Delta-Normal Basis Risk from Correlated Interest Rate Changes
Summary
The document asks how to estimate portfolio risk across correlated interest-rate tenors when the factor series can cross zero. The questioner uses absolute daily changes rather than percentage returns, forms a covariance matrix, and asks how to interpret portfolio standard deviation and map a portfolio tail estimate back to individual curve points. The response gives the delta-normal variance formula: combine factor exposures with their covariance matrix, then take the square root and scale by a chosen quantile from the inverse normal distribution. With DV01 exposures as the factor weights, the result is expressed in cash-risk terms.
This approach summarizes joint linear exposure under a normal approximation; it does not provide a rule for converting an aggregate portfolio tail move into a separate shock for each tenor. The result depends on correctly defined exposures and a covariance matrix estimated on compatible units and time horizons. Normal-tail estimates also omit nonlinearity and non-normal behavior, and the brief reply notes that other tail-risk measures and methodologies are available.
Key ideas
- Correlated rate-factor changes can be combined using a covariance matrix and a vector of factor exposures.
- The delta-normal portfolio variance is the exposure vector multiplied through the covariance matrix and back through the exposure vector.
- Scaling portfolio volatility by a normal quantile gives a cash-denominated risk estimate when exposures such as DV01 are used.
- An aggregate portfolio tail estimate does not by itself specify a unique shock for each tenor.
- The calculation relies on a linear exposure approximation and a normal distribution assumption.
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# Covariance Interest Rate Risk Time Series
# Covariance Interest Rate Risk Time Series
Apologies in advance if this question has been asked already.
I am estimating basis risk for different term points in the curve. Imagine i have three time series (1-month, 3-month, 1-year). I believe they are NOT independent of each other. I have estimated the absolute daily change for each one of them (% change didn't work as some time series switch from negative to positive, or the other way around). I could generate a variance/covariance matrix of the absolute daily change in these three time series, and calculate some sort of a 'portfolio' standard deviation (assuming equal weights). This standard deviation would be in basis points, as mentioned above. For instance, 2 bps. I could also generate some sort of an expected value (average) for this portfolio, for instance 10 bps.
The question that I have is: how do I interpret the standard deviation in the context of each separate time series? If i want to estimate tail risk for this portfolio, is it accurate to (assuming normal distribution), to just estimate 2 standard deviations from the mean and analyze the tail risk from that data point? What would 4 basis points mean for each time series, i.e., how do I transform this portfolio tail risk to each time series, given that it is not proportional (%) change, but rather absolute change?
## Answer by Bram (score 1)
https://quant.stackexchange.com/a/35603
If $v$ is a vector of DV01 ezposures versus your various risk factors/buckets and $\Sigma$ is your covariance matrix tegen portfolio (delta-normal) Var is given by $$\alpha \sqrt{v^T \Sigma v }$$ where $\alpha$ is some point from the inverse cumulative normal.this gives you risk in cash (money) terms rather than % losses.
There are other ways to measure tail risks of course, both in metrics and methodology, but this looks like what you're afterShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.