Question

You own a fixed income portfolio with a single 10-period zero-coupon bond with a face value of $100 million and a current yield of 6% per period. During the past 100 trading days there were 50 days when the yield on these bonds did not change, 15 days when the yield increased 1 basis point, 15 days when the yield decreased by 1 basis point, 9 days when the yield increased by 5 basis points, 9 days when the yield decreased by 5 basis points, 1 day when the yield increased by 10 basis points, 1 day when the yield decreased by 10 basis points. During this 100 day estimation period, the estimated standard deviation of daily interest rate changes equals 2.36 basis points.

A) What is 1-day 99% VAR using historical simulation?

B) What is 1-day 95% VAR using historical simulation?

C) What is the 99% 1-day Delta-Normal VAR?

Answer 1

A). 1-day 99% VAR using historical simulation: The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective. As a historical example, let's look at the Nasdaq 100 ETF, which trades under the symbol QQQ (sometimes called the "cubes"), and which started trading in March of 1999.2 If we calculate each daily return, we produce a rich data set of more than 1,400 points. Let's put them in a histogram that compares the frequency of return "buckets." For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%. You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:

• With 99% confidence, we expect that the worst daily loss will not exceed 7%.
• Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.

B) 1-day 95% VAR using historical simulation: Notice the red bars that compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VAR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:

• With 95% confidence, we expect that our worst daily loss will not exceed 4%.
• If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).

C). 99% 1-day Delta-Normal VAR: This method assumes that stock returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve. Here we plot the normal curve against the same actual return data. The idea behind the variance-covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve. They are a function of our desired confidence and the standard deviation.

Confidence # of Standard Deviations (σ)

99% (really high)   - 2.33 x σ

The average daily return happened to be fairly close to zero, so we will assume an average return of zero for illustrative purposes. Here are the results of plugging the actual standard deviation into the formulas above: