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Question on CML and Market Portfolio

Thread starter #1
Hi David,

I am studying the diagram of Expected Return vs Volatility of a portfolio of risk free asset + risky asset. The diagram showed CML and the market portfolio which I have a question on. Points along the CML (Between risk free return and market portfolio) is a portfolio consisting of a % of risk free asset and a % of Market portfolio. How can we create a portfolio which lays on points along the CML (beyond the market portfolio) which has a expected return greater than those on the efficient frontier? I thought the maximum expected return is a portfolio of 100% in risky asset.


Best Regards,
Dennis
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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#4
Yes, because it's a percentage of portfolio funds; e.g., 130/30 funds are short 30% and long 130%

You might find helpful the associated XLS: http://www.bionicturtle.com/premium/spreadsheet/1.a.2._cml_sml/
(first tab is CML)

Note in this XLS, the expected return of market portfolio is about 12.2%
(i.e., here the MP is merely the mix of Asset A&B with highest sharpe ration. Very simple universe of two assets)
So, if you have $100, and you invest 100% in MP, E(return) = 12.2%

if you look down to rows 37+ you see the *leveraged* allocations; e.g., 150% = E(return) of 14.8% because:

you invest $100 plus you borrow $50:
Future value of ($150 in market portfolio) = 150*1.122 = about $168.3
then payoff your loan @ $53.5 (= 50* 1.07 riskfree rate)
= about $114.8 in future value
= 114.8/100 = 14.8% return on your original $100

i think this is worth a meditation because this is leverage-in-action. Not unlike Jorion's leverage, Hull's, McDonald's;
e.g., you could borrow 90% and invest only $10 of your own to buy $100 of the Market portfolio - very highly leveraged,
or borrow (theoretically) 100% to buy $100 of the market portfolio, and we have created a synthetic forward: no initial investment. And then we have Jorion's "double edged sword" as this is very "efficient" (i.e., zero cost) but, on the other hand, zero "downpayment" is not our exposure so our exposure isn't as obvious as when we invest $100

David
 
#6
Hi, David,

In the Excel file (CML tab), what if the riskless rate is 9%. The CML will not carve a tangent to the efficient frontier, but pass through the point of market portfolio. CML will lie below the upward-sloping part of the the efficient frontier . How to interpret the result ? The investor will not borrow to invest in the makret portfolio ? The line of CML will not dominates the efficient frontier at any give level of risk ? and the CML will not be a straight line?


Daniel
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#7
Hi Daniel,

if you increase the RF rate, you have to recompute the market (optimal) portfolio as the market portfolio maximizes Sharpe ratio.
So, with my inputs (Asset A = 10% return & 10% sigma; Asset B = 15% return & 20% sigma; rho = 0.2), when I use SOLVER to maximize cell N3 (Sharpe ratio) by changing cell K3 (% allocated to Asset A), I get 22.2% to Asset A with Sharpe ratio of 0.3028,

And the CML line (thankfully) adjusts and the market portfolio point (red dot) moves. Visually, it looks to work as expected.
Screenshot below...
... so i think as long as the RF rate is < either risky rate (by definition), the CML will always connect the RF y-intercept with the market portfolio, but the market portfolio is influenced by the RF rate only because the numerator is excess return. David


http://www.bionicturtle.com/images/forum/0216_cml.png
 
#8
Hi David,

Your excel file once again prove that it is very helpful to illustrate concept by numbers and graph. It is very nice to visualize it. I find it so unique in BT that no other souce of training can be compared with.

Daniel
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#9
Daniel,

I am thrilled to hear this perspective, truly, because (i) it is a core belief of mine that learning risk/finance is enabled by a concrete dive into math and spreadsheets (where spreadsheets are just the common tool we use; I feel the same way about mathematica or matlab ... so, IMHO, the point is really about hands-on engagement with the concrete math that underlies finance) and (ii) because it has taken quite a bit of work/time to build them ... feedback like this encourages me to maintain and improve them!
...FWIW, in my own process to learn the FRM curriculum, one of the fascinating insights for me was that whenever I've knocked up against a hard idea (e.g., some of the Grinold stuff), the best way I found to "overcome" or learn the ideas was to try and replicate the illustrations in excel ... the excel like other math-based tools don't really tolerate fuzzy thinking in the way that sentences and paragraphs often do...

