“Each efficient portfolio represents weights on each of the asset classes. The only constraint was that the sum of all the weights should be equal to one. If we add an additional constraint of no short sales, we restrict the value of the weights to be greater than or equal to zero. Here we have an additional constraint that the weight of each asset class be non-negative. In case of a constrained optimization (no short sales in this case), the minimum variance frontier changes from a smooth curve to one with corner portfolios.” I do not understand this concept of corner portfolios. I am fine till minimum variance frontier, then I’m lost. Please help
If there are short sales allowed: You pick the portfolio with the highest Sharpe ratio and combine it with the risk-free asset. Solve for the weight of the portfolio and the risk-free asset. If there are NO short sales allowed: Pick 2 corner portfolios in which the desired return falls in between the expected returns of the 2 corner portfolios. Solve for (Desired return) = (weight of P1) x (corner portfolio 1) + (weight of P2) x (corner portfolio 2) , where (weight of P1) + (weight of P1) = 1
So when do we choose the two corner portfolios and when do we choose the one with highest sharp?
Either if there’s short sales allowed or no short sales allowed
From my understanding, you choose two corner portfolios when there’s a restriction on short sale. Otherwise, choose the one with the highest sharp ratio, and have it combined with the rf instrument.
One caveat, if the highest sharpe ratio portfolio does not exceed the expected return and borrowing is not allowed then you choose the highest sharpe ratio that does exceed the expected return
i am praying this is on the exam. plug and chug baby.
seems derswap started out wanting to know why corners exist - ie why the EF is not a smooth curve. Here’s my take - if shorting is allowed, then you get a smoooth curve as the asset weights swing smoothly from positive to negative weights as you go up the return/risk scale (often huge numbers like +200% and -300%, etc) - silly values really, but it’s a theoretical model so you get silly numbers for the asset weights if there are no restrictions on shorting. Since most investors & mandates can’t short, the weights of each asset get clipped at zero, so they just drop out of the mix rather than being shorted (which would produce negative asset weights). As soon as an asset weight drops to zero and drops out of the mix, the EF drops below the curve and becomes a straight line. The corners are the points along the curve when asset weights drop to zero and just sit at zero weight, or where an asset moves from zero weight to a positive weight. The straight line section between each pair of adjacent corners represents a particular set of assets - where the asset weights move in a linear manner. At each corner the line changes direction slightly as an asset either drops to nil or comes back into the mix from nil. So if corner 4 has assets A, B & C in it, and corner 5 has assets B, C, D in it, then the line between corner 4 and 5 has assets that are common to both - ie assets B & C. At any point on that straight section you can work out the weights of B & C by simple linear interpolation from their weights B & C for any given target return or risk. that’s the basics - now for the exam tricks - here’s what I have picked up from going through past cfai exams back to 2003: 1. the question usually sets out a table showing the corner portfolios in order - usually starting from the highest risk/reward at the top - so it’s easy to pick the right section (between 2 corners) that meets the return and risk objectives 2. when it comes to interpolating the point on the straight line between the 2 selected corners - the trick is in the exact wording of the targets in the IPS: - if the IPS sets a MINIMUM return goal (eg “at least 10%”) and a fixed risk goal (eg “10% st.dev” or variance) then start with the stated RISK target and interpolate the portfolio return. - but if the IPS sets a single return goal (eg “10%”) and a MAXIMUM risk goal (eg “no more than 10% st.dev”) - then start with the stated RETURN target and interpolate the portfolio risk. 3. only use lend/invest along the capital allocation line if the table has a risk-free asset in it, and there are other hints - eg if the IPS says that the investor considers the Sharpe ratio to be a primary measure. In that case - pick the portfolio with the higest sharpe, then lend (invest) at the RFR (which will be given) to move down to the target return or risk goal - then interpolate the risk and return in the total portfolio. This will be more efficient than any corner or straight section. all the questions I did cover the same old things every year - so I’m hoping they come up again this time… hope this helps…
n&n, This was exactly what I was looking for. Thanks for the long explanation and tips. Bless you.
