolio is only of the 2 assets, do you really need the weight of each? The weight of one only is enough.

There’s a point…

Waoh I really like this thread! Very insightful!

gdiddy Wrote: ------------------------------------------------------ > > We know in this case that r = 1, so let’s leave > out r > > How would you know it? Just because they both have the same STDV and expected values?