David
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#11
Hi Martin,

Thanks for asking about that: I made a note to clarify this non-obvious step on revision.
It applies property of covariance http://en.wikipedia.org/wiki/Covariance
... specifically: COV(aX + bY, cW + dV) = ac*COV(X,W) + ad*COV(X,V) + bc*COV(Y,W) + bd*COV(Y,V)
... not in Gujarati per se but generalizes from Gujarati's properties
... and we are here simplistically only using two assets where both the portfolio and the market are two-asset portfolios.

so given:
a = %weight Asset A in Market Portfolio M
b = %weight Asset B in Market Portfolio M; i.e, b = 1 - a
c = %weight Asset A in Portfolio P
d = %weight Asset B in Portfolio P; i.e., d = 1 -c
Ra = Return (Asset A)
Rb = Return (Asset B)

Unrealistically, both portfolios P & M are 2-asset portfolios, same assets, but different mixes:
Return (Market Portfolio M) = a*Ra + b*Rb
Return (Portfolio P) = c*Ra + d*Rb
Cov (M, P) = Cov(a*Ra + b*Rb, c*Ra + d*Rb), and per the above covariance property
= a*c*Cov(Ra,Ra) + a*d*COV(Ra,Rb) + b*c*COV(Rb, Ra) + b*d*COV(Rb, Rb)
...since COV(Ra,Ra) = variance (Ra) and COV(Rb,Rb) = variance(Rb)!
= ac*variance(Ra) + bd*variance(Rb) + covariance(Rb,Ra)*[ad + bc]

hope that helps, appreciate you noticed that step b/c i can include this explain in next XLS version...David
 

sl

Active Member
#13
Hi David

I am not sure how you arrived at the riskfree rate of 1.07 from original 7% while computing the loan on the leveraged amount. Why was 1 added to the 7%.

Thanks in advance

sundeep
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#14
Hi sl,

For the CML, I have (col E): weight_market portfolio%*E[return to market portfolio] + (1-weight_market portfolio%)*riskfree rate

So the second term uses (1-weight%) to arrive at (weight to riskfree%)*riskfree return
i.e., I think it agrees with you, rather it's only a weighting function per the CML which says your portfolio can only be invested in two places: either the market portfolio or, if not that, then (1-) to riskfree asset

David
 

sl

Active Member
#15
Hello David,

I am refering to the below portion of the calculation in your initial reply to one of the customers who posted the original question

.---part of your reply
if you look down to rows 37+ you see the *leveraged* allocations; e.g., 150% = E(return) of 14.8% because:


you invest $100 plus you borrow $50:
Future value of ($150 in market portfolio) = 150*1.122 = about $168.3
then payoff your loan @ $53.5 (= 50* 1.07 riskfree rate)

The rates (1.122 and 1.07) highlighted in blue are the ones i am having trouble figuring out. Thanks
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#16
Hi sl,

(apologies for delay) I see what you are saying. My explain comes at the calc from a slightly different angle. As you know, the calc itself @ 150% is: E(return) = 150%*12.2% + (-50%)*7%;
Where 12.2% is the E(return) of the market portfolio (the mix of Asset A & B that maximizes the Sharpe ratio in this unrealistic universe of only two assets).

My explain just tried illustrate how that plays out in practice: you borrow 50% to "lever up" your position in the market portfolio. The return is 0.122 in any case, so that means X would grow to X*1.122 to give the 0.122 return; e.g., to figure ending wealth on $1, it grows to $1*1.122 = $1.12 and this is an annual 12.2% return. (but I think you realize the 7% itself is merely an input). Or, if this is not too silly (it's not for me!):

annual return of (1.122x - x)/x = 12.2%

Hopefully that explains.

David
 
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