Optimal management of immunized portfolios


We generalize the contribution of Fong and Vasicek (Financ Anal J 39:73–78, 1983a; Innov Bond Portf Manag Durat Analy Immun 1983:227–238, 1983b; J Financ 39:1541–1546, 1984) by developing a risk-return optimization problem for immunized life insurance portfolios. The M2 measure of risk for arbitrary changes of the term structure of interest rates is used for a bond portfolio with duration matched to a given liability horizon. The immunization case becomes a “passive” strategy among an entire menu of active management decisions in which a partial risk minimization is exchanged for more return potential. As in the classical Markowitz (Portfolio Selection, New York, Wiley, 1959) approach, an efficient frontier at the given horizon provides the optimal tradeoff between risk and return. An empirical application to insurance companies shows that such a perspective may be proved useful to highlight which segregated funds can be re-positioned over the efficient frontier, at the chosen level of the firm’s risk appetite.

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  1. 1.

    See Messmore [20] who shows that if E > 0 (the present value of assets greater than the present value of liabilities) the immunization of \(E\), i.e. \(D_{E} = 0\), implies \(D_{A} < D_{L}\). In the following we assume a fixed income portfolio with \(A = L\).

  2. 2.

    This multiple asset—single liability case is equivalent to the case of an asset only bond portfolio with target horizon H and target value \(\bar{L} = A(0)\exp \left( {\int \limits_{0}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)\).

  3. 3.

    The duration measure was independently obtained by Hicks [13] with the name of “average period”. Earlier developments could be seen in Lidstone [16].

  4. 4.

    Clearly, interest rate dynamics implying constant shifts are not arbitrage-free. See Boyle [3].

  5. 5.

    As duration is linked to the first derivative of the price with respect to the interest rate, convexity represents the second derivative. It is easy also to show that \(M^{2} = - \frac{\partial D}{\partial r}\).

  6. 6.

    This multiple liability immunization has been generalized by Shiu [28].

  7. 7.

    The EIOPA calculation of the liability side in Fig. 1 does not take into account all the contract optionality available in different countries so that the country relative positions could be altered.

  8. 8.

    For instance, see the EIOPA official presentation of the 2014 Stress Test in EIOPA [6].

  9. 9.

    See EIOPA [7], section C, “Low Yield Module Description and Results”, paragraph 2, sub-paragraph 56.

  10. 10.

    See Martellini, Priaulet and Priaulet [19], ch. 6 and the references therein.

  11. 11.

    Apart from EIOPA Stress Tests, convexity effects are still ignored in many empirical applications as well as popular textbooks in banking and finance (e.g. Mishkin and Apostolos [21]). Moreover, under the Basel III Capital Requirement Regulation (CRR, in force since 1 January 2014), the Standardised Approach includes a calculation for the own funds requirement for the general risk on debt instruments only based on the duration. See European Banking Authority (EBA), Interactive Single Rulebook, article 340 in https://www.eba.europa.eu/regulation-and-policy/single-rulebook. A partial justification can be found in the relevance of the first order effect (duration) as documented in Schaefer [26] and in the empirical literature on Principal Components (Martellini, Priaulet and Priaulet [19], ch. 3) where the first factor (the parallel shift of the term structure) accounts for 50% to 90% of total variability of interest rate changes, especially in the long side of the yield curve.

  12. 12.

    Note that using (realistically) bonds instead of cash flows has no effect on the calculation of duration but it affects the calculation of risk. In fact, the duration of a portfolio of bonds is simply the “portfolio” (average) of bond durations; however, the M2 of a portfolio of bonds is the “portfolio” of M2 (within variance) plus the variance of durations (between variance). See the Appendix 4.

  13. 13.

