Abstract
Dynamic portfolio strategies, as opposed to static or “buy and hold” ones, covers a wide variety of short-term and long-term management styles that can be classified according to different criteria. Strategic allocation concerns a small number of asset classes (typically stocks, bonds and monetary securities) and depends on the investors’ risk aversion and wealth. Its horizon is long, typically several years. The first approach of strategic allocation is the so-called common sense management rules (allocation according to age, risk profile…). The second approach is portfolio insurance, based on automatic rules and designed to hedge the portfolio against bearish conditions while partially benefiting from bullish conditions. The last one is Merton’s dynamic optimization model which theoretically allows the investor’s portfolio to be adapted at any time to their own objectives.
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Notes
- 1.
Passive management is essentially static, except under certain circumstances, as described in the following chapter.
- 2.
Possibly with leverage (debt) that can be obtained through forward transactions.
- 3.
See Canner, Mankiw and Weil (1997).
- 4.
See Bajeux, Jordan and Portait (2001) for this explanation.
- 5.
This is why it is tempting to reason in terms of annualized returns which seem (falsely) to allow for the comparison of investments of different horizons.
- 6.
See Bodie, Merton and Samuelson (1992). To simplify, human capital is defined as the present value of all expected labor income until retirement.
- 7.
H. Leland and M. Rubinstein developed Portfolio Insurance based on the theoretical and practical plans within the framework of their corporation LOR (Leland, O’Brien and Rubinstein) which was to experience a rapid and spectacular development then a sudden drop due to the crash of October 1987, which highlighted the limits of the method. Other academics (Perold, Sharpe, etc.) have also contributed to the development of Portfolio Insurance.
- 8.
At each moment, a portfolio which duplicates a call includes: \( \left\{\begin{array}{ll}C-{\delta}_CS& \mathrm{euros}\ \mathrm{in}\ \mathrm{monetary}\ \mathrm{asset}\mathrm{s}\\ {}{\delta}_CS& \mathrm{euros}\ \mathrm{in}\ \mathrm{underlying}\ \mathrm{asset}\end{array}\right. \).
- 9.
Despite the fact that the price of the call is a positive function of the interest rate.
- 10.
The dynamics of the underlying security price s(t) and the value invested in the security S(t) are different since the number of the underlying security included in the portfolio changes. However, in the infinitesimal interval separating t and t+dt, we have dS/S = ds/s (= μ dt + σ dW), due to the self-financing condition imposed on the portfolio; see Chaps. 10 and 11 concerning self-financing strategies.
- 11.
See Prigent and Bertrand (2003), for example.
- 12.
Or the sale of forward (futures) contract on the underlying asset, which in turn involves, through cash-and-carry transactions, the sale of underlying asset.
- 13.
See Brennan and Schwartz (1989) and Genotte and Leland (1990) in particular.
- 14.
Efficient portfolios in this sense are obtained by maximizing the expectation of a quadratic utility function. This function is only a special case of a class of utilities called HARA (Hyberbolic Absolute Risk Aversion ); see Chap. 21.
- 15.
For dRi to represent the return of i, any potential dividend it distributes must be reinvested in i and incorporated into Si as if it were the share of a capitalization mutual fund composed solely of security i.
- 16.
Except for an investor who can influence security prices because of their size, hence future rates of return.
- 17.
Which means: \( X(0)=E\left(\frac{X(T)}{H(T)}\right)={E}^Q\left(\frac{X(T)}{S_0(T)}\right) \).
- 18.
We can indeed show that LimT –>∞ Proba [H(T) > X(T)] = 1 does not imply E(U(H(T)) > E(U(X(T)) for any U and for T large enough. Thus, Kelly’s criterion is incompatible with the principle of maximizing expected utility, and is unfounded in this respect.
- 19.
In fact, the Merton model addresses a more general case than the problem of simple portfolio optimization, since it also includes optimal consumption/saving trade-offs. The objective function is written as:
\( \underset{\underline{x}}{\mathit{\operatorname{Max}}}E\left\{{\int}_0^Tu\Big(c(t),t\Big) dt+U\left(X(T)\right)\right\} \) where c(t) represents the consumption density from which the consumer-investor derives, between t and t+dt, a utility u(c(t),t)dt. The function U is interpreted as the utility of the bequest. The dynamics of the portfolio value (constraint C1) writes: \( dX=X\left\{ rdt+{\underline{x}}^{\hbox{'}}\left(d\underline{R}- rdt\underline{1}\right)\right\}- cdt \).
- 20.
The investment opportunity set is defined as the set formed by the coefficients α(t, Y); Ω(t, Y); ϕ(t, Y); q’(t, Y); μ(t, Y); Σ(t, Y).
- 21.
No dRi return can be replicated between t and t+dt by a portfolio containing the N assets 0, …, i – 1, i + 1, …, N (see Chap. 21). In a diffusion model, this assumes that the variance-covariance matrix V(τ) = ΣΣ’ is invertible, which is true if and only if Σ is of full rank (N).
- 22.
The demonstration is based on the calculus of variations and not on the simple calculation of a Lagrangian.
Suggestions for Further Reading
Books
Bodie, Z., Kane, A., & Marcus, A. (2010). Investments (9th ed.). Irwin.
Elton, E., Gruber, M., Brown, S., & Goetzman, W. (2010). Modern portfolio theory and investment analysis (8th ed.). Wiley.
*Merton, R. (1999). Continuous-time finance. Basil Blackwell.
Articles
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Poncet, P., Portait, R. (2022). Strategic Portfolio Allocation. In: Capital Market Finance. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-84600-8_24
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