## Hedge Ratios – CFA L3 Fixed Income Walkthrough

# Hedge Ratios – CFA L3 Fixed Income Walkthrough

As we know when we are hedging, we come across different sets of risk. The main one of which is *basis risk.* Basis risk is “the risk that
offsetting investments in a hedging strategy will not experience price changes in entirely opposite directions from each other.” [1]

This is especially common in *cross-hedges, *where the underlying asset in the futures bond is different than the asset being hedged. The most
common cross-hedge example is using a corporate bond futures to hedge a treasury bond.

**
Basically, if the futures contract and the bond have different sensitivities or volatilities we need to account for that in our calculation of how many
contracts we need using a hedge ratio.
**

The hedge ratio is basically telling us that the number of contracts we need to effectively hedge the underlying bond is a function of the relative
sensitivity of the bond and futures to a given risk factor. **So if the bond has greater risk exposure than the futures, you would need more futures contracts to fully cover the risk**.
The second equation on the right just adds the conversion factor because the pricing will invariably be in terms of the CTD bond. We can also rewrite this
equation in terms of duration:

The final piece of the puzzle is *yield beta. *Yield Beta measures the sensitivity to interest rate changes of the CTD bond versus the original
bond. The equation for yield beta is a simple regression:

A yield beta of 1.2 for example would imply that for a yield change of 100 bps on the CTD the underlying bond spread would change 120 bps. Think of this way: if the bond yield has greater volatility than the CTD than more hedges are needed.

The key takeaway for the exam is that different sensitivities to underlying movements may impact the amount of contracts you need to adequately hedge your exposure. Thus we fold yield beta into our equation to solve for the number of futures contracts:

Note, on the exam if they don’t include a yield beta then it just = 1. You MUST memorize this last equation.

To recap we’ve gone through a pretty in-depth walk through of the concepts underlying constructing a hedge ratio and figuring out how many futures contracts we need to hedge a bond position. At a minimum memorize this last equation, and be able to relate the above discussion to the deeper section on risk management with futures section where they give us this equation for modifying a bond portfolio’s duration:

**Hedging Errors**

Errors in hedging arise from three sources:

1. A wrong forecast at the time of the hedge

2. A mistake in estimating durations

3. A mistake in estimating yield beta

[1]
http://www.investopedia.com/terms/b/basisrisk.asp
. One common example is *price basis *which refers to the difference between the spot price and the futures delivery price at the time of
delivery.