    Note that Fong and Vasicek [10] assume, in the variance calculation, the asymptotic approximation: \(\mathop \sum \nolimits_{t = 0}^{H} M_{t}^{4} = \frac{{M_{0}^{4} }}{{H^{6} }}\mathop \sum \nolimits_{t = 0}^{H} (H - t)^{6} = \frac{{M_{0}^{4} }}{{H^{6} }}\frac{{H(H + 1)(2H + 1)(3H^{4} + 6H^{3} - 3H + 1)}}{42}\mathop \to \limits_{{for H\mathop \to \nolimits_{ } \infty }} \frac{H}{7}M_{0}^{4}\)

    so that the variance is proportional to \(\frac{{M_{0}^{4} }}{7H}\). We do not use this asymptotic approximation.

  14. 14.

    Because of the profit participation mechanism, the liability cash-flows partly depend on the asset cash-flows, through the returns realized by the assets in the segregated funds. Usually, this effect is negligible.

  15. 15.

    Note that the second order (convexity) condition is always satisfied in the single liability case.

  16. 16.

    See for example, Smith [30], p. 1670, formula (9) or de La Grandville [5], p. 163, formula (19). With continuous compounding, the formula simplifies into M2 = C – D.

  17. 17.

    Bloomberg data include in the price P accrued interest (full price = clean price + accrued interest) and assume a variation of ± 1 basis point w.r.t. the yield implicit in the price P.


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We thank Antonio De Pascalis, Lino Matarazzo, Enzo Orsingher, Stefano Pasqualini, Fabio Polimanti, Giovanni Rago, Arturo Valerio and the participants to a seminar of IVASS and Bank of Italy for their comments. A special thank to Oldrich Vasicek and two anonymous referees for their detailed comments and suggestions to a previous version of the paper. All remaining errors are ours.

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Correspondence to Riccardo Cesari.


Appendix 1: Proof of the theorems

Proof of Theorem 1

The current budget constraint expressed at the forward time H is:

$$\mathop \sum \limits_{j = 1}^{m} A_{j} \exp \left( {\int \limits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right) \equiv \mathop \sum \limits_{j = 1}^{m} a_{j} = \bar{L}$$


$$G\left( {r_{FW} + \Delta_{0} } \right) \equiv \mathop \sum \limits_{j = 1}^{m} A_{j} \exp \left( {\int \limits_{{s_{j} }}^{H} \left( {r_{FW} \left( {0,\tau } \right) + \Delta_{0} } \right)d\tau } \right) - \bar{L} \equiv \Delta A_{0} \left( H \right)$$

We have:

$$\frac{\partial G}{{\partial \Delta_{0} }} = \mathop \sum \limits_{j = 1}^{m} A_{j} \left( {H - s_{j} } \right)\exp \left( {\int \limits_{{s_{j} }}^{H} \left( {r_{FW} \left( {0,\tau } \right) + \Delta_{0} } \right)d\tau } \right)$$
$$\frac{{\partial^{2} G}}{{\partial \Delta_{0}^{2} }} = \mathop \sum \limits_{j = 1}^{m} A_{j} \left( {H - s_{j} } \right)^{2} \exp \left( {\int \limits_{{s_{j} }}^{H} \left( {r_{FW} \left( {r,\tau } \right) + \Delta_{0} } \right)d\tau } \right) > 0$$

so that \(G\left( {r_{FW} + \Delta_{0} } \right) = 0\) for \(\Delta_{0} = 0\) and \(G\left( {r_{FW} + \Delta_{0} } \right) \ge 0\) for every \(\Delta_{0}\) in a neighbourhood of \(0\), if \(\Delta_{0} = 0\) is a (local) minimum i.e. if:

$$\frac{\partial G}{{\partial \Delta_{0} }}_{|\Delta_0 = 0} = \mathop \sum \limits_{j = 1}^{m} A_{j} \left( {H - s_{j} } \right)\exp \left( {\int \limits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right) = 0$$

where the derivative is taken at Δ0 = 0. Solving for H and multiplying and dividing by \(P\left( {0,H} \right)\), we have the duration conditionFootnote 15:

$$D\left( A \right) = \frac{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} s_{j} \exp \left( { - \int \nolimits_{0}^{{s_{j} }} r_{FW} \left( {0,\tau } \right)d\tau } \right)}}{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( { - \int \nolimits_{0}^{{s_{j} }} r_{FW} \left( {0,\tau } \right)d\tau } \right)}} = H$$


