Module 6
Portfolio risk: Value-at-Risk
Once you hold more than one bond, the question becomes: how much could this whole book lose on a bad day? Value-at-Risk (VaR)answers it with a single number — e.g. “we're 99% confident we won't lose more than $X tomorrow.” Its companion, Expected Shortfall, answers the follow-up: “and on the days we doblow through VaR, how bad is it on average?”
There are three ways to compute it, and they disagree in instructive ways:
- Parametric — assume a bell curve and use the portfolio's volatility (from each position's risk, the factor vols, and their correlations). Fast, but thin-tailed.
- Monte-Carlo — simulate thousands of random days and read the loss off the distribution. Flexible.
- Fat-tailed — same, but with a Student-t that has bigger tails, because real interest-rate moves crash harder than a bell curve predicts. It reports a larger, more honest loss.
And the most important free lunch in finance shows up here too: diversification. Because your positions aren't perfectly correlated, the portfolio's VaR is lessthan the sum of each position's standalone VaR. Lower the correlation and watch the benefit grow.
🎛 VaR lab
Your book — DV01 per factor ($/bp; negative = long, loses when rates rise)
Parametric VaR
$93,225
Expected Shortfall
$106,907
Monte-Carlo VaR
$92,308
Fat-tailed VaR
$102,235
Simulated daily P&L distribution (15,000 paths)
VaR is the loss you shouldn't exceed on a normal bad day (here 99% of days). Parametric assumes a bell curve; Monte-Carlo simulates it; the fat-tailed version uses a Student-t to reflect that real rate moves have bigger tails than a bell curve — so it reports a larger loss. Educational tool — not investment advice.
Things to try
- • Drop the correlation toward 0 — the diversification benefit jumps, and portfolio VaR falls well below the sum of parts.
- • Compare parametric vs fat-tailed VaR — the Student-t reports a bigger loss. That gap is why 2008 “shouldn't have happened.”
- • Switch to 10-day and 99% — how regulators size capital.