Beta/Asset's weight

[hawayi_vgo]

63.2. Your colleague Robert uses a two-factor model in order to estimate the volatility of a Portfolio. He specifies the covariance matrix as follows:

equity factor, bond factor
equity factor 0.09, 0.072
bond factor 0.072, 0.16


The Portfolio has the following factor sensitivities (i.e., betas): 0.60 to the Global Equity Factor and 0.25 to the Global Bond Factor. The volatility of the Portfolio is nearest to which value?

The answer:
Variance(Portfolio) = 0.60^2*0.090 + 0.25^2*0.160 + 2*0.60*0.25*0.0720

Are we using beta as asset's weight?
Are they the same?
Thank you!

 

[ShaktiRathore]

Hi there,
According to CAPM: E(Rp)= Rf+ beta(ERP) is a one factor model. where beta is sensitivity of portfolio to only one factor market.

Generalizing to 2 factors the expected portfolio return can be return as, we can do for n factors also..
E(Rp)= Rf+ beta1[F1]+beta2[F2] where beta1 and beta2 are sensitivity of portfolio to factors F1 and F2 . APT model!!! remember for n factors


taking variance on both sides,
variance(p)=variance[Rf+ beta1[F1]+beta2[F2]]
variance(p)=variance[beta1[F1]+beta2[F2]] risk free asset has variance and covariance with any asset as 0 so we remove it.
Now we know that Variance(A+B)=Variance(A)+Variance(B)+2 CoVariance(A,B)

so variance(p)=variance[beta1[F1]+variance[beta2[F2]]+2 CoVariance(beta1[F1],beta2[F2])
variance(p)=variance(beta1*F1)+variance(beta2*F2)+2 beta1*beta2*CoVariance([F1],[F2])
variance(p)=beta1^2*variance(F1)+ beta2^2*variance(F2)+2 beta1*beta2 CoVariance([F1],[F2])

here F1= equity factor and F2=Bond factor deducing values from var-covar matrix,
variance(equity factor )=0.09, beta1=0.60
variance(Bond factor) =0.16, beta2= 0.25
CoVariance([(equity factor],[Bond factor])=0.072
substituting values in formulae above,
variance(p)=0.60^2*0.09+ 0.25^2*0.16+2 0.60*0.25 0.072

hope u understood, thanks

 

[David Harper CFA FRM]

Thanks ShaktiRathore, you are really helpful, as usual!

hawayi_vgo I think it's really important to understand how Shaki derived the answer. You can see that it's the same idea/ingredients as "weights" in a two-asset portfolio, but too many candidates just memorize the two-asset variance formula without understanding how it applies variance properties, then are stuck when asked about variations that apply the same ideas.

Here you can see from Shakti's derivation the applied ideas ingredients, which are highly testable, are:

• variarance(consant;e.g,, riskfree rate) = 0
• variance[a*x] = a^2*variance(x), and
• variance(x + y) = var(x) + var(y) + 2*cov(x,y), and combine these last two into:
• variance (ax + by) = var(ax) + var(by) + 2*cov(ax,ay) = a^2*var(x) + b^2*var(y) + 2*cov(ax,by)

BTW, my question is a modeled on this GARP 2011 sample question, which in typical FRM fashion, asks the question that mere memorization won't be ready for:

this question thread is http://forum.bionicturtle.com/threa...nd-arbitrage-pricing-theory-foundations.4104/

 

[ShaktiRathore]

If people are having hard time where above formula came from i would give u a brief derivation based on var formulas suggested by david
E(RA)=Rf+ betaA,1[F1]+betaA,2[F2]
E(RB)=Rf+ betaB,1[F1]+betaB,2[F2]
Covariance(A,B)= Covar[(Rf+ betaA,1[F1]+betaA,2[F2])*(Rf+ betaB,1[F1]+betaB,2[F2] )]
Covariance(A,B)= Covar[( betaA,1[F1]+betaA,2[F2])*( betaB,1[F1]+betaB,2[F2] )]
Covariance(A,B)= Covar[( betaA,1[F1]*( betaB,1[F1]+betaB,2[F2] )+betaA,2[F2]*( betaB,1[F1]+betaB,2[F2] )]
Covariance(A,B)= Covar[( betaA,1[F1]*betaB,1[F1]+betaA,1[F1]*betaB,2[F2] )+betaA,2[F2]*betaB,1[F1]+betaA,2[F2]*betaB,2[F2] )]
Covariance(A,B)= Covar( betaA,1[F1]*betaB,1[F1])+Covar(betaA,1[F1]*betaB,2[F2] )+Covar(betaA,2[F2]*betaB,1[F1])+Covar(betaA,2[F2]*betaB,2[F2] )
Covariance(A,B)= betaA,1*betaB,1 *Covar( [F1]*[F1])+betaA,1*betaB,2* Covar([F1]*[F2] )+betaA,2*betaB,1 *Covar([F2]*[F1])+betaA,2*betaB,2* Covar([F2]*[F2] )
Covariance(A,B)= betaA,1*betaB,1* var( F1)+betaA,1*betaB,2* Covar(F1*F2 )+betaA,2*betaB,1 *Covar([F2]*[F1])+betaA,2*betaB,2 *var(F2 )
Covariance(A,B)= betaA,1*betaB,1* var( F1)+betaA,2*betaB,2 *var(F2 )+betaA,1*betaB,2* Covar(F1*F2 )+betaA,2*betaB,1 *Covar([F2]*[F1])
Covariance(A,B)= betaA,1*betaB,1* var( F1)+betaA,2*betaB,2 *var(F2 )+(betaA,1*betaB,2+betaA,2*betaB,1 )* Covar(F1*F2 ) which is the same formula represented above in the garp sample paper.thanks

 

[hawayi_vgo]

got it.
thank you!