Proof of Theorem 2

Using the notation introduced in the proof of Theorem 1, define:

$$f\left( s \right) \equiv \exp \left( {\int \limits_{s}^{H} \Delta_{0} \left( \tau \right)d\tau } \right)$$

so that:\(f\left( H \right) = 1\), \(f^{'} \left( s \right) = - f\left( s \right)\Delta_{0} \left( s \right)\), \(f^{''} \left( s \right)\left[ {\Delta_{0}^{2} \left( s \right) - \Delta_{0}^{'} \left( s \right)} \right] \ge 0\) (by the “convexity condition”)

Then the change in value is:

$$\Delta A_{0} \left( H \right) \equiv G\left( {r_{FW} + \Delta_{0} } \right) - G\left( {r_{FW} } \right) = G\left( {r_{FW} + \Delta_{0} } \right) = \mathop \sum \limits_{j = 1}^{m} a_{j} \left( {f\left( {s_{j} } \right) - 1} \right)$$

and by Taylor (exact) formula:

$$f\left( {s_{j} } \right) = f\left( H \right) + \left( {s_{j} - H} \right)f^{'} \left( H \right) + \frac{1}{2}\left( {s_{j} - H} \right)^{2} f^{''} \left( {\xi_{j} } \right)$$

so that, substituting:

$$\begin{aligned}G\left( {r_{FW} + {{\Delta }}_{0} } \right) =& - \mathop \sum \limits_{j = 1}^{m} a_{j} \left( {s_{j} - H} \right){{\Delta }}_{0} \left( H \right) + \frac{1}{2}\mathop \sum \limits_{j = 1}^{m} a_{j} \left( {s_{j} - H} \right)^{2} f\left( {\xi_{j} } \right)\left[ {{{\Delta }}_{0}^{2} \left( {\xi_{j} } \right) - {{\Delta }}_{0}^{'} \left( {\xi_{j} } \right)} \right] \\ =&\frac{1}{2}\mathop \sum \limits_{j = 1}^{m} a_{j} \left( {s_{j} - D} \right)^{2} f\left( {\xi_{j} } \right)\left[ {{{\Delta }}_{0}^{2} \left( {\xi_{j} } \right) - {{\Delta }}_{0}^{'} \left( {\xi_{j} } \right)} \right] \ge 0\end{aligned}$$

where the second equality comes from the duration matching hypothesis D = H and the inequality from the “convexity condition”.\(\square\)

An alternative proof is given by Montrucchio and Peccati [22].

Proof of Corollary of Theorem 2

Using Taylor’s formula around \(H\):

$$f\left( {s_{j} } \right) \simeq f\left( H \right) + \left( {s_{j} - H} \right)f^{'} \left( H \right) + \frac{1}{2}\left( {s_{j} - H} \right)^{2} f^{''} \left( H \right) = 1 - \left( {s_{j} - H} \right)\Delta_{0} \left( H \right) + \frac{1}{2}\left( {s_{j} - H} \right)^{2} \left[ {\Delta_{0}^{2} \left( H \right) - \Delta_{0}^{'} \left( H \right)} \right]$$

so that, for \(H = D\):

$$\frac{{\Delta A_{0} \left( H \right)}}{{A_{0} \left( H \right)}} \equiv \frac{{G\left( {r_{FW} + \Delta_{0} } \right)}}{{\bar{L}}} \simeq \frac{1}{{2\bar{L}}}\mathop \sum \limits_{j = 1}^{m} a_{j} \left( {s_{j} - D} \right)^{2} \left[ {\Delta_{0}^{2} \left( D \right) - \Delta_{0}^{'} \left( D \right)} \right] \equiv \frac{1}{2}M_{0}^{2} \left( D \right)\left[ {\Delta_{0}^{2} \left( D \right) - \Delta_{0}^{'} \left( D \right)} \right]$$


