Sizing Multi-Leg PCA Hedges and Calculating Spread Value Changes
Summary
The document considers how to build a hedge from quarterly short-term interest-rate futures using weights from a first principal component. It normalizes the component weights to one contract month to create a spread representation, then shows how equal tick values and price increments permit a direct contract ratio for a two-leg hedge. For a hedge with three contracts, it allocates the target exposure across the other two legs according to their normalized weights, so contract quantities can differ from the displayed spread coefficients.
The question asks how to translate movement in such a multi-leg spread into total dollar value change. Its key observation is that multiplying the change in the displayed weighted spread by one contract’s tick value and the target quantity can be misleading when the hedge quantities do not match the spread coefficients. The excerpt provides hypothetical weights and a proposed sizing calculation, but it does not supply the requested general P&L formula. A complete calculation must combine each contract’s own price change, tick value, and signed quantity; PCA neutrality also depends on the selected component and assumptions about the contracts.
Key ideas
- PCA component weights can be normalized to express a spread relative to a chosen contract month.
- For two contracts with equal tick values and increments, the normalized coefficient implies a hedge ratio.
- A multi-leg hedge can distribute exposure among several contracts, producing quantities that differ from spread coefficients.
- Dollar P&L should account for each contract’s price movement, tick value, and signed position size.
- The document poses the general multi-leg value-change question but does not derive a complete formula.
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# Price display of weighted spreads via PCA and value changed # Price display of weighted spreads via PCA and value changed Using PCA I have the below PC1, first component weights, for 4 quarterly expiries of short term interest rate future. These are hypothetical values used to help my question. March: 0.005542604, June: 0.028545363, Sep: 0.043833371, Dec: 0.049176453, Assuming tick value for all 4 products is $12.50 and each price increment is 0.005 (half of a basis point), to construct a hedge between durations of say, June versus Dec, one would: - Normalize the function to June such that (0.028545363 / 0.028545363) June * (0.028545363 / 0.049176453 ) * Dec, which is equal to: 1 June - 0.580468 Dec. This implies that for every 1 unit of June, we need less of Dec to the tune of approximately 0.58. Likewise, if a hedger is looking to neutralize risk described by the first component PCA, for every 100 June, he/she would sell 58 Dec. (tick value and price increment the same for June and Dec as mentioned above) To display this spread function in normalized fashion , one would input June market price / 0.005 - 0.580468 * Dec market price / 0.005. This way, if we observe the function and saw a 50% change in the function as a whole, we know the total value change of the above spread for every 100 units is 0.50(spread change) * $12.50 (change per 1.00) * 100 units. However, the above relates to 2 products hedging each other. How does this change if we have 3 products? Say to hedge 100 units of June, we will use a combination of Sep and Dec instead of just Dec only. We go through this process again first by creating a spread function to display price movement. - standardizing Sep and Dec to June, we have 1 June - 0.651224 * Sep - 0.580468 * Dec - Let's say for 100 units of June, we will use 30 units of Sep and 31.31 units of Dec to be risk neutral. This is derived from 30 units of Sep / 0.651224 = 46.06707, so the remaining units for Dec needs to be (100 - 46.06707) / 0.58046, which gets us our quantity of 31.31 units. Like for two products in the June vs Dec example, one can easily calculate the value changed by the change in the function * 12.50 * 100 units. However, this is not the same for spread functions that has more than 2 product such as the June vs Sep and Dec. One cannot simply say 50% change observed in 1 June - 0.651224 * Sep - 0.580468 * Dec * 12.50 * 100 units is the total change in the spread function. This is due to the uneven weighting of quantities used for hedging for spread functions that has more than 2 symbols. My question is: How does one formulaically represent the total value change for spread functions with more than 2 symbols in a way that accounts for the different quantities used in an elegant way similar to our spread function with just 2 symbols? I am beginning to think that for spread functions with more than 1 product as a hedge, another coefficient needs to be applied for each term to describe the relative proportion of the hedge quantity or contribution...
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