n&n Please help me to understand point 2. Even if quesiton sets a fixed target risk (eg “10% st.dev” or variance) it is kind of hard to work our the weights of the 2 corners portfolio as the equation for total risk is not as simple as for expected return. i.e SD^2= (W1* SD1)^2 + (W2*SD2)^2 + 2 (W1*W2*SD1*SD2) W1+W2=1 Do we really have to solve for W1 and W2 from this equation in the exam? Thanks
for corner portfolios…you just use the weighted average to calculate std deviation … similar to the expected return…
it’s 11pm and i’ve had a few beers so here goes… the aim is to either go for the maximum return for a given level of risk - or go for a given return for the lowest risk. So the first thing is to decide which is paramount - return or risk. In most examples I have done the return goal is fixed and the aim is to do it with the lowest risk - eg foundations/endowments - need to earn the spend rate (plus fees & inflation, etc), at the lowest risk, or individuals need to earn the living expenses (plus inflation, etc). So you look at the table to find the 2 corners that the required return falls between. Let’s say: corner 4: Return of 13% at risk of 15% - consists of 50% asset A + 50% asset B corner 5: return of 9% at risk of 10% - consists of 80% asset A + 20% asset B if the return goal = 10% you can work out the risk and the weights of A + B by simple linear interpolation along the straight line between the 2 corners: - Return goal = 10% which is 25% of the distance between corner 5 (return 9%) and corner 4 (return 13%) so the return goal = 75% of corner 5 + 25% of corner 4 = 10%. - portfolio weights for a return of 10% = 75% of corner 5 weights + 25% of corner 4 weights. so: - weight of asset A = 75% of corner 5 asset weight A (80%) + 25% of corner 4 weight A (50%) = 72.5% - weight of asset B = 75% of corner 5 asset weight B (20%) + 25% of corner 4 weight B (50%) = 27.5% - so the target portfolio = 72.5% asset A + 27.5% asset B = 100% total - Risk (st.dev) of target portfolio - can work out by one of 2 ways: 1) linear interpolation using the same weights (because it is a straight line you can use the linear interpolation approximation) - ie portfolio risk = 75% of corner 5 risk (10%) + 25% of corner 4 risk (15%) = 11.25%. 2) can also calculate it the long way if you know the risk and correlations (or covariances) of the individual asset classes. (ie weightA^2 * varianceA + weightB^2 * varianceB + 2* weightA*weightB*covarianceAB). If the line was a curve (ie a regular non-sign constrained efficient frontier) this variance/covariance method would be the only way to calculate portfolio risk - but since the restriction on shorting generates a straight line between pairs of corner portfolios, you can use the linear method. in most questions you are not given the individual asset class variances or covariances, so you just use the linear interpolation method which is much quicker anyway Sharpe ratio and the Capital Allocation Line— If corner 4 has the highest sharpe (or maybe another corner has the highest sharpe) and you are specifically given a risk free asset to invest in then you don’t travel down the line from corner 4 toward corner 5 - you draw a straight line betwen corner 4 (or the highest sharpe portfolio) and the risk free rate on the y-axis. Then travel down that line to work on the weights needed to achieve the required return (or risk) target. That way you retain the highest sharpe ratio rather than compromising the sharpe by mixing in part of the inferior corner portfolio 5. eg if Corner 4: Return of 13% at risk of 15% - consists of 50% asset A + 50% asset B (same as above) Risk free asset = return of say 3% (given) at a risk of 0% (by definition) then the target Return (10%) is 30% of the way down the capital allocation line between corner 4 and the RFR on the y-axis - so the weights of the portfolio = 70% of corner 4 portfolio + 30% of the risk free asset. Risk of target portfolio on the Capital Allocation Line = 70% of corner 4 risk (15%) + 30% of the risk free asset risk (0%) = 10.5% so, by traveling down the capital allocation line (and maintaining the highest sharpe ratio- which is the slope of the CAL) you can get the same target return (10%) at a lower portfolio risk (10.5%) than if you mixed corner portfolios 4 and 5 (risk of corner 4/5 mix = 11.25%). So using the CAL is superior because the sharpe (slope of the CAL) is higher. In the past exams I have worked through this stuff pretty much comes up all the time. NB all of this is much easier with a white-board - or the back of a beer coaster! hope this helps… now for another beer…!
mumu has it right on the money For corner portfolios you assume no correlation between them