Proof of Theorem 3

From the first order approximation for \(f\left( s \right) = e^{x\left( s \right)} \ge 1 + x\left( s \right)\) we have:

$$G\left( {r_{FW} + \Delta_{0} } \right) = \mathop \sum \limits_{j = 1}^{m} a_{j} \left( {f\left( {s_{j} } \right) - 1} \right) \ge \mathop \sum \limits_{j = 1}^{m} a_{j} \int \limits_{{s_{j} }}^{H} \Delta_{0} \left( \tau \right)d\tau$$

By the integral formula:

$$\Delta_{0} \left( H \right) - \Delta_{0} \left( \tau \right) = \int \limits_{\tau }^{H} \Delta_{0}^{'} \left( u \right)du$$

so that, integrating and using the Fubini theorem to change the order of integration:

$$\begin{aligned}\int \limits_{{s_{j} }}^{H} \Delta_{0} \left( \tau \right)d\tau =& \left( {H - s_{j} } \right)\Delta_{0} \left( H \right) - \int \limits_{{s_{j} }}^{H} \int \limits_{\tau }^{H} \Delta_{0}^{'} \left( u \right)dud\tau = \left( {H - s_{j} } \right)\Delta_{0} \left( H \right) - \int \limits_{{s_{j} }}^{H} \Delta_{0}^{'} \int \limits_{{s_{j} }}^{u} d\tau du \\=& \left( {H - s_{j} } \right)\Delta_{0} \left( H \right) - \int \limits_{{s_{j} }}^{H} \Delta_{0}^{'} \left( u \right)\left( {u - s_{j} } \right)du\end{aligned}$$

If \(K_{0} \equiv \mathop {\hbox{max} }\limits_{u} \left\{ {\Delta_{0}^{'} \left( u \right)} \right\}\).,

$$\int \limits_{{s_{j} }}^{H} \Delta_{0} \left( \tau \right)d\tau \ge \left( {H - s_{j} } \right)\Delta_{0} \left( H \right) - K_{0} \int \limits_{{s_{j} }}^{H} \left( {u - s_{j} } \right)du = \left( {H - s_{j} } \right)\Delta_{0} \left( H \right) - \frac{1}{2}K_{0} \left( {H - s_{j} } \right)^{2}$$

so that, using the duration matching condition \(D = H\):

$$G\left( {r_{FW} + \Delta_{0} } \right) \ge - \frac{1}{2}K_{0} \mathop \sum \limits_{j = 1}^{m} a_{j} \left( {D - s_{j} } \right)^{2} = - \frac{1}{2}K_{0} M_{0}^{2} \left( D \right)\bar{L}$$


Shiu [27] uses an exact Taylor formula for \(\int \limits_{{s_{j} }}^{H} \Delta_{0} \left( \tau \right)d\tau\) and Nawalkha and Chambers [23] use higher-order approximations for \(f\left( s \right)\).

Appendix 2: The shift function

The Fong–Vasicek [11] sufficient condition for immunization is a constraint on the shift function:

$$\begin{aligned} \Delta_{t} \left( \tau \right) \equiv r_{FW} \left( {t + dt,\tau } \right) - r_{FW} \left( {t,\tau } \right): \hfill \\ \Delta_{t}^{2} \left( \tau \right) - \Delta_{t}^{'} \left( \tau \right) = c \ge 0,\forall {{\tau }} \hfill \\ \end{aligned}$$

Note that the term “convexity condition” stems from the fact that under this condition the function:

$$f\left( s \right) \equiv \exp \left( {\int \limits_{s}^{H} \Delta_{t} \left( \tau \right)d\tau } \right)$$

is convex (\(f^{''} \left( s \right) \ge 0\)).

