# Analytical Duration and Empirical Duration

## Differences between Analytical Duration and Empirical Duration

Analytical duration refers to estimating duration and convexity using mathematical formulas (as done in the previous learning objectives). Analytical duration approximates the effect of changes in benchmark yields on bond prices by assuming that the government bond yields and spreads are independent and uncorrelated variables.

On the other hand, empirical duration refers to constructing statistical models using historical data to estimate duration. The statistical data includes diverse factors impacting bonds prices. Since empirical duration gives estimates over a period of time, it is normally used in the decision-making process.

## Choice between Analytical Duration and Empirical Duration

The decision to either use analytical or empirical duration approaches depends on the correlation between benchmark yields and the credit spreads. For instance, when the correlation between benchmark yields and the credit spreads is negative, the empirical duration estimates will be lower since wider credit spreads will partly or fully offset the decrease in government benchmark yields.

As such, for a bond portfolio consisting of only government bonds (from the same issuer), the benchmark yields are the primary determinant of the bond yields. Thus, the outcomes of analytical duration and empirical duration are majorly the same. However, for a bond portfolio consisting of corporate bonds from different issuers, the interaction between benchmark yield changes and credit and liquidity spreads offset each other (especially when the market is stressed), leading to lower empirical duration estimates than analytical duration.

## Question

Empirical duration utilizes statistical and historical data to explain the price-yield relationship for particular bond or bond portfolios, especially in stressed market conditions. Which of the following is the most appropriate portfolio where empirical duration is applicable?

A portfolio of:

1. highly rated corporate bonds all from the same issuer.
2. highly rated sovereign bonds, all from the same issuer.
3. 30% highly rated sovereign bonds, 40% highly rated corporate bonds, and 30% speculative-grade corporate bonds, all from different issuers.

Solution

Empirical duration is appropriate compared to analytical duration in measuring the effect of yield changes to portfolio value, particularly under stressed market conditions and a bond portfolio consisting of different bonds from different issuers. Given the combination in option C, credit spreads on the high yield bond may be offset by yield changes on the highly-rated corporate and sovereign bonds.

Option A and B are incorrect.  The bond portfolios described are of the same type and from the same issuers. Therefore, empirical and analytical duration estimates should be broadly the same.

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