Lesson 4 of 25 · 7 min
Beta and residual variance, and which one the lender is pricing
A market hedge addresses one half of the variance; a concentrated schedule is discounting mostly for the other half.
Beta is the more famous number and residual variance is the more relevant one, at least for a pledged single holding. Working out which dominates is a two-line calculation and it changes how the rest of this track reads.
total variance = beta squared × market variance + residual variance
- total variance
- Volatility squared. Variance is used here rather than volatility because the two components add only in squared form.
- beta
- The holding's sensitivity to a 1 percent market move.
- market variance
- The broad market's volatility, squared.
- residual variance
- The company-specific volatility from lesson 2, squared.
Take a single large-company holding with a beta of 1.15, a market at 15 percent volatility, and residual volatility of 26 percent. The market piece is 1.15 squared × 225 = 297.56. The residual piece is 26 squared = 676. Total variance is 973.56, so total volatility is 31.20 percent. The market's share of variance is 297.56 ÷ 973.56 = 30.6 percent, leaving 69.4 percent residual.
That ratio is the practical content of the lesson. Roughly seven tenths of what can happen to this holding is not a market event at all, and no instrument written on a broad index can offset it. A hedge on the index addresses the 30.6 percent. The other 69.4 percent is untouched, and it is the part that produces overnight gaps and trading halts.
This is also why a lender's concentration schedule looks harsh next to a beta figure. Beta 1.15 sounds close to the market and reads as unremarkable. The schedule is not pricing the 1.15. It is pricing a 31.20 percent total volatility of which two thirds arrives from a direction the lender cannot see coming and cannot hedge itself.
Key terms
- Variance
- Volatility squared. It is the form in which unrelated risk components can be added together and split into shares.
- Residual variance
- The company-specific part of total variance. For a single holding it is usually the larger part.
- Variance share
- One component's variance divided by the total. It answers what fraction of what can happen comes from that source.
Worked example
Splitting one holding's variance into shares
One large-company holding. Beta 1.15, market volatility 15%, residual volatility 26%. Illustrative figures.
- Market component of variance
- 297.56
- Residual component of variance
- 676.00
- Total variance
- 973.56
- Total volatility
- 31.20%
- Market share of variance
- 30.6%
- Residual share of variance
- 69.4%
1.15 squared × 15 squared.
26 squared.
Square root of 973.56.
297.56 ÷ 973.56.
100% − 30.6%.
A beta near 1 concealed a holding whose behaviour is roughly two thirds company and one third market.
Check yourself
A holding has a beta of 0.9, the market has 14% volatility, and the holding's residual volatility is 31%. What share of its variance is market risk?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.