PROBLEM: Minimum-variance hedge ratio (MVHR)

D

daniellmy

New member
Joined
Jun 18, 2026
Messages
0
Reaction score
0
Reference: 2016 SchweserNotes Level III Book 3 - Assigned Reading:18 (Currency Management: An Introduction) - LOS18.h - page 175-176
Strong positive correlation between Rfx and Rfc increases the volatility of Rdc. A hedge ration greater than 1.0 would reduce the volatility of Rdc.
Strong negative correlation between Rfx and Rfc naturally decreases the volatility of Rdc. A hedge ratio less than 1.0 would reduce the volatility of Rdc.
Can anyone help explain the concept and application of MVHR in simple words and the above statements? I dont get the idea behind all of these.
Lets say, for example, Rfx and Rfc are negatively correlated in Japan which is heavily dependent on export. I bought 100 shares at 100JPY and need to convert it back to my USD portfolio after 3 months. So I should literally short JPY10,000 forward, right? But since the hedge ratio is less than 1.0 as Statement 2 says, I should ideally short less than JPY10,000? I dont get it.
 
If the correlation is negative (not perfectly) then what happens when the shares rise in price? The JPY goes down.
Let’s say the correlation is -0.5, and the shares rise 100%, then the JPY/USD should also rise by 50%.
If you hedge only JPY100,000 forward, you end up with JPY100,000 that are not hedged, of which their return has gone down by 50%. So your total return for the move was return on forward (let’s say -10%) *.5 + -50%*.5 + 100% = 70%
If the correlation was positive, you would find an exaggerated return instead. What you should understand from it is this, if the returns move together, then you should increase the hedge ratio, because you would likely find big moves in either direction, and hedging a bigger portion reduces volatility (and return in absolute value), while underhedging would not as much. And lower the hedge ratio if the correlation is negative, since the volatility is already reduced between the Rfx and Rfc, so overhedging would puts you at risk of higher returns when the stock market returns are positive, since you buy a portion of the FC at the depreciated marktet.
 
sorry, just want to clarify, Rfx is measured in DC/FC right? Basically it means that if there is a positive relationship between the foreign currency value and the return on foreign asset, then we should use a hedge ratio of greater than 1? For example, when foreign currency appreciates and the foreign asset yields a positive return, a hedge ratio of greated than 1 should be used to reduce the volatility. Correct?
I just saw a statement on another forum saying “For example, if when the local currency appreciates the underlying asset is more likely to appreciate too then you would want to hedge less than the principal amount. I guess the optimal amount will be function of the strength of the relationship.”.
Doesn’t this mean the opposite? “hedge less than the principal amount” = hedge ratio of less than 1?
 
zxfmontreal wrote:
sorry, just want to clarify, Rfx is measured in DC/FC right? Basically it means that if there is a positive relationship between the foreign currency value and the return on foreign asset, then we should use a hedge ratio of greater than 1? For example, when foreign currency appreciates and the foreign asset yields a positive return, a hedge ratio of greated than 1 should be used to reduce the volatility. Correct?
I just saw a statement on another forum saying “For example, if when the local currency appreciates the underlying asset is more likely to appreciate too then you would want to hedge less than the principal amount. I guess the optimal amount will be function of the strength of the relationship.”.
Doesn’t this mean the opposite? “hedge less than the principal amount” = hedge ratio of less than 1?
Click to expand...
Yes.
If you overhedge on a positve correlation, two things can happen (assume fwd rate = spot rate)
1) Both return postively, you earn the stock market retrun plus minimal fx gain/loss
2) Both return negatively, you are clearly overhedged in this situation, you lose the stock market return, and gain the remainig principal on the hedge not covered by the stock market proceeds. By changing the DC to FC at a lower rate, and getting it back at the higher fwd rate, or simply sell the contract for a gain.
In both instances, the return volatility is reduced, because they are banded.
 
How is the principal calculated then? For a hedge ratio greater than 1, why should we hedge “less than the principal amount”?
 
MrSmart wrote:
If the correlation is negative (not perfectly) then what happens when the shares rise in price? The JPY goes down.
Let’s say the correlation is -0.5, and the shares rise 100%, then the JPY/USD should also rise by 50%.
Click to expand...
Are you saying that if the correlation was -0.3, for example, a 100% rise in share price would be accompanied by a rise in JPY/USD by 30%?
 
tickersu wrote:
MrSmart wrote:
If the correlation is negative (not perfectly) then what happens when the shares rise in price? The JPY goes down.
Let’s say the correlation is -0.5, and the shares rise 100%, then the JPY/USD should also rise by 50%.
Click to expand...
Are you saying that if the correlation was -0.3, for example, a 100% rise in share price would be accompanied by a rise in JPY/USD by 30%?
Click to expand...
Overly-simplified, in the abscence of residual error and a zero coefficent, yes.
 
daniellmy wrote:
Lets say, for example, Rfx and Rfc are negatively correlated in Japan which is heavily dependent on export. I bought 100 shares at 100JPY and need to convert it back to my USD portfolio after 3 months. So I should literally short JPY10,000 forward, right? But since the hedge ratio is less than 1.0 as Statement 2 says, I should ideally short less than JPY10,000? I dont get it.
Click to expand...
Yes, let’s assume MVHR is 0.8 (less than 1 due to negative correlation between Rfx and Rfc), then you should short JPY 8000 = 0.8 x JPY 10000.
The idea being if Rfx and Rfc are negatively correlated, they are effectively hedging each other and reducing the volatility of the portfolio, so you do not need to hedge the full amount of the investment.
 
You must log in or register to reply here.
Back
Top