# Portfolio Standard Deviation

Portfolio standard deviation is the standard deviation of a portfolio of investments. It is a measure of total risk of the portfolio and an important input in calculation of Sharpe ratio.

One of the most basic principles of finance is that diversification leads to a reduction in risk unless there is a perfect correlation between the returns on the portfolio investments. Owing to the diversification benefits, standard deviation of a portfolio of investments (stocks, projects, etc.) should be lower than the weighted average of the standard deviations of the individual investments.

## Formula

Portfolio standard deviation for a two-asset portfolio is given by the following formula:

$$ Portfolio\ Standard\ Deviation (σ_p) \\ = \sqrt{{ω_A}^2{σ_A}^2+{ω_B}^2{σ_B}^2+2ω_Aσ_Aω_Bσ_B.ρ_{AB}} $$

In case of three assets, the formula is:

$$ Portfolio\ Standard\ Deviation (σ_p) \\ = \sqrt{ {ω_A}^2{σ_A}^2 + {ω_B}^2{σ_B}^2 + {ω_C}^2{σ_C}^2 \\+ 2ω_Aσ_Aω_Bσ_B.ρ_{AB} \\+ 2ω_Bσ_Bω_Cσ_C.ρ_{BC} \\+ 2ω_Cσ_Cω_Aσ_A.ρ_{CA} } $$

Where,

ω_{A} = weight of asset A in the portfolio;

ω_{B} = weight of asset B in the portfolio;

σ_{A} = standard deviation of asset A;

σ_{B} = standard deviation of asset B; and

ρ_{AB} = correlation coefficient between returns on asset A and asset B.

ρ_{BC} = correlation coefficient between returns on asset B and asset C.

ρ_{CA} = correlation coefficient between returns on asset C and asset A.

## Example

Okoso Arden is your friend. A year back he started following the stocks. In a finance article published in a magazine in those days, he read that the not-all-eggs-in-one-basket approach to investing is useful because it helps reduce risk. He started a portfolio with $2,000, invested 50% in Black Gold Inc., an energy company, and 50% in Bits and Bytes, an information technology firm.

Following statistics relate to these two investments:

Black Gold | Bits & Bytes | |
---|---|---|

Return | 7% | 15% |

Standard deviation | 10% | 20% |

Correlation coefficient between returns of BG & B&B is 0.6.

Okoso requested you to calculate for him the extent to which the risk was reduced by the strategy.

**Solution:**

We can illustrate the fact that diversification indeed reduces the risk level by finding the weighted average standard deviation of the investments and then finding the portfolio standard deviation after taking into account the correlation between the two investments.

Weighted portfolio standard deviation

= ω_{BG}×σ_{BG} + ω_{BB}×σ_{BB}

= 50%×10% + 50%×20% = 15%

Where,

ω_{BG} = weight of Black Gold;

ω_{BB} = weight of Bits & Bytes;

s_{BG} = standard deviation of Black Gold; and

s_{BB} = standard deviation of Bits & Bytes

The portfolio standard deviation after consideration of correlation:

$$ Portfolio\ Standard\ Deviation \\ = \sqrt{50\%^2×10\%^2+50\%^2×20\%^2+2×50\%×50\%×10\%×20\%×0.6} \\ =13.6\% $$

The portfolio standard deviation is 13.6%. The less than perfect correlation has reduced the standard deviation from 15% to 13.6% which indicates a reduction in risk: the benefit of diversification.