Portfolio standard deviation for 5 assets

CFA Level I
#1

Guys,

CFA Institute pretty clear talks about 2 and 3 assets in the portfolio, but what shall I do if I need to calculate standard deviation of a portfolio with 5 assets?

Which formula should I use?

Thanks in advance

#2

No specific formula given in the for that . So, I think you should use same formula in extended form for the calculation of 5 assets.

#3

Shall I use Covariation for all pairs?

#4

The general formula for the variance of an n-asset portfolio has two sets of terms (each is added):

  1. the weight of each asset times its variance

  2. twice the cross-weights of each pair of assets times the pair’s covariance

-think of it as (A+B+C+…)^squared – A^2 +B^2 +…+N^2 + 2AB + 2AC+ 2AD +…

So, adding an additional asset adds one “weighted variance” term to the formula but 2x the covariance of the asset with each existing portfolio asset. So, once you have more than a few assets, the important determinant of the impact of an additional asset on the portfolio’s risk is roughtly the asset’s average covariance with the existing portfolio assets. In fact, this concept is tested regularly on the L2 exam.

#5

First, it’s “covariance”, not “covariation”.

Second, CFA Institute will not give you a portfolio with five assets and ask you to compute the variance or standard deviation of returns; it will take you far too long and will not be better at showing that you’ve mastered the material than giving you a portfolio with two or three assets.

Third, the general formula for the variance of returns for a portfolio is:

σ²(port) = ΣΣw(i)w(j)σ(i)σ(j)ρ(i,j)

where the first sum is taken over all _i_s, and the second over all _j_s. Thus, for a 5-asset portfolio, the formula would be:

σ²(port) = w1²σ1² + w2²σ2² + w3²σ3² + w3²σ3² + w5²σ5²

  • 2w1w2σ1σ2ρ(1,2) + 2w1w3σ1σ3ρ(1,3) + 2w1w4σ1σ4ρ(1,4) + 2w1w5σ1σ5ρ(1,5)

  • 2w2w3σ2σ3ρ(2,3) + 2w2w4σ2σ4ρ(2,4) + 2w2w5σ2σ5ρ(2,5)

  • 2w3w4σ3σ4ρ(3,4) + 2w3w5σ3σ5ρ(3,5)

  • 2w4w5σ4σ5ρ(4,5)

In general, for n securities, there will be n terms with a single security’s variance, and n(n – 1)/2 terms with two securities’ standard deviations (and the correlation of returns for the pair); thus, the number of cross terms grows much faster than the number of securities.

#6

In real life - I have no idea.

For the test - they won’t ask you to calculate stdev for a 5-asset portfolio. You need to know how to do a 2-asset portfolio like the back of your hand. A three-asset portfolio might show up. They won’t do more than three.

#7

Okay, thanks!

#8

You’re welcome.

#9

i don’t think 3 assets ever shown up, correct me if i’m wrong, for calculating covariance anyway. but 2 assets for sure! actually you almost need to know the formula for L1 to 3.

#10

URGENT!!! - Remember, you must be able to do the 2-asset portfolio variance in your sleep. And you will need to remember how to do it in Levels 2 and 3, too. If you can’t crank this out in less than a minute, then save yourself a Saturday in June and stay home. (Ditto CAPM and Gordon Growth.)

I can’t stress this enough.

#11

I guess they would not give it for more than 3 assets because it would become extremely time consuming. Any ways, the general formula would require N variances and {N(N-1)}/2 pair wise co-variances or correlations. (Source: Kaplan Scheweser)

Hence we will require, 5 variances, and {5(5-1)}/2= 10 pair wise correlations or covariances

We will require following pair wise correlations or covariances, where the 1 is stock no 1, and 2 is stock no 2 and so on,

1,2

1,3

1,4

1,5

2,3

2,4

2,5

3,4

3,5

4,5

(Remember, the correlation between 1,3 is same as 3,1 and 2,4 is same as 4,2 etc.)

For five assets, the formula would become something like this

5 portfolio variance= {(w1^2)(s1^2) + (w2^2)(s2^2) + (w3^2)(s3^2) + (w4^2)(s4^2) + (w5^2)(s5^2)} + {2(w1)(w2)Cov1,2 + 2(w1)(w3)Cov1,3 + 2(w1)(w4)Cov1,4 + 2(w1)(w5)Cov1,5 + 2(w2)(w3)Cov2,3 + 2(w2)(w4)Cov2,4 + 2(w2)(w5)Cov2,5 + 2(w3)(w4)Co3,4 + 2(w3)(w5)Cov3,5 + 2(w4)(w5)Cov4,5}

Sorry if I get some part wrong in the formula, it is starting to look like Greek to me.

Consider the parts in italics and non-italics separately and it will become really easy.

This will give you the variance for 5 assets portfolio. Square root the result to get the standard deviation

#12

Yes it is extended form of formula you can use it for the calculation of 5 assets.

#13

[quote=“Finkid”]

I would definitely not recommend this, if ever you needed to do it in real life.

The simplest way is MATRIX MULTIPLICATION in a SPREADSHEET.

Will have a go at putting it in simple terms (WARNING -----may sound a little Math’y for those not interested):

  1. Say you have N assets with weights W1, W2…,WN and the usual correlations RHO(i,j) where i & j are 1, 2, 3….N.
  2. Create an Nx1 column vector of WiSi where i = 1, 2,…N, and W & S are respectively weights and standard deviations of asset i. Call this vector V
  3. Create an NxN correlation matrix , that is all RHO (i,j), and ensuring that the leading diagonal contains all 1 (i.e. RHO(i,i) =1). Call this matrix CORR.
  4. Calculate Portfolio Variance = Transpose (V) x CORR x V.
  5. If you parameterize your spreadsheet with V & CORR you can use the MMULT combined with Transpose functions in excel to get the answer almost immediately. DO NOT FORGET TO ENTER IT AS AN ARRAY FORMULA ( i.e. after entering the formula, then press "Ctrl+Shift+Enter")

Hopefully you find some use in this but it is acceptable to find it boring.