Subject 8. Portfolio Expected Return and Variance

The expected return on a portfolio of assets is the market-weighted average of the expected returns on the individual assets in the portfolio. The variance of a portfolio's return consists of two components: the weighted average of the variance for individual assets and the weighted covariance between pairs of individual assets.

σ2(Rp) = w12σ2(R1) + w22σ2(R2) + 2w1w2Cov(R1, R2)

You have a portfolio of two mutual funds, A and B, with 75% invested in A.
E(RA) = 20%; E(RB) = 12%.
Covariance Matrix:

The values on the main diagonal are the variances and the other values are the covariances.

The expected return on the portfolio is:
E(Rp) = wA E(RA) + (1 - wA) E(RB) = 0.75 x 20% + 0.25 x 12% = 18%

The correlation matrix:
σ(RA) = (625)1/2 = 25, σ (RB) = (196)1/2 = 14
ρ(RA, RB) = Cov(RA, RB) / [σ(RA) x σ(RB)] = 120 / (25 x 14) = 0.342857, or 0.34

The variance of the portfolio is:
σ2(RP) = wA2σ2(RA) + wB2σ2(RB) + 2wAwBCov(RA, RB)
= (0.75)2(625) + (0.25)2(196) + 2(0.75)(0.25)(120) = 408.8125

The standard deviation is σ(RP) = (408.8125)1/2 = 20.22%.

It's also possible that you could be given a correlation matrix, which is simply a matrix that shows the correlation between any two assets in the portfolio. Consider the following correlation matrix for assets A, B and C.

Note that the matrix is symmetrical about its main diagonal (top left to bottom right). The entries on this diagonal are all 1, as the correlation between any variable and itself is obviously 1. Similarly, the correlation between RA and RB is 0.53, the correlation between RA and RC is 0.78, and the correlation between RB and RC is 0.6.

The steps that would now be involved would be:

  • Calculate expected values and variances for the return on each asset.
  • Square-root your variances in each case to get standard deviations.
  • Use the standard deviations together with the correlations from the matrix above to calculate covariances using the link formula.
  • Calculate the values of the portfolio weights.
  • Now calculate E(Rp) and Var(Rp) using the above formula.

Essentially, the processes are the same. In each case, we need to obtain expected values, variances and covariances, in order to calculate E(Rp) and Var(Rp). How we obtain them depends on how the data are presented to us.

Familiarize yourself with the two different types of matrices, as explained in this section, and know what each term represents in each covariance formula.

User Contributed Comments 11

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anup5: Could somebody explain the detailed steps for the steps when we have a correlation matrix. eg : How to calculate variances ?
julescruis: I dont understand the covariance matrix, anyone? (first graph way up)
epizi: It is quite simple
Remember Cov(Ra,Ra)is same as Var(Ra)
Also Cov(Ra,Rb)=Cov(Rb,Ra)
So at the diagonals you will always find this common Cov(Ra,Ra),Cov(Rb,Rb) and Cov(Rc,Rc).
Note that you can always have this in mind able and estimate the number of covariance that can be found in the set of various variables.

Hope is clear now
epizi: Sorry I explain something else,I think the first table is given, but if you must caculate the entries you may have to go back to the previous note just before this.
CraigSaba: covariance matrix from example above - the covariance b/w A & B is 120. The variance of A is 625, of B is 196
challenge10: Anyone know how to derive the correlation of Rc with the portfolio Return of A & B in order to calculate the portfolio expected return of all 3 assets? Cheers as I'm wondering whether it would come up in the exam.
jorgeman81: As far as I understand, in this case the correlation matrix is given you can't calculate it with the given data.
Morginador: How do you calculate the variance out of the correlation matrix?
ybavly: I think jorgeman81 is correct. we need more information to calculate expected values and variance from the correlation matrix.
onyameba: Can anybody help solve these questions?
1.If 100 random samples of size n were drawn and, if, for each, a 99% confidence interval for u was computed, then exactly 99 of these confidence intervals would contain the true population mean within their limits.
2.A dormitory on campus houses 200 students, 120 are male, 50 are upper division students, and 40 are upper division male students. A student is selected at random. Find the probability of selecting a lower division student, given the student is a female.
3. It is desired to test the claim that a steady diet of a medicine will cause a male to lose 10 lbs over 5 months. A random sample of 49 males was taken yielding an average weight loss over 5 months of 12.5 pounds, with s = 7 lbs.
a. Identify the critical value suitable for conducting a two-tail test of the hypothesis at the 2% level.
b. Is there any significant evidence that the pharmaceutical company claim about the drug performance is not valid, given an p=0.02? Show your work.
CJHughes: Based on the notes given, I am a little confused how to back-out all of the last steps from the Correlation Matrix? If anyone could offer some assistance it would be greatly appreciated. Thanks