Controling ex-post volatility by ex-ante limits

Question

In the context of mutual funds the KID directive forces us to calculate 5 year ex-post volatility of a (market) fund (weekly returns). Thus each week we look back in the past and calculate volatility (annualized) using the last $5*52 = 260$ weekly returns.

Given a $5$ year history and thus a recent ex-post volatility how can we set an ex-ante limit for a market fund in order not to breach a certain level of ex-post volatility in the next period?

To start a chain of thoughts: We are given returns $r_1,\ldots, r_{260}$ now and one week later we are given returns $r_2\ldots r_{260}$ and a new return which we assume ex-ante to be normally distributed with volatility $\sigma$ then the future ex-post variance is given by (assume an average return of $0$ for simplicity) $$ 1/259 \sum_{i=2}^{260} r_i^2 + 1/259 X^2 \sigma^2/52, $$ where $X^2$ is chi-squared distributed with $1$ df (the factor $1/259$ is the usual factor for the variance estimator and $1/52$ scales annualized variance to weekly). Thus if we want that the ex-post vol is below a threshold $t$ we need that $$ 1/259 \sum_{i=2}^{260} r_i^2 + 1/259 X^2 \sigma^2/52 \le t^2 $$ and thus $$ X^2 \sigma^2/52 \le 259*t^2-\sum_{i=2}^{260} r_i^2, $$ and we could use this equation to set a limit on ex-ante vol.

Answer 1

The chain of thoughts above seem to work. I post some R code where I implemented this. Seems to work well. Comments are welcome!

r.hist = rnorm(5*52,0,0.18/sqrt(52)) ## 18% vola
limit.vol = 0.2 ## limit 20% vola

## next period 5% confidence

gap = 0.2^2/52*259-sum(r.hist[2:259]^2)
var.treshold = gap/qchisq(0.95, 1)
target.vol = sqrt(var.treshold*52)
cat("Target vola",round(target.vol*100),"%")
## simulate
ex.post.vol = NULL
N = 1000
new.ret = rnorm(N,0,target.vol/sqrt(52))
for (i in 1:N){
r.new = c(r.hist[2:259],new.ret[i])
ex.post.vol = c(ex.post.vol,sd(r.new)*sqrt(52))
}

plot(ex.post.vol,main="Sampled ex-post vol one week later")
quantile(ex.post.vol,0.95)
quantile(ex.post.vol,0.99)