How to calculate volatility (variance od returns) from total derivative of asset
From: Krzysiek (kszlasa_at_neostrada.pl)
Date: 07/31/04
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Date: Sat, 31 Jul 2004 17:00:42 +0200
>From article of David Blake "Pension schemes as options on pension fund
assets" Published in Insuance:Mathematics and Economics 23 (1998)
Expected value of a scheme member's pension assets at any date t is equal
the expected value of the financial assets Ft plus Xt - the expected
discounted value of the remaining contributions until retirement date.
At =Ft+Xt==SUM from k=1 to t{ [pk*y*Y0*(1+gy)^(k-1]/(1-rho)}*{PRODUCT from
j=k+1 to t (1+rFj)} + SUM from k=t+1 do T{ [pk*y*Y0 * (1+gy)^(k-1)] /
[(1-rho) * PRODUCT form j=t+1 to k {(1+rFj)}] ,t=1,...,T
Y0- starting income; y-the contribution rate as a proportion of income;
gy-the expected future growth in income(constant oer time); rFt-the expected
yields on the investments in financial assets purchased with contributions;
T-number of years of pensionable service; pt- one year survival
probabilities from date t=0
t=0
assume pk=p=const
rFj=rF=const
for simplicity d=p*y*Y0 / (1 - rho )
A0=X0=SUM form k=1 do T{ [d * (1+gY)^k] / (1+rF)^k}
the elasticity of A0 with respect to (1+rF) is given by:
( I )
partial derivative(A0 / (1+rF)) * (1+rF) / A0 = - 1 / A0 * SUM from k=1 to
T{[k*d * (1+gy)^k] / [(1+rF)^k] } = - DA0
DA0 - duration of A0
the elasticity of A0 with respect to (1+gy) is given by:
(II)
partial derivative(A0 /(1+gY)) * (1+gy)/A0 = 1 / A0 * SUM from k=1 to T
{ k*d* (1+gy)^k] / [(1+rF)^k] } = DA0
The total differential of A0 can be written:
dA0/A0 = ( I ) * drF/(1+rF) + ( II ) * dgy/(1+gy) + epsilonA = -
DA0*(drF/(1+rF) - dgy/(1+gy)) + epsilonA
where we include serially and contemporaneously uncorrelated specific risk
component (epsilonA) to the rate of return on pension assets. The volatility
of the
rate of return on pension assets is given by:
sigmasquaredA0 = DurationsquaredA0 * (sigmasquared rF + sigmasquared gy) +
somesymbolsquared A0
Note that sigmasquared rF and sigmasquared gy are the volatilities of the
rate of change
in interest rates and growth rates, rather than the volatilities of their
levels so these will take relatively low values
Any ideas how differentiating A0 produced epsilonA??
Does anybod know the logic underlying the derivation of the volatility of A0
from total derivative of A0?
I will appreciate any suggestion...Please help:)) Thanks
PS If anybody wants the full article text in pdf email me.
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