Learning outcomes. After watching this video,

you will be able to: explain why we cannot diversify away all risk,

explain the difference between idiosyncratic and systematic risk.

Diversification revisited.

We earlier saw that having risky assets that are not perfectly

positively correlated helps us reduce or diversify risk.

Does that mean that we can endlessly reduce risk by

simply adding more risky assets to our portfolio?

This is going to be the focus of this video.

Continuing with our portfolio with three risky assets,

we can move the efficient frontier to

the left by adding more risky assets to our portfolio.

Will the efficient frontier eventually touch

the vertical axis as we keep adding additional risky assets to our portfolio?

That is, can we completely diversify away our risk?

In other words, can we make it zero?

In general, the answer is no.

We can prove this in a couple of steps.

The variance of a portfolio with n assets is

sigma-squared_sub_P (σₚ²) which equals the summation of i equal to

one to n w_i-squared sigma_i-squared plus

the summation of i equal to one to n and the summation of j equal to one to n,

i not being equal to j w_i times w_j times sigma_ij,

where w_i is the weight of asset i in the portfolio,

sigma_i-squared is the variance of its returns,

and sigma_ij is the covariance between asset i and asset j returns.

Now let's assume that the weight of each asset in the portfolio is the same,

which means that each w_i is equal to one over

n. Let's further assume that all assets have the same variance v,

and each pair of assets has the same covariance c.

This simplifies the expression of the portfolio variance to n times one

over n-squared times v plus n times n minus one times one over n-squared times

c. This can further be reduced to v over n

plus n minus one over n times c. As n goes to infinity,

a really large number,

the first term becomes zero and the second one simply

becomes c. This tells us two things about large,

well-diversified portfolio of assets.

One, only covariances matter and not so much the variances.

Two, if the covariance is not zero,

all risk cannot be diversified away.

That is, we cannot make risk zero

even if we add a large number of assets to our portfolio.

This risk that cannot be diversified away is called systematic or non-diversifiable risk.

The part that can be diversified away is called idiosyncratic or diversifiable risk.

As you can see in the figure now,

increasing the number of stocks in the portfolio reduces its variance.

The blue curve represents the total variance,

while the horizontal black line represents the systematic risk.

The gap between the blue curve and the black line

is the idiosyncratic or diversifiable risk.

As you can see, the diversifiable risk drops

pretty quickly as the number of stocks in the portfolio increases,

but beyond 50 to 60 stocks,

one cannot achieve any additional diversification.

At this point, the idiosyncratic risk is close to zero.

Diversification is akin to the old saying,

"Don't put all your eggs in one basket.

If you drop the basket, all the eggs break."

Here, diversification says don't put all your money in one asset.

If that asset performs poorly,

you will lose all your wealth.

Spread your wealth across various assets.

The less-than-perfect positive correlation between

these assets will help reduce your risk.

So far, we have looked at investment opportunity sets that involve only risky assets.

What happens to the investment opportunity set if we introduce a risk-free asset?