Articles from Cass Knowledge

Streamlined mean-variance analysis for multi-period models

With reference to recent research paper Numeraire-Invariant Quadratic Hedging and Mean-Variance Portfolio Allocation

The methodology is based on maximisation of mean-variance utility according to Markowitz [4], i.e., the investor wishes to maximise her mean return for a given level of portfolio variance. The researchers work in a multi-period setting with a fixed time horizon 𝑇 and assume that the investor pre-commits to the optimal strategy as perceived at the start of the planning horizon at time 0.

The research identifies three quantities that conveniently encapsulate all information needed for the construction of the efficient investment frontier.

1) Opportunity process 𝐿 measures the smallest second moment of a fully invested portfolio.

2) Tracking process 𝑉(1) records intermediate wealth that minimizes the hedging distance to the constant payoff 1.

3) Squared hedging error 𝜀2(1) gives the smallest hedging distance to the constant payoff 1.

With the three quantities in hand, the formula for the variance of a fully invested portfolio 𝑅 on the efficient frontier is given by Formula (#1), which can be seen in this summary document.

Formula (#1) has several advantages.

  • It is robust to specification of asset price processes.
  • It does not rely on Markovian assumptions.
  • It holds whether one works in discrete or continuous time.
  • It applies equally in models with and without a risk-free asset.

Thanks to its versatility, formula (#1) brings together multiple disparate results such as [2–4, 6, 7].

In discrete time, the suggested computational strategy has an additional advantage in that the three quantities L,V(1), and ε2 (1) can be obtained by a simple backward recursion. For practical illustrations along these lines, see [1, Section 6].


[1] Aleš Černý, Christoph Czichowsky, and Jan Kallsen (2021, Oct). Numeraire-invariant quadratic hedging and mean-variance portfolio allocation

[2] Aleš Černý and Jan Kallsen (2008). Mean--variance hedging and optimal investment in Heston's model with correlation. Mathematical Finance 18(3), 473–492.

[3] Duan Li and Wan-Lung Ng (2000). Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Mathematical Finance 10(3), 387–406.

[4] Andrew E. B. Lim (2004). Quadratic hedging and mean–variance portfolio selection with random parameters in an incomplete market. Mathematics of Operations Research 29(1), 132–161.

[5] Harry Markowitz (1952). Portfolio selection. Journal of Finance 7(1), 77–91.

[6] Haixiang Yao, Zhongfei Li, and Shumin Chen (2014). Continuous-time mean--variance portfolio selection with only risky assets. Economic Modelling 36, 244–251.

[7] Xun Yu Zhou and Gang Yin (2003). Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model. SIAM Journal on Control and Optimization 42(4), 1466–1482.