Omitting the subscript, we can check that \(\Delta \left( \tau \right) =\) const (uniform additive positive or negative shift) satisfies the convexity condition. The same holds for any shift for which \(\Delta^{'} \left( \tau \right)\) is negative (decreasing term structure of shifts): for example \(\Delta \left( \tau \right) = \exp \left( { - a\tau } \right)\) for \(a = \sqrt c \ge 0\). In general, the convexity condition is an ordinary, nonlinear differential equation of the Riccati type.

Using the substitution \(\Delta = - u'/u\) we obtain:

$$u^{''} \left( \tau \right) - c\left( \tau \right)u\left( \tau \right) = 0$$

The special case \(c\left( \tau \right) = \tau\) is well known because one solution is the Airy function (Abramowitz and Stegun (eds.) [1], ch. 10, p.446):

$$u\left( \tau \right) = \frac{\sqrt \tau }{3}\left[ {I_{{ - \frac{1}{3}}} \left( {\frac{2}{3}\tau^{3/2} } \right) - I_{{\frac{1}{3}}} \left( {\frac{2}{3}\tau^{3/2} } \right)} \right]$$

where \(I_{\nu } (x)\) is the modified Bessel function of order ν (Abramowitz and Stegun (eds.) [1], ch. 9 p. 375).

The integral representation is:

$$u\left( \tau \right) = \frac{1}{\pi }\int \limits_{0}^{\infty } \cos \left( {\frac{{x^{3} }}{3} + x\tau } \right)dx$$

so that:

$$\Delta \left( \tau \right) = \frac{{\int \nolimits_{0}^{\infty } \sin \left( {\frac{{x^{3} }}{3} + x\tau } \right)xdx}}{{\int \nolimits_{0}^{\infty } \cos \left( {\frac{{x^{3} }}{3} + x\tau } \right)dx}}$$

In the stochastic case, the shift function is the stochastic change in the forward rate \(dr_{FW} \left( {t,\tau } \right)\). In the Vasicek term structure model it is given by \(e^{ - r\tau } dr\left( t \right)\), i.e. the change in the short term rate smoothed by a negative exponential. Using the shift function implied by the EIOPA Stress Test (see Fig. 2) we obtain the convexity condition displayed in Fig. 4.

Fig. 4

EIOPA shift function \(\Delta \left(\varvec{\tau}\right)\) and convexity condition

Appendix 3: The stochastic approach to immunization

Under the no-arbitrage approach of financial markets, the case of “flat shifts” and the immunization property \(A\left( H \right) \ge \bar{L}\) imply a riskless arbitrage and are therefore not compatible with equilibrium [3, 15]. In continuous time, under stochastic term structure models, the immunization condition means that the value of assets in the next instant must be equal to the value of liabilities, so that, in differential terms: \(dA\left( t \right) = dL\left( t \right)\) and immunization is guaranteed if the value of assets and liabilities have the same sensitivity to the state variables [2]. This has suggested the definition of a generalized or stochastic duration concept [4].

As an example of this stochastic approach to the asset-liability problem, let us consider the case of the extended-Vasicek [31], no arbitrage term structure model [14]:

$$dr\left( s \right) = \left[ {\hat{a}\left( s \right) - kr(s)} \right]ds + \sigma d\hat{Z}(s)$$

where the time-dependent drift \(\hat{a}(s)\) is obtained in order to guarantee a perfect fit of the theoretical term structure with the current yield curve:

$$\hat{a}\left( s \right) = kr_{FW} \left( {t,s} \right) + \frac{{\partial r_{FW} (t,s)}}{\partial s} + \frac{{\sigma^{2} }}{2k}(1 - e^{{ - 2k\left( {s - t} \right)}} )$$

