Standard Deviation Increase and Decrease

Hi, I have answers to the question, but I don’t know how to figure out how they came up with it.

A portfolio gives at 10% return with a standard deviation of 18%. You would like the standard deviation to drop to 14%. What should you do? What should you do if you want the standard deviation to rise to 23%. The answer to the first part is to add more risk-free assets until they account for 4/18 of the portfolio.
The answer to the second part is use debt to finance an increase in the size of this portfolio by 5/18. I’ve been working on this for over 2 hours and cannot understand.

You’re simply moving right or left along the capital allocation line (CAL).

Is there some kind of formula to use?

\sigma_{target} = w\left(\sigma_{risk-free\ asset}\right) + \left(1 - w\right)\left(\sigma_{portfolio}\right)
\sigma_{target} = w\left(0\right) + \left(1 - w\right)\left(\sigma_{portfolio}\right) = \left(1 - w\right)\left(\sigma_{portfolio}\right)


Recall from Probability Theory that
Variance (aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2*a*b*Cov(X,Y)
In this type of problem, X is the return of a risky portfolio, Y is the return on risk-free securities, and a, and b are the percentages invested in them.
If we let
R = expected return of the risky portfolio,
r_f = expected return of the risk-free security
x = percent of your funds invested in the portfolio, with the remaining funds invested in the risk-free securities.
note: I’m calling x what S2000magician is calling 1-w

Then the variance of this
Variance (xR + (1-x)r_f) = x^2Var(R) + (1-x)^2Var(Rf) + 2x(1-x)Cov(R, Rf)
Now, because risk-free securities have no risk, its variance is zero, which means the last two terms drops out. This means:
Variance (xR + (1-x)r_f) = x^2Var(R)
or Volatility(xR + (1-x)r_f) = x*Volatility(R)

The left hand side is the resulting volatility of the portfolio with x percent invested in risky portfolio.

This problem gives Volatility® = .18
and is asking for (1-x) such that Volatility(xR + (1-x)r_f) is .14 or .23.
All you have to do is solve for x.
For example, to get a volatility of .14:

.14 = x*.18
thus x = 14/18. therefore you need to invest 1-x = 4/18 of your funds in risk-free securities.

If you try to solve .23 = x*.18, you will find that 1-x is negative. This means that you have to borrow funds to achieve the desired volatility.

on a side note, maybe I’m still half-awake, but the 1st line in @s2000magician’s derivation doesn’t seem quite right. It just happens to give the correct result because the risk-free has zero volatility.

I just happened to take that into account when I wrote it.

Simplify the complicated side. Don’t complify the simplicated side.

Thanks, you really helped me to understand how this is done! I really want to understand these concepts so it makes me feel good when I can see and grasp it but I’ll admit I struggle with a lot of it. One more question. How do you calculate the total risk and coefficient of a share? I was able to calculate the return pretty easily.

If by “total risk” you mean the standard deviation of returns, n_a_C told you for a portfolio that contains two securities: S_1 and S_2:

\sigma_{w_1S_1+w_2S_2}^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2 \rho_{1,2}


\sigma_{w_1S_1+w_2S_2} = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{1,2}}

I’m not sure what you mean by “coefficient of a share”. Which coefficient, exactly?

Here is the problem.
Calculate the return on the ENI share and on the Italian index over 13 months until 1 Jan 2011.

ENI: JAN 10: 16.93 FEB 10: 16.57 MAR 10: 17.37 APR 10: 16.86 MAY 10: 15.2 JUN 10: 15.19
JUL 10: 15.69 AUG 10: 15.67 SEP 10: 15.83 OCT 10: 16.19 NOV 10: 15.50 DEC 10: 16.34 JAN: 17.30

What is the total risk of the ENI share? What is the beta coefficient of ENI? What portion of the total risk is explained by market risk?

I was able to figure out the return on the ENI share by subtracting the beginning price from the end price - 1 = 2.2% I don’t understand how to figure out the rest.

I was also able to figure out the periodic returns for the ENI shares by taking the previous month’s closing price and subtracting the current closing price / previous closing price x 100

Without the market prices you cannot compute beta.

So here, would that be the Italian index?

JAN 10: 21896 FEB 10: 21068 MAR 10: 22847 APR 10: 21562 MAY 10: 19543 JUN 10: 19311 JUL 10: 21021 AUG 10: 19734 SEP 10: 20505 OCT 10: 21450 NOV 10: 19105 DEC 10: 20173 JAN: 22050

With those, you can calculate the monthly returns on the index and the monthly returns on ENI. Use the latter to calculate the standard deviation of returns for ENI. Use both to calculate ENI’s beta. You know the formula for beta, yes?

I think so. I was able to calculate the returns for both pretty easily. I got stuck on the beta and risk.

Use Excel.

You can use the formula for beta, or use the built-in SLOPE function.

I was able to figure it out with your help, thanks for your guidance!