Buy Low-Sell High
Calories in Calories out
get in early then leave earlier yer friends
Someone must lose for someone to gain.
But why can't we win?
Suppose your portfolio has value VPV\_PVP and return RPR\_PRP. You add a hedge instrument with return RHR\_HRH and hedge weight hhh.
The hedged portfolio return is:
Rhedged=RP+hRHR\_{\\text{hedged}} = R\_P + hR\_HRhedged=RP+hRH
Here:
* h>0h>0h>0: you are long the hedge instrument.
* h<0h<0h<0: you are short the hedge instrument.
* h=−1h=-1h=−1: the hedge has the same dollar size as the portfolio, but in the opposite direction.
The variance of the hedged portfolio is:
Var(Rhedged)=σP2+h2σH2+2hCov(RP,RH)\\operatorname{Var}(R\_{\\text{hedged}}) = \\sigma\_P\^2+h\^2\\sigma\_H\^2+2h\\operatorname{Cov}(R\_P,R\_H)Var(Rhedged)=σP2+h2σH2+2hCov(RP,RH)
Because:
Cov(RP,RH)=ρPHσPσH\\operatorname{Cov}(R\_P,R\_H)=\\rho\_{PH}\\sigma\_P\\sigma\_HCov(RP,RH)=ρPHσPσH
we can write:
σhedged2=σP2+h2σH2+2hρPHσPσH\\sigma\_{\\text{hedged}}\^2 = \\sigma\_P\^2+h\^2\\sigma\_H\^2 +2h\\rho\_{PH}\\sigma\_P\\sigma\_Hσhedged2=σP2+h2σH2+2hρPHσPσH
To find the hedge ratio that minimizes variance, differentiate with respect to hhh:
∂σhedged2∂h=2hσH2+2Cov(RP,RH)\\frac{\\partial \\sigma\_{\\text{hedged}}\^2}{\\partial h} = 2h\\sigma\_H\^2+2\\operatorname{Cov}(R\_P,R\_H)∂h∂σhedged2=2hσH2+2Cov(RP,RH)
Set that equal to zero:
2hσH2+2Cov(RP,RH)=02h\\sigma\_H\^2+2\\operatorname{Cov}(R\_P,R\_H)=02hσH2+2Cov(RP,RH)=0
Therefore:
h∗=−Cov(RP,RH)Var(RH)\\boxed{ h\^\*=-\\frac{\\operatorname{Cov}(R\_P,R\_H)} {\\operatorname{Var}(R\_H)} }h∗=−Var(RH)Cov(RP,RH)
Or:
h∗=−ρPHσPσH\\boxed{ h\^\*=-\\rho\_{PH}\\frac{\\sigma\_P}{\\sigma\_H} }h∗=−ρPHσHσP
This is the **minimum-variance hedge ratio**.
# 2. Minimum-variance example
Assume:
VP=$1,000,000V\_P=\\$1,000,000VP=$1,000,000
Portfolio volatility:
σP=24%\\sigma\_P=24\\%σP=24%
Hedge volatility:
σH=18%\\sigma\_H=18\\%σH=18%
Correlation:
ρPH=0.75\\rho\_{PH}=0.75ρPH=0.75
The optimal hedge ratio is:
h∗=−(0.75)0.240.18h\^\* = -(0.75)\\frac{0.24}{0.18}h∗=−(0.75)0.180.24 h∗=−1.00h\^\*=-1.00h∗=−1.00
Therefore, the mathematical model says to short approximately:
$1,000,000\\$1,000,000$1,000,000
of the hedge instrument.
The resulting variance is:
σhedged2=0.242+(−1)2(0.18)2+2(−1)(0.75)(0.24)(0.18)\\sigma\_{\\text{hedged}}\^2 = 0.24\^2+(-1)\^2(0.18)\^2 +2(-1)(0.75)(0.24)(0.18)σhedged2=0.242+(−1)2(0.18)2+2(−1)(0.75)(0.24)(0.18) =0.0576+0.0324−0.0648= 0.0576+0.0324-0.0648=0.0576+0.0324−0.0648 =0.0252=0.0252=0.0252
Therefore:
σhedged=0.0252≈15.87%\\sigma\_{\\text{hedged}} = \\sqrt{0.0252} \\approx 15.87\\%σhedged=0.0252≈15.87%
The hedge reduces estimated annual volatility from:
24%→15.87%24\\%\\rightarrow15.87\\%24%→15.87%
It does not remove all risk because the correlation is only 0.750.750.75, not 1.001.001.00.
The remaining risk is called **basis risk** or **idiosyncratic risk**.