The solution for \(r\left( s \right)\) is:

$$r\left( s \right) = r_{FW} \left( {t,s} \right) + \frac{{\sigma^{2} }}{2}G^{2} \left( {t,s} \right) + \sigma \int \limits_{t}^{s} e^{{ - k\left( {s - u} \right)}} d\hat{Z}\left( u \right)$$
$$G\left( {t,s} \right) = \frac{{1 - e^{ - k(s - t)} }}{k}$$

and the relative difference between ex post and ex ante portfolio value is:

$$\begin{aligned}\frac{{A\left( H \right) - A_{0} \left( H \right)}}{{A_{0} \left( H \right)}} = &\frac{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( {\int \nolimits_{{s_{j} }}^{H} r\left( u \right)du} \right) - \mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( {\int \nolimits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)}}{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( {\int \nolimits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)}} \\=& \frac{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( {\int \nolimits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)\left[ {\exp \left( {\int \nolimits_{{s_{j} }}^{H} r\left( u \right)du - \int \nolimits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right) - 1} \right]}}{{\mathop \sum \nolimits_{j = 1}^{m} A_{j} \exp \left( {\int \nolimits_{{s_{j} }}^{H} r_{FW} \left( {0,\tau } \right)d\tau } \right)}} \\ \approx& \frac{{\mathop \sum \nolimits_{j = 1}^{m} a_{j} \left[ {\int \nolimits_{{s_{j} }}^{H} \left( {r\left( u \right) - r_{FW} \left( {0,u} \right)} \right)du} \right]}}{{\mathop \sum \nolimits_{j = 1}^{m} a_{j} }} \equiv \mathop \sum \limits_{j = 1}^{m} b_{j} X_{0j}\end{aligned}$$

The variable \(X_{{t_{j} }}\) has an explicit form in this case:

$$X_{tj} = \int \limits_{{s_{j} }}^{H} \left( {r\left( u \right) - r_{FW} \left( {t,u} \right)} \right)du = \frac{{\sigma^{2} }}{{2k^{2} }}\int \limits_{{s_{j} }}^{H} \left( {1 - e^{{ - k\left( {u - t} \right)}} } \right)^{2} du + \sigma \int \limits_{{s_{j} }}^{H} \int \limits_{t}^{u} e^{{ - k\left( {u - v} \right)}} d\hat{Z}(v)du$$

with mean:

$$E\left( {X_{tj} } \right) = \frac{{\sigma^{2} }}{{2k^{2} }}\int \limits_{{s_{j} }}^{H} \left( {1 - e^{{ - k\left( {u - t} \right)}} } \right)^{2} du = \frac{{\sigma^{2} }}{{k^{3} }}\left[ {\frac{k}{2}\left( {H - s_{j} } \right) + e^{ - k(H - t)} - e^{{ - k\left( {s_{j} - t} \right)}} - \frac{{e^{{ - 2k\left( {H - t} \right)}} }}{4} + \frac{{e^{{ - 2k(s_{j} - t)}} }}{4}} \right]$$

and covariance

$$Cov\left( {X_{ti} ,X_{tj} } \right) = \sigma^{2} E\left[ {\int \limits_{{s_{i} }}^{H} \int \limits_{t}^{u} e^{ - k(u - v)} d\hat{Z}\left( v \right)du\int \limits_{{s_{j} }}^{H} \int \limits_{t}^{s} e^{{ - k\left( {s - v} \right)}} d\hat{Z}\left( v \right)ds} \right] = \sigma^{2} \int \limits_{{s_{i} }}^{H} \int \limits_{{s_{j} }}^{H} \int \limits_{t}^{{{ \hbox{min} }(u,s)}} e^{{ - k\left( {u - v} \right) - k(s - v)}} dvdsdu$$

so that a risk-return approach can be analytically developed.

Appendix 4: Practical aspects of the computations of some financial variables

In this Appendix we give some technical details of the computations involving financial variables.

