## Limits: Trading Limits – Duration, Convexity and PVBP Limits

*in*Middle Office and VaR

# Other Limits

## Duration Limits

Duration measures the sensitivity of the price of the product/ value of the portfolio to changes in the interest rate. In order to limit the sensitivity the company needs to decide what the acceptable level of duration for the product/ portfolio should be. For example it may want the portfolio duration to be not more than 2 years. In other words it would like to limit its exposure to interest rate changes so that an increase in interest rates of 1% would not result in a decline in the portfolio value of greater than 2%. For example if the portfolio value was AED 1 million then the limit will be breached if the portfolio’s value fell below AED 980,000.

## Convexity Limits

Duration does not account accurately for large changes in the interest rates. So in order to assess sensitivity more accurately the convexity measure may also be included in the calculation. The duration & convexity metric will then be used in setting the exposure limits. This metric is calculated as follows for duration and convexity:

**A 1% increase in interest rates leads to an x% decrease in price where x = absolute value of {-1 × Duration**** × (0.01) x 100 + Convexity**** × (0.01) ^{2 }x 100}. **

This is illustrated in the following example for 3-year bond issues. The duration limit is set at 2.3% whereas the duration& convexity limit is set at 2.2%:

3-year Bond | P_{0} | P- | P+ | Duration metric (%) | Duration & Convexity metric (%) | Duration Limit | Duration & Convexity Limit |

Issue 1 | 97.51 | 99.95 | 95.14 | 2.46 | 2.42 | Breach | Breach |

Issue 2 | 97.56 | 99.93 | 95.27 | 2.39 | 2.35 | Breach | Breach |

Issue 3 | 97.67 | 99.92 | 95.47 | 2.28 | 2.24 | Safe | Breach |

Issue 4 | 98.02 | 99.94 | 96.15 | 1.93 | 1.91 | Safe | Safe |

Figure 9: Setting Duration & Convexity limits

## PVBP Limits

PVBP is the change in present value of an instrument brought about by a 1 basis point change in interest rates. PVBP limits may be set separately for each maturity bucket or may be evaluated across all buckets. The PVBP metric is calculated to determine the change in present value by increasing the underlying interest rates by 0.01%. If the PVBP metric calculated for the portfolio exceeds this limit, a limit violation occurs. This is illustrated below:

Interest Rate | 6% | Maturity Bucket | ||||

Total Across Buckets | 0-1 | 1-2 | 2-3 | 3-4 | 4-5 | |

Amount | 125,000 | 25,000 | 25,000 | 25,000 | 25,000 | 25,000 |

PV @ time zero | 105,309 | 23,585 | 22,250 | 20,990 | 19,802 | 18,681 |

PVBP metric | 29 | 2.22 | 4.20 | 5.94 | 7.47 | 8.81 |

Limit as a % of PV | 0.02% | 0.01% | 0.02% | 0.03% | 0.03% | 0.04% |

PVBP Limit | 27 | 2.17 | 3.53 | 6.38 | 6.93 | 7.52 |

Limit Status | Breach | Breach | Breach | Safe | Breach | Breach |

Figure 10: Setting PVBP limits