Update: I am correcting another mistake in approximating R^M_i. I had argued it is affected by the level of interest rates r, but overlooked it contains a yield term r_i describing value accrued through time. Unfortunately this accrual yield depends on the cohort. Below I apply
\(R^M_i = r_i \delta t + \Delta_i \delta r + \frac{1}{2}\Gamma_i (\delta r)^2\)
and introduce (and recall)
\(V_{Xr} = \sum_j x_j r_j V_j; \quad\text{recalling }\;
V'_i = V_i\left(1 + (1-x_i)R^c + x_i R^M_i + y_i R^O \right)\)
the two middle terms together form the hedge return R^H_i in DNB formulas.
In a previous post on allocation management I derived portfolio sensitivities to market factors under new policy for WTP. That was getting long and I was getting tired so I left a question open at the end, that is an itch I want to scratch today.
In particular, I had left excess return R^O in the expressions for interest rate sensitivities. But this depends on other quantities (when we account for convexity):
\(\begin{split}
R^O V_Y = O &= CFR - \sum_j R^H_j V_j = CFR - \sum_j V_j \left( (1-x_j)R^c + x_j R^M_j \right)\\
&= (cfr- r^c(V-V_X) - V_{Xr})\delta t - \delta r V_{X\Delta} - \frac{1}{2}(\delta r)^2 V_{X\Gamma}
\end{split}\)
(Where cfr \delta t = CFR for notation)
The aggregated first-order sensitivity to interest rates was shown to be given by:
\(V'_{X\Delta} = \sum_j x_j \Delta_j V'_j = V_{X\Delta}
+\left(r^c (V_{X\Delta}-V_{X^2\Delta})
+ V_{X^2\Delta r} \right) \delta t
+\delta r V_{X^2\Delta^2} + (\delta r)^2 V_{X^2\Delta\Gamma}
+R^O V_{X\Delta Y}\)
but substituting R^O from above this becomes:
\(\begin{split}
V'_{X\Delta} &= V_{X\Delta}
+r^c \left(V_{X\Delta}-V_{X^2\Delta} - \frac{V_{X\Delta Y}}{V_Y}(V-V_X) \right) \delta t
+cfr \frac{ V_{X\Delta Y}}{V_Y} \delta t \\
& +\delta t \left(V_{X^2 \Delta r} - \frac{ V_{X\Delta Y}}{V_Y} V_{Xr} \right)
+\delta r \left( V_{X^2\Delta^2} -\frac{ V_{X\Delta Y}}{V_Y} V_{X\Delta}\right)
+ \frac{1}{2}(\delta r)^2 \left( V_{X^2\Delta\Gamma} -\frac{ V_{X\Delta Y}}{V_Y} V_{X\Gamma}\right)
\end{split}
\)
Similarly, the aggregated second-order sensitivity was given by
\(V'_{X\Gamma} = \sum_j x_j \Gamma_j V'_j = V_{X\Gamma}
+\left(r^c (V_{X\Gamma}-V_{X^2\Gamma})
+V_{X^2\Gamma r} \right) \delta t
+\delta r V_{X^2\Delta\Gamma}
+\frac{1}{2} (\delta r)^2 V_{X^2\Gamma^2}
+R^O V_{X\Gamma Y}\)
and subsituting R^O this becomes
\(\begin{split}
V'_{X\Gamma} &= V_{X\Gamma}
+r^c \left(V_{X\Gamma}-V_{X^2\Gamma} - \frac{V_{X\Gamma Y}}{V_Y}(V-V_X) \right) \delta t
+cfr \frac{ V_{X\Gamma Y}}{V_Y} \delta t\\
&+\delta t \left( V_{X^2\Gamma r} -\frac{ V_{X\Gamma Y}}{V_Y} V_{X r}\right)
+\delta r \left( V_{X^2\Delta\Gamma} -\frac{ V_{X\Gamma Y}}{V_Y} V_{X\Delta}\right)
+ \frac{1}{2}(\delta r)^2 \left( V_{X^2\Gamma^2} -\frac{ V_{X\Gamma Y}}{V_Y} V_{X\Gamma}\right)
\end{split}\)
Now this may give readers headaches, but notice the structure of the expression. A portfolio manager knows CFR = V’-V and observes (\delta r) and R^c in the market (and \delta t on their calendar). All other V_{xyz} terms are fixed actuarial quantities—the above expressions allow PMs to estimate target hedge sensitivities for LDI management.
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