The computation of term structure changes

In the empirical application we use daily data and we calculate, from the 1 year-forward rates, rFW(t,τ, τ+1) for τ = 1,…,25 the daily changes:

$$\begin{aligned} \Delta_t (\tau ) \equiv r_{\text{FW}} ( {\text{t}}\, + \, 1,\tau ,\tau + 1 )- r_{\text{FW}} ( {\text{t}}\,,\tau ,\tau + 1 )\hfill \\ \Delta_t^{\prime } (\tau ) \equiv \Delta_t({\tau}\, + \, 1) - \Delta_{\text{t}} (\tau ) \hfill \\ \Delta {\text{S}}_{\text{t}} (\tau ) \equiv 1/2[\Delta_t^{2} (\tau ) - \Delta_t^{\prime } (\tau )] \hfill \\ \end{aligned}$$

The analytic, closed form computation of the portfolio cash-flow dispersion

It is possible to compute in closed-form, i.e. via an analytic formula, the cash-flow dispersion of a securities portfolio. Portfolio cash-flow dispersion turns out to be a quadratic function of portfolio duration, of its yield-to-maturity and its convexityFootnote 16:

$$M^{2} (D) = C^{Mod} \cdot \left( {1 + y} \right)^{2} - D \cdot \left( {D + 1} \right)$$

\(M ^{2} (D)\) = variance of the asset cash-flow maturities around their duration; y = portfolio yield-to-maturity, \(D\) = portfolio duration (average time-to-maturity), \(C^{Mod}\). = portfolio modified convexity.

The variance with respect to horizon H is obtained as:

$$M^{2} (H) = M^{2} \left( D \right) + (D - H)^{2}$$

Yield, duration and convexity aggregation at portfolio level

  • The computation of the portfolio aggregate yield-to-maturity

Traditionally one computes the average portfolio yield, weighted by the amount invested (w.r.t. either book-value or market value). This estimate represents an approximation to the yield-to-maturity as seen from the valuation date. A better alternative seems to consist of computing an estimate that takes into account the different time distance of cash-flows from the valuation date: one weights each yield by the “dollar duration” of its security. One obtains the so called WADD yield (Weighted Average Dollar Duration Yield, see Grondin [12]). The Dollar Duration is the product of duration and price of a security.

  • The computation of average portfolio Duration and Convexity

The (modified) duration was downloaded from Bloomberg, for each security in the data set (at the date 31/12/2014). Portfolio duration is the average (weighted by market value amounts) of the durations of single securities.

Convexity was also downloaded from Bloomberg, for each security (at the date 31/12/2014). Bloomberg approximated the second derivative (of price w.r.t. yield) as the “central difference” by varying yield-to-maturity by ± 1 basis point, that isFootnote 17:

$$C^{Mod} \cong \frac{1}{P} \cdot \frac{{\frac{{P^{ + } - P}}{\Delta y} - \frac{{P - P^{ - } }}{\Delta y}}}{\Delta y} = \frac{1}{P} \cdot \frac{{P^{ + } - 2 \cdot P^{ } + P^{ - } }}{{\underbrace {{(\Delta y)^{2} }}_{\Delta y = 0.01\% }}} = 10,000 \cdot \frac{{P^{ + } - 2 \cdot P^{ } + P^{ - } }}{P} .$$

Convexity is a “quadratic” measure and hence it can be linearly aggregated (by summation): by weighting with the amounts (at market value) one obtains portfolio convexity.

