So, now, what I want to illustrate is
using the risk budgeting stuff for
volatility, we're going to come up with a,
a nice summary measure of an asset's
contribution of portfolio volatility.
And the summary measure's going to be
called Beta.
Beta plays a very key role in, in finance
as, as we'll see.
And, and So, how, how does this work?
So, we were looking at a portfolio of
assets right, just a weighted average with
our portfolio weights, and we.
Derive, you know, the following result.
We use Euler's theorem to additively
decompose volatility into its component
pieces.
And We showed that the derivative of
portfolio volatility with respect to the
portfolio weights is the co-variance
matrix times the vector weights divided by
volatility.
And an assets marginal contribution was
the eye throw of this thing.
Okay.
Now, what we're going to do is, with a
little bit of algebra, we're going to
derive an alternative expression for an
asset's marginal contribution to risk.
Okay?
So we wanna get a, a, an easier expression
for this de-, partial derivative, rather
than saying it's the ith row of this
thing.
Okay?
And, and so I'm going to show you the
result first and then give the derivation.
So the result is going to be that, an, an
asset's, marginal contribution to risk is
going to be directly proportional to
something, called its beta.
So the beta of asset I, with respect to,
the portfolio is defined as the covariance
between asset i, and the return on the
portfolio divided by the variance of the
return on the portfolio.
Okay?
So it's, And this is key.
So, when we talk about a beta.
And.
The beta is always with respect to
something else.
So this is beta with respect to portfolio
with portfolio weights X, okay?
And in this context asset I is actually a
member of this portfolio, okay?
And it's just co variance divided by
variance.
Now next week we'll see that beta is in
fact a linear regression coefficient. If
you regress the returns of S at I, on the
portfolio returns, the regression
co-efficient beta is this covariance
divided by variance, so.
And so that's the definition of beta.
And the result that we're going to see is
the following.
Beta for asset I, measures asset I's
contribution to portfolio volatility.
That is, we will show that the marginal
contribution to risk of asset I is equal
to its beta multiplied by portfolio
volatility.
An asset's contribution to risk is the
portfolio allocation times beta, times
volatility.
And the percent contribution to risk is
its allocation weight multiplied by beta.
So the key thing here, when we think about
percent contribution to risk of asset i
it's, it's allocation weight multiplied by
it's beta.
So beta is really the, the key thing that
summarizes an asset contribution.
So I'm going to make some comments.
So by construction, the beta of the
portfolio is one.
So we talk about.
You know, well, what would be the beta of
the portfolio?
The beta of the portfolio would be the
co-variance of the return on the portfolio
with itself divided by the variance of the
portfolio.
So the co-variance of the, of the
portfolio with itself is just the
variance, so we get the variance divided
by the variance which is one.
Right?
So if we use the definition of beta and we
talk about what is the beta of a portfolio
with respect to itself it's one.
And intuitively that should make sense.
So you want to think of well, you know,
what is the contribution of the portfolio
to its volatility.
Well, it's equal to its allocation which
is one, times its beta, which is one.
So, this is important because when we talk
the, the value of one for beta is kind of
a benchmark.
If an asset has a beta equal to one, then
its risk from a portfolio context is the
same as the portfolio.
Because its contribution to the portfolio
volatility, you know, is, is propor-, is
its beta times its allocation weight.
Okay.
So, when we think about the case when an
asset beta is equal to one,
Then it's marginal contribution risk is
equal to, you know, beta times portfolio
volatility.
And when beta is equal one, so then the
marginal contribution risk is just
portfolio volatility.
I mean, so notice that, you know, this is
saying how much does the portfolio
volatility change when we incrementally
increase its allocation in the portfolio.
Well, the portfolio volatility changes by
exactly the, the, the, the portfolio
volatility itself.
The contributional risk of an asset when
its beta is one, is its allocation weight
times portfolio volatility, right?
So if an asset is you have ten percent of,
of, of an asset in a portfolio its
contribution risk would then be
ten percent of the portfolio volatility.
And then its percent contribution risk
when beta is equal to one, the percent
contribution risk is its allocation
weight.
Alright.
So if you have beta one, ten percent
portfolio allocation, its percent
allocation, its percent contribution to
the risk of the portfolio is, is, is ten%.
Now, if a be-, if an asset beta's greater
than one, then essen-, then intuitively
you should be thinking that the asset has
greater risk than the portfolio.
Now.
