Lesson 1 of 25 · 7 min
Why a normal distribution is used at all, and what it buys
The bell curve survives because it makes portfolios addable in closed form, and that convenience outlives every demonstration of its failure.
Track 4 lesson 2 said finance uses the normal distribution for reasons that are mostly not about accuracy. This track spends twenty-four lessons on the inaccuracy, so it is worth starting by pricing the convenience. Knowing what the assumption buys is the only way to judge what abandoning it costs.
The property that does the work is closure under addition. A weighted sum of normal variables is itself normal, which means a portfolio of forty positions has a fully specified distribution the moment you know the volatilities and the correlations — 820 numbers for forty holdings, and no further information required. Nothing else about the joint behaviour needs to be supplied, because under this assumption there is nothing else.
Work one case. Two sleeves at 62 percent and 38 percent of the account, with annual volatilities of 22 and 31 percent and a correlation of 0.45. The variance is (0.62 × 0.22)² + (0.38 × 0.31)² + 2 × 0.62 × 0.38 × 0.45 × 0.22 × 0.31 = 0.018605 + 0.013877 + 0.014461 = 0.046943, so the portfolio volatility is √0.046943 = 21.67 percent. The weighted average of the two volatilities is 25.42 percent. The gap is the correlation term, and it came out of one line of arithmetic.
Two more things come free. Uncertainty scales with the square root of time, so an annual figure becomes a daily one by dividing by √252. And any percentile is a single multiplication: 2.3263 times the daily standard deviation gives the level exceeded one day in a hundred. On a $3,120,000 account at 21.67 percent annual volatility, the daily standard deviation is 1.3648 percent and that level is $99,064.
A lender revalues thousands of pledged accounts overnight. Closed-form arithmetic is what makes that possible at all, which is why the assumption survives long after everyone involved knows where it fails.
Key terms
- Closure under addition
- The property that combining normal variables gives another normal variable. It is what lets a portfolio be described without describing every position's joint behaviour.
- Covariance matrix
- The table of volatilities and correlations that, under a normal assumption, contains everything there is to know about a portfolio's risk.
- Closed form
- An answer obtainable by direct calculation rather than by simulation. Fast, exact given its inputs, and available only under restrictive assumptions.
Worked example
What two parameters buy, on a two-sleeve account
62 percent at 22 percent volatility, 38 percent at 31 percent volatility, correlation 0.45. Illustrative figures.
- First sleeve contribution
- 0.018605
- Second sleeve contribution
- 0.013877
- Cross term
- 0.014461
- Portfolio volatility
- 21.67%
- Weighted average of the two volatilities
- 25.42%
- Work required
- One line
(0.62 × 0.22)².
(0.38 × 0.31)².
2 × 0.62 × 0.38 × 0.45 × 0.22 × 0.31.
√0.046943.
0.1364 + 0.1178.
No simulation, no path, no random numbers.
The whole distribution of the combined account came from three numbers. That is the purchase, and the rest of this track is the bill.
Check yourself
Two sleeves, 62 percent and 38 percent of an account, with volatilities of 22 and 31 percent and a correlation of 0.45. Why is the portfolio volatility 21.67 percent rather than 25.42 percent?
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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.