Question

You own a fixed income portfolio with a single 10-period zero coupon bond with face value of $100 and a current yield of 6% per period. During the past 100 trading days, there were 50 days when the yield on this bond did not change; 15 days when the yield increased by 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 increase 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 2.46 basis points. The 1-day 99% VaR using historical simulation method is ______________ . The modified duration of this zero coupon bond is ___________ . The 1-day VaR using Delta-Normal approach is _____________ . (Only keep two decimal places for this question.)

Answer 1

Historical Simulation Var:

The lowest return represents the 1% lower tail of the “distribution” of 100 historical returns. The lowest return (–0.0010) is the 1% daily VaR We would say there is a 1% chance of a daily loss exceeding 0.1%, or $1.

Modified Duration of ZCB: Macaulay Duration of ZCB = Time period of ZCB = 10

Modified Duration = Mac Duration / (1 + r) = 10 / (1 + 6%) = 9.43

Delta-Normal approach:

To locate the value for a 1% VaR, we use Cumulative z-Table. When using this table, you can look directly for the significance level of the VaR for example, if you desire a 1% VaR, look for the value in the table which is closest to (1 − significance level) or 1 − 0.01 = 0.9900. You will find 0.9901, which lies at the intersection of 2.3 in the left margin and 0.03 in the column heading.

Adding the z-value in the left-hand margin and the z-value at the top of the column in which 0.9901 lies, we get 2.3 + 0.03 = 2.33, so the z-value coinciding with a 99% VaR is 2.33.

where:

= expected 1-day return on the portfolio

= [50 * 0 + 15 * 0.0001+ 15 * (-0.0001) + 9 * 0.0005 + 9 * (-0.0005) + 1 * 0.0010 + 1 * (-0.0010)] / 100

= 0%

VP = value of the portfolio = 100

z = z-value corresponding with the desired level of significance = 2.33

σ = standard deviation of 1-day returns = 0.000246

Var: [0 – 2.33 * 0.000246] * 100

= - 0.057318

The interpretation of this VaR is that there is a 1% chance the minimum 1-day loss is 0.057318

Feel free to ask any Query in Comment Section

Please provide feedback.

Cheers