How we want to view this is, is the
following.
So think about, beta's a measure of
portfolio risk because it's looking at the
covariance between the return on the asset
and the portfolio.
Well the portfolio has N assets in it.
So this covariance between asset i and the
portfolio is really the covariance between
asset i and asset one, and asset i and
asset two, and asset one and asset N.
So beta is really summarizing the variance
N contribution and all the covariance
contributions as well.
Okay.
Now if a, if a Beta's greater than one,
then that's saying, like, asset i has a
large covariance with all of the acids in
the portfolio.
So, that when you increase it's
allocation, you kind of increase the
average covariance in the portfolio and
that's gonna increase it's riskiness.
So when an asset beta is greater than one,
its marginal contribution to risk is
greater than portfolio volatility.
Its contribution risk is greater than its
allocation weight times portfolio
volatility and it's percent contribution
risk is going to be bigger than its, its
asset allocation.
And similarly when an asset beta's less
than one.
Its marginal contribution is, is less than
portfolio volatility.
Its contribution to risk is less than the
asset allocation times volatility.
And its percentage contribution to risk is
less than its allocation rate.
So the key thing about beta is it, it's a
measure of an asset contribution portfolio
risk and a, and a, the benchmark value of
one you know, kind of tells you, you know,
is the asset riskier or less risky than
the portfolio?
So when you have a,
You know, assets with beta less than one
are less risky to the portfolio, and in
summary sense or like diversifiers.
Right, so if you have an asset with a beta
less then one and, and, and you increase
your allocation to the portfolio, you're
actually going to decrease the portfolio
risk.
Where if you have an asset with a beta
greater than one and you increase the
allocation then you're going to increase
the volatility of the portfolio.
So this really emphasizes the, the notion
of,
Of riskiness in a portfolio context.
Because, I mean, when you form a portfolio
you're, you're diversifying, right.
You're trying to spread your risks out,
but not all assets are diversifiers.
If an asset is very high correlated with
everything else it's not a diversifier,
you know.
It, it is actually risk increaser, and so
beta is kind of measuring when an asset is
a diversifier versus when it isn't.
Okay.
So that's the intuition about beta and so
now we'll do the derivation of the result,
which is, it's, it's a little tedious but
it's not difficult.
But it's informative to go through the
derivation because it enforces some of the
I don't know, some of the rules of working
with random variables that we've done so
far throughout the course.
All right.
So.
The, what we're, again, what we're trying
to derive here is.
Is this result.
That the marginal contra risk to asset I
is equal to its beta multiplied by
portfolio volatility.
So this is, this is the key result that
was stated that we now want to actually
derive.
So when you take the vector of partial
derivatives, we did this, the derivative
of sigma P with respect to X is the
covariance matrix times X divided by
volatility.
Now if we write this thing out, you know,
longhand.
Let's actually look at the numerator.
If we look at Sigma times X what is that?
Well, we have the co-variance matrix that
looks like this, and X is just the vector
the portfolio weights.
Okay.
Now what I'm going to do is I'm going to
concentrate on the first row of this, of
this of this vector.
You know, so sigma one times X one, sigma
one that should be a sigma one squared.
What the heck did I do?
So the first row is going to be.
Sigma one squared times x1, sigma twelve
times x2 plus sigma 1n times xn.
So, that's, if we want to know what is the
marginal contribution of asset one to
portfolio volatility, then it's going to
be the first row of this divided by
portfolio volatility.
Consider the first row of sigma time X is
X1 sigma1 squared X2 sigma 1,2 plus XN
sigma 1N.
Okay.
So turns out that this here is, is in fact
the co-varince between asset one and the
portfolio Okay.
So why is that?
So, if we look at the co-variance between
asset one and the portfolio, the port, the
port, it's equal to the co-variance
between asset one, and X one R one plus X
two R two plus X N R N.
Right?
Because the portfolio has N pieces to it.
Now you had a homework assignment, I don't
know if you remember, that covariance is
additive.
So this covariance can be written as the
covariance between R1 and X1 R1 plus the
covariance between R1 and X2 plus the
covariance between R1 and X N R N.
And the covariance between R1 r! and X1 R1
is X1 times its variance.
Right?
Cuz the co-variance between R1 and itself
is its variance, and the co-variance of AX
and BY is AB times a co-variance between X
and Y.
And then the covariance between R1 and X2
R2 is X2 times sigma one two, and then the
covariance between R1 and XN RN is XN
times sigma one N.