  • The bond benchmarks for maturity classes

The four bond classes used in the empirical application are represented by the following benchmark bonds:

Bond index Bond ISIN Time-to-maturity (years) Duration (years) Convexity M-squared
3 IT0004867070 2.84 2.72 10.31 0.21
5 IT0003644769 5.09 4.58 27.08 1.54
10 IT0004513641 10.17 8.27 86.32 9.59
20 IT0003535157 19.60 13.46 241.60 46.84
  • M-squared for a portfolio of bonds

Let \(B_{T} = B_{1} + B_{2}\) a portfolio of 2 bonds with cash flows \(c_{11} ,c_{12}\) and \(c_{21} ,c_{22}\) respectively at times \(t_{11} ,t_{12}\) and \(t_{21} ,t_{22}\) respectively.

Using a flat term structure:

$$\begin{aligned} B_{1} & = c_{11} \exp \left( { - rt_{11} } \right) + c_{12} \exp \left( { - rt_{12} } \right) \\ B_{2} & = c_{21} \exp \left( { - rt_{21} } \right) + c_{22} \exp \left( { - rt_{22} } \right) \\ D_{1} & = t_{11} c_{11} \exp \left( { - rt_{11} } \right)/B_{1} + t_{12} c_{12} \exp \left( { - rt_{12} } \right)/B_{1} \\ D_{2} & = t_{21} c_{21} \exp \left( { - rt_{21} } \right)/B_{2} + t_{22} c_{22} \exp \left( { - rt_{22} } \right)/B_{2} \\ D_{T} & = t_{11} c_{11} \exp \left( { - rt_{11} } \right)/B_{1} \left( {B_{1} /B_{T} } \right) + t_{12} c_{12} \exp \left( { - rt_{12} } \right)/B_{1} \left( {B_{1} /B_{T} } \right) + t_{21} c_{21} \exp \left( { - rt_{21} } \right)/B_{2} \left( {B_{2} /B_{T} } \right) \\+& t_{22} c_{22} \exp \left( { - rt_{22} } \right)/B_{2} \left( {B_{2} /B_{T} } \right) =D_{1} B_{1} /B_{T} + D_{2} B_{2} /B_{T} \\ M_{1}^{2} & = \left( {t_{11} - D_{1} } \right)^{2} c_{11} \exp \left( { - rt_{11} } \right)/B_{1} + \left( {t_{11} - D_{1} } \right)^{2} c_{12} \exp \left( { - rt_{12} } \right)/B_{1} \\ M_{2}^{2} & = \left( {t_{21} - D_{2} } \right)^{2} c_{21} \exp \left( { - rt_{21} } \right)/B_{2} + \left( {t_{22} - D_{2} } \right)^{2} c_{22} \exp \left( { - rt_{22} } \right)/B_{2} \\ M_{T}^{2} & = \left( {t_{11} - D_{T} } \right)^{2} c_{11} \exp \left( { - rt_{11} } \right) /B_{1} \left( {B_{1} /B_{T} } \right) + \left( {t_{12} - D_{T} } \right)^{2} c_{12} \exp \left( { - rt_{12} } \right) /B_{1} \left( {B_{1} /B_{T} } \right) \\+& \left( {t_{21} - D_{T} } \right)^{2} c_{21} \exp \left( { - rt_{21} } \right) /B_{2} \left( {B_{2} /B_{T} } \right) + \left( {t_{22} - D_{T} } \right)^{2} c_{22} \exp \left( { - rt_{22} } \right) /B_{2} \left( {B_{2} /B_{T} } \right)\\ =& \left[ {M_{1}^{2} + \left( {D_{1} - D_{T} } \right)^{2} } \right]\left( {B_{1} /B_{T} } \right) + \left[ {M_{2}^{2} + \left( {D_{2} - D_{T} } \right)^{2} } \right]\left( {B_{2} /B_{T} } \right) \\ \end{aligned}$$

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Cesari, R., Mosco, V. Optimal management of immunized portfolios. Eur. Actuar. J. 8, 461–485 (2018). https://doi.org/10.1007/s13385-018-0174-6

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  • Insurance companies
  • Immunization
  • Convexity
  • Fong–Vasicek theorem
  • Efficient frontier

JEL Classification

  • G22
  • G11
  • G12