(While this procedure for valuation and duration estimation is not strictly correct, a number of firms use it.) The liability duration is assumed to be 1 1. The final, two columns in the upper half of the table give the value of surplus (net worth is the analog for non insurance companies), S, and the duration of surplus, _DS._ In this example, the duration of surplus is zero.<sup>5</sup> The problem with the duration estimates shown in the upper half of the table is that each is derived with respect to a different reference rate of interest.

**Exhibit** **_I_ Balance Sheet Conversion of Duration to Effective Duration**

|Grade|_A_|_DA_||_L_|_DL_|||_S_|_DS_||
|---|---|---|---|---|---|---|---|---|---|---|
|AA|_55_|10|||||||||
|BB|55|10||100|<br>11|||10|0||
|_Total:_|110|10||100|11|||10|0||
|Grade|_A_|Basis Risk<br>Adjustment<br>_DA_<br>Pactor|_EDA_|_L_|_DL_|Basis Risk<br>Adjustment<br>Factor|_EDL_|_S_<br>|_DS_|_EDS_|
|AA|55|10<br>0.95|9.5||||||||
|RB|55|10<br>0.80|8.0|100|11|<br>1.15|12.65|10|0|-30.25|
|_Total:_|110|10|8.75|100|11||12.65|10|0|-30.25|
|||Liabilities adjuste<br>correct discount|d for<br>rate:|103|11|<br>1.15|12.65|7||-48.641|

In the lower half of the table, the previous elements are repeated, but adjustments are introduced to account for basis risk in the duration measures. These adjustment factors reflect the fact that yields on AA bonds may fluctuate only 95% as much as Treasury yields, whereas yields on BB bonds may fluctuate only 80% as much as Treasury yields. (The adjustment numbers used were chosen to illustrate the issue, and are not precise.) When the adjustments are applied to the duration measures, we produce effective duration of assets measures, _EDA,_ for the two grades of assets held by the firm. The overall asset portfolio effective duration, defined with respect to movements in the reference Treasury rate, is now 8.75. Furthermore, the effective duration of liabilities, _EDL,_ when taken with respect to the reference Treasury rate, is now increased by 15% and is 12.65. (For guaranteed insurance liabilities, the appropriate discount rate is very close to Treasury rates and independent of the asset portfolio yield.) When taken together, and levered by the assets-to-surplus ratio, the effective duration of surplus, _EDS,_ is now -30.25. Thus, what initially looked like an immunized balance sheet is actually highly exposed to interest rate risk.

But we must not stop there. Not only does the liability duration measure need to be augmented by 15% to reflect the fact that the driving interest rate for liabilities is the Treasury rate, but the value of liabilities themselves must also reflect the lower discount rates.<sup>6</sup> Since the hypothetical insurer was using the blended yield on its asset portfolio to discount the liabilities, the valuation using lower Treasury rates will result in a higher lia5The duration of surplus is equal to the difference in asset and iiability durations, multiplied by the assets-tosurplus ratio, and augmented by the duration of liabilities.

6 This applies equally whether a single discount rate is used to value insurance liabilities, or an entire lattice of stochastic interest rates is used to value them.

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