So we see that the first row of sigma X is
the same as the covariance between R1 and
RP.
Alright.
Now, let's remember the definition of
beta.
Beta for asset one is the covariance
between asset one and the portfolio,
divided by the variance of the portfolio.
This probably should be a.
Okay.
And so we can write the numerator here.
The co-variance between R1 and the
portfolio is the beta multiplied by the
variance, okay?
So.
Again, by algebra, the co-variance between
R1 and RP is beta one times the variance.
But the co-variance between R1 and RP is
this,
Which is the first row of sigma X.
So we can get that the first row of sigma
times X is equal to beta one times
portfolio variance.
And so the marginal contribution to risk
of asset one which is the first row of
sigma x divided by portfolio volatility is
beta one times variance divided by
volatility.
And so that's beta one times volatility,
Okay. So this is the, the result that the
marginal contributional risk of asset one
is directly proportional to its beta.
And then using a similar analysis you can
get the marginal contribution whereas with
asset I, is just it's the Ith row of sigma
X, sigma times X divided by volatility,
and then we see that's beta I time times
volatility.
So again, what, what's key, what's nice
about this example where we do risk
budgeting with volatility, is that we can
re-express an assets marginal contribution
risk in terms of its beta.
And beta is a measure that summarizes an
asset's variance plus all the co-variance
contributions to, to the portfolio.
So it's a summary measure of how an asset
contributes to the variability of a
portfolio.
Alright.
Alright so some key points to take away,
about beta as a measure of portfolio risk.
So, the first point is that asset specific
risk can be diversified away by forming
portfolios.
So, the reason why we create a portfolio
is to try to spread our risks around.
And when we diversify a portfolio
diversification eliminates certain types
of risk but you don't eliminate all risk
when you form a diversified portfolio.
Some risks still remains, right?
So , you can have a portfolio of, of 5,000
assets but that portfolio is not
risk-free.
You know, the portfolio still has risk.
Alright and so what is the risk that
remains.
It turns out that.
Beta measures the risk that remains after
you do diversification.
So, what remains after you diversify a
portfolio, is, is called portfolio risk,
right.
[laugh].
And it's kind of like the, you know, in
any a sense, so, so, this portfolio risk
is actually measured by beta.
And beta again, is a measure , you know,
it has asset, it has a variance
contribution and the covariance
contributions.
So, that's that, that's what remains.
Now, when we think about.
The risk of a single asset.
Portfolio theory tells us the following.
If certain types of risks can be
diversified away.
Then when we view the riskiness of an
asset.
You know, if we think of a portfolio
context.
We should really think about how the asset
influenced the risk of the portfolio.
Not necessarily the it's stand alone risk.
Alright?
Because when you know, certain assets can
be you know, good in a portfolio context,
that is they could be a diversifier.
And some portfolios, some assets can be
bad in a portfolio context, that is you
know, they don't diversify very well at
all.
So when you think of you know, the
riskiness of an asset you can also think
of risk from a portfolio context.
And this is what beta does for us.
And the last one is beta measures the
portfolio risk of an asset.
Now.
This was recognized early on.
You know, I guess you know, people
developing finance theory in the 1960's.
And in particular William Sharpe who is
one of the Nobel Prize winners for his
work in the Capital Asset Pricing Model.
You know, again recognized that in the
context of forming portfolios you can
summarize an asset's portfolio risk using
data.
And this lead to the following conjecture.
And If Beta is, the appropriate measure of
risk, you know from the context of a
portfolio, and if we believe there is a
risk return trade off, that is assets with
higher risks should generate higher
expected returns, then perhaps the
expected return on assets should be
related to its beta, right'cause assets
with high beta are risky from a portfolio
context, so they should generate a higher
risk premium, and assets with low betas
are less risky in a portfolio context, so,
you know they should not.
You know, generate a high-risk premium.
And so.
Willam Sahrpe and others essentially
conjectured if, if the data of an asset
with respect to a portfolio is the
appropriate measure of the risk of an
asset then the asset's expected return
should, should depend upon its data.
So, the expected return on the asset
should be some function of data.
And this is what essentially formalized in
the capital asset pricing model Capital
asset pricing model says the risk return
trade-off between assets, is dictated by
its beta.
Assets with high beta should, should give,
give you high risk premiums, assets with
low betas should give you low risk
premiums.