Diversification Benefit - a practical question

[Raj_S]

Hi David et al,

From VAR topics we know if portfolio A is added to B there will some divesification benefit if correlation between A & B is not 1. Let me add some spice to it.

Portfolio A loss distribution approximately resembles normal distribution ( say retail portfolio)
Portfolio B loss distribution has heavy fat tail ( say large corporate portfolio with big single name concentrations. ie big loans to a single obligor)
Say correlation between portfolio A and B is low ( say 0.4)

Q1. In this case will portfolio A enjoy more diversification benefit than B ( given that B has huge tail losses)
Q2. Assuming above is true, if we hedge some of the big tail events (obligor level risk concentrations) in portfolio B will portfolio B get a decent share of diversification benefit.

Regards
Subin

 

[ShaktiRathore]

Hi,
You said correlation between A and B is not 1.
sigma(A)<sigma(B) and corr(A,B)=.4
let suppose, wA=wB=.5 and sigma(B)=.60 very risky due to heavy tails and sigma(A)=.3
now for portfolio sigma(p) will be such that,
sigma(p) =sqrt[(.5*.6)^2+(.5*.3)^2+2*(.5*.6)*(.5*.3).4]
=>sigma(p) =sqrt[.09+.0225+.036]= sqrt[0.1485]=0.385
so , .3<.385<.6 or that sigma(A)<sigma(p)<sigma(B) we can prove this for any two portfolios or assets with correlation less than 1.
So we conclude that risk for portfolio A increases after the addition to B while risk of B has decreased. So in a way B enjoys more diversification benefit at the expense of A.

hedging the risk in B will surely decrease its volatility sigma B. Now if this volatility falls below .3 than A now enjoys more diversification benefits than B with reduced risk in form of portfolio risk.
Now suppose sigma(A)=.3 and sigma(B)=.25 then sigma(B)<sigma(p)<sigma(A) so that A's risk has fallen while that of B has risen.
However if the volatility still remains above the risk of A than B's diversification benefit still remains with reduced risk but its share in benefit will go down!! e.g. if risk of B is now 40% then suppose portfolio risk now is 35% then its risk falls down by 12.5% as compared to previous fall of 22.5/60=375%. so B's diversification benefit has certainly reduced.

[Its like mixing two liquids of concentrations A and B so that the mixture has concentration intermediate of A and B while concentration of B falls and that of A rise provided concentration of B>concentration of A in whatever proportion they are mixed.]
thanks

 

[Raj_S]

Hi Shakti,

Thanks for the reply. When two portfolios are combined both enjoy diversification benefit ( if correlation if any number other than 1). Hence riskiness of both A and B decreases.

May be a bit difficult to comprehend so here are some cooked up nos to make my question clear.
EC = economic capital.

EC for Protfolio A = 100
EC for portfolio B = 150
correlation = =.4
EC for portfolio(A+B) = sqrt[(100)^2+(150)^2+2*(100)*(150)*.4] = 210
Now the tricky bit. The combined EC 210 is reallocated back to portfolio A and portfolio B to understand their individual contributions
After reallocation
EC for Portfolio A = 70 ( diversification benenfit = 30%)
EC for Portfolio B = 140 (diversification benefit = 6.6%)
( the sum is still 210)


Note that EC for both has decreased but at different magnitudes.

Repharsing my first question : Is the above case (in green) possible (given the difference in the loss distribution - ie fat tale in B)?
Q2 remains same : Assuming above is possible, if we hedge some of the big tail events (obligor level risk concentrations) in portfolio B will portfolio B get a decent share of diversification benefit.

Probably this is a bit advanced / practical question and sincere apologies if this is not relevant for this forum

Regards
Subinx2

 

[ShaktiRathore]

May be i was confusing ,
Diversification benefit happens when the overall risk is reduced for the portfolio. The isolated assets A and B are combined to form a lower risk portfolio.
Let 500 be total value of portfolio A and 500 be total value of portfolio B . wA=wB=.5
Now EC(A)=sigmaA*z*500 and EC(B)=sigmaB*z*500
Assume that sigma A=30% and sigma B=40% => EC(A)=.30*1*500=150 and EC(B)=.40*1*500=200 when this portfolios are in isolation
Now lets combine A and B in a new portfolio C,
sigma(A+B)=sqrt[(.5*.4)^2+(.5*.3)^2+2*(.5*.4)*(.5*.3)*.4]=sqrt[.04+.0225+.024]=29.41%
So EC for A+B=C is sigma(A+B)*z*1000=.2941*1*1000=294.1
Applying weight of A and B the , EC(A)=.5*294=147 and EC(B)=147
So diversification benefit for B=53/200=26.5% and diversification benefit for A=3/150=2%
So diversification benefit to B is much more as compared to A. as can be seen.
I think you have taken the special case where diversification is reducing risk of both A and B with correlation .4 but you can try with correlation .6 and see what happens while risk of one is reducing that of other is in fact increasing.
Diversification is seen in context of the portfolio and the overall risk of the portfolio.
May be you understand this and i don't know how you bought the weights there in allocating EC to A and B from the portfolio.

thanks

 

[Raj_S]

Thanks Shakti. The first part is clear - ie portfolio EC calculation when portfolio A and Portfolio B are combined. The part which is not clear is the reallocation bit. Is the reallocation purely based on weights or do is it done in a more risk sensitive way?. If so how the characterstic heavy tail (concentrations) or Credit quality (PD) in portfolio B will affect it .

