effectiveness of hedging

[ajsa]

Hi David,

I think there are 2 ways to express the effectiveness of hedging
1. effectiveness of hedging = 1 - (variance(basis) / variance(spot price)
2. in the minimum variance hedging, effectiveness of hedging=square(R)=square(corrlation coefficient)

Are they consistent?

Thanks.

 

[David Harper CFA FRM]

Hi asja,

Yes, it is a *great* observation...
they are consistent as they derive from the same measure of basis risk: variance (futures - spot)

if you take the 1st definition of hedge effectiveness and plug in the minimum variance hedge rate (h*); i.e.,
1 - variance(basis) / variance (spot), where under this singular condition
variance(basis) = variance(spot) + (h*)^2*variance(futures) - 2*(h*)*(correlation)*(spot volatility)*(future volatiliy), then
this formula for hedge effectiveness solves for correlation^2

so, the (1) is a general measure which can give different results;
e.g., if you hedge 1 futures contract per 1 spot exposure, that is unlikely to be optimal, so you will get a different result

whereas (2) is solving for the optimal number of futures contract...put another way, if you use (2) to solve for the optimal hedge, then this will produce the highest possible hedge effectiveness under (1)
...so we might view #2 as a special case of #1 where the hedge is optimal, but otherwise #1 can give many different answers

David

 

[ajsa]

Hi David,

sorry it seems i have some difficulty to follow you:

effectiveness of hedging = 1 - variance(basis) / variance (spot) = 1 - [(var(S) + var(F) - 2 * correlation * SD(S) * SD(F))]/var(S) = -var(F)/var(S) + 2*correlation*SD(F)/SD(S)
since h* = correlation*SD(S) /SD(F)
effectiveness of hedging = -(correlation /h*)^2 + 2*(correlation)^2/(h*)^2 = var(F)/var(S) <>correlation..

Do I miss anything?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

that starts by using a hedge ratio that is implicitly 1.0, you want to test for the hedge effectiveness under the special assumption that the hedge ratio (h) = the optimal hedge ratio (h*):

hedge effectiveness =
= 1 - (var(s) + h^2*var(f) - 2*h*rho*SD(s)*SD(f))/var(s)
= (2*h*rho*SD(s)*SD(f) - h^2*var(f)) / var(s)

if h = h* = rho*SD(s)/SD(f),
= (2*rho*SD(s)/SD(f)*rho*SD(s)*SD(f) - [rho*SD(s)/SD(f)]^2*var(f)) / var(s)
= (2*rho*SD(s)/SD(f)*rho*SD(s)*SD(f) - rho^2*SD(s)^2/SD(f)^2*var(f)) / var(s)
= (2*rho*SD(s)*rho*SD(s)- rho^2*SD(s)^2) / var(s)
= (2*rho^2*SD(s)^2- rho^2*SD(s)^2) / var(s)
= rho^2*SD(s)^2 / var(s)
= rho^2

David

 

[ajsa]

Hi David,

I understand variance(basis) = var(S-F) = var(s) + var(F) - 2 * rho * sd(s) * sd(F). But you used var(s) + h^2*var(f) - 2*h*rho*SD(s)*SD(f). do I miss anything?

thanks.

 

[David Harper CFA FRM]

Hi asja,

That formula (from assigned Geman) is implicitly for hedge ratio = 1.0. It really should make explicit the hedge ratio (h) such that we are looking for:
var (s - h*f); i.e., we probably are not hedging 1 future:1 spot,

such that:
var (s - h*f) = var(s) + var (h*f) - 2*cov(s,h*f)
=var(s) +h^2*var(f) - 2*h*cov(s,f); i.e., h is a constant

this is what the Geman reading really should instead show b/c it's the general case that includes h=1...

moral of the story (IMO): those few variance and covariance properties in Gujarati are always "paying dividends"

David

 

[ajsa]

Hi David,

so the defintion of effectiveness of hedging should not be 1 - (variance(basis) / variance(spot price). It should be 1 - (variance(s - h*f) / variance(s), instead, since basis = s-f <> s-hf, right?

Thanks.

 

[David Harper CFA FRM]

well ....

1. yes, IMO, that would be a better (more generalized) formula...
the basis is still spot - futures...but this is in regard to the position which is, presumably: the spot asset and some fraction/multiple of futures contracts used to hedge


2. However, that's not what the assignment has...we went down this path, instructive i think (for me, it was!) to show that Hull's MV hedge ratio is compatible with Geman's hedge effectiveness ratio ... i don't think that's cause for re-writing Geman's formula, I just think it highlights for us that Geman's hedge effectiveness is a limited test of when the forward:spot are 1:1:

for example, if we are a corn farmer planning to hedge the anticipated sale of 5,000 buschels of corn.
if we hedge with 1.0 *short* forward contract, that's what Geman's formula means to test (h=1); i.e., we ar hedging 5,000 spot with 5,000 futures (h = 1)

but if we hedge with 2.0 forward contracts, Geman's "breaks" and we need the generalized version as our hedge ratio, h = 2.0
maybe that's an overhedge...we are hedging 5,000 spot with 10,000 futures (h=2)

but there is some h that maximizes the generalized Geman hedge effectiveness, and that is the optimal hedge ratio (h*)

David

 

[ajsa]

Thank you David so much for this detailed explanation and your great patience!

 

[David Harper CFA FRM]

sure thing, it's no trouble...in case you didn't notice, I did not see this connection until you raised it, so this was a learning for me... The Geman is newly assigned and I wondered when I saw it, but it took your query to force an exploration...Thanks! David