 

[Raj_S]

Shakti ,

Using risk contribution in your example will give following result

ECA = Economic capital for portfolio A
ECB = Economic capital for portfolio A
EC(A+B) = Economic capital for combined portfolio A +B
RC means Risk Contribution ( not reg cap)

RCA = [ RCA * ( RCA + ρAB * RCB ) ] / EC(A+B) à 1​

RCB = [ RCB * ( RCB + ρBA * RCA ) ] / EC(A+B) à 2

Where

ECA+B = RCA + RCB

From your example

ECA = 150
ECB = 200
EC(A+B) = RCA + RCB = 294.1
ρAB = ρBA = 0.4


Solving equation 1 and 2

RCA = 117.3

RCB = 176.81

But my question still remains. say portfolio B is fat tailed( eg. large corporate portfolio). does that mean its diversification benefit is low from a practical standpoint. The RC calculated above dont say that story though as it is just dependent on correlation of two loss ditributions only.

David , any expert opinion?

 

[ShaktiRathore]

From you Equations:
RCA = [ ECA * ( ECA + ρAB * ECB ) ] / EC(A+B) à 1
RCB = [ ECB * ( ECB + ρBA *ECA ) ] / EC(A+B) à 2
As portfolio B is more riskier than A it should have more risk contribution in A+B, so that ECB > ECA
RCB -RCA = [ ECB * ( ECB + ρBA *ECA ) ] / EC(A+B) - [ ECA * ( ECA + ρAB * ECB ) ] / EC(A+B)
RCB -RCA = [ ECA * ( ECA + ρAB * ECB ) ] / EC(A+B)- [ ECB * ( ECB + ρBA *ECA ) ] / EC(A+B)
RCB -RCA ={ [ ECA * ( ECA + ρAB * ECB ) ] - [ ECB * ( ECB + ρBA *ECA ) ] }/ EC(A+B)
RCB -RCA ={ [ ECA * ECA -ECB * ECB+ ρAB *ECA *ECB -ρBA *ECA*ECB ) ] }/ EC(A+B)
RCB -RCA ={ [ (ECA-ECB) * (ECA+ECB ) ] }/ EC(A+B)
RCB -RCA =[ (ECA-ECB) * (ECA+ECB ) ] / EC(A+B)
which is independent of correlation , now since ECA<ECB
RCB -RCA =[ (ECA-ECB) * (ECA+ECB ) ] / EC(A+B)>0
RCB -RCA>0=> RCB>RCA so that risk contribution of B the more risky portfolio is greater than A the less risky one for any correlation between them.
RCB/RCA>1 =>RCB+RCA/RCA>2 =>RCA/RCB+RCA<.50 => % risk contribution of A is less than 50% while that of B is greater than 50% of total economic capital.
Diversification benefit of A=DA=ECA -RCA
Diversification benefit of B=DB=ECB -RCB
from above equations, DA-DB=(ECA -RCA)-(ECB -RCB )
DA-DB=(ECA -ECB+RCB-RCA)
DA-DB=(ECA -ECB)+[ (ECA-ECB) * (ECA+ECB ) ] / EC(A+B)
DA-DB=(ECA -ECB)[1+(ECA+ECB )/EC(A+B)]
Now since ECB > ECA=> DA-DB<0=> DA<DB
So we conclude that diversification benefit of A is less than the diversification benefit of B. You see from the above equation that it depends on the riskiness of the portfolio that is if it more risky like B its risk contribution is more in A+B while if its less risky then its risk contribution is less to A+B. So that B brings more risk to the overall position A+B. The diversification benefit depends on the same notion that if a position is more risky then it gains at the expense of less risky position inequality ECB > ECA is satisfied. So B gains more.
If you are hedging the B portfolio so that you are reducing the EC for B so that now ECB < ECA which reverses our results above,
RCB<RCA and DA>DB so that A now enjoys more diversification benefit and has more risk contribution to total portfolio.

thanks

 

[David Harper CFA FRM]

Hi subin,

I agree with all the calculations (although, in your first example, i get a decomposition of 100*(100+0.4*150)/210.95 = 75.85 for A + 135.1 for B = 210.9 in total volatility/VaR), but all of the illustration above, as far as i can tell, assumes normality (mean-variance); e.g., fat tails are not captured with a higher volatility input. The risk contribution (RC) is valid but it's just a mean-variance-based decomposition of and presumes normality throughout; $75.85 is decomposition of marginal risk that utilizes only volatility and correlation. So, the "problem" with the illustration is that it never leaves the assumptions of mean-variance and the intense assumptions of (linear) correlation. More simply, these models (RC, marginal volatility) don't really allow a fat-tail distribution to enter in so far as producing back for us statistics like RC.

I *think* that's the point of Miller's cross-central moments; e.g., co-skew, co-kurtosis. The tail benefits of diversification, due to the tail hedge, presumably would manifest in joint (e.g.,) bivariate higher-moment statistics, thanks,

 

[Raj_S]

Thanks David. My first example was not based on RC but was more as illustration to show the punitive effect (so they say) of tail risk on portfolio B (140 vs 135.1) . I will try to do a bit of research on this

 

[David Harper CFA FRM]

Hi Subin, okay, thanks. I would be interested in whether there is another method to decompose if we are given only variance-covariance inputs (no 3rd or 4th moments). The above methods, to my thinking all employ a natural "marginal volatility" (first derivative, dPortfolioVol/dPosition) only because they have the typical two-asset portfolio variance to work with (from which to take a first derivative). Of course, if the portfolio risk is employs a different measure, then RC can be computed as function of its first derivative, whatever it may be (so i think the "problem" is not so much to find the first derivative but i'd think to specify the portfolio risk, instead of typical variance, as a metric that incorporates 3rd and 4th moments) that , thanks,