BUY AND HOLD STRATEGIES IN OPTIMAL PORTFOLIO SELECTION PROBLEMS: COMONOTONIC APPROXIMATIONS J. Marín-Solano (UB), M. Bosch-Príncep (UB), J. Dhaene (KUL), C. Ribas (UB), O. Roch (UB), S. Vanduffel (KUL)
MULTIPERIOD OPTIMAL PORTFOLIO SELECTION PROBLEM 1. Investment strategies. 2. Buy and hold strategy. Terminal wealth. 3. Upper and lower bounds for the terminal wealth. 4. Optimal portfolio.
1. INVESTMENT STRATEGIES There are (m+1) securities. One of them is riskfree: There are m risky assets: By defining B i (t) = 1 σ i d j=1 dp i (t) P i (t) = µ idt + σ ij W j (t), d j=1 σ ij dw j (t). dp i (t) P i (t) = µ idt + σ i db i (t), i = 1,..., m. dp 0 (t) P 0 (t) = rdt.
From the solution to this equation, [( P i (t) = p i exp µ i 1 ) 2 σ2 i ] t + σ i B i (t), we obtain that the random yearly returns of asset i in year k, Y i k, are independent and have identical normal distributions with E [Y i k ] = µ i 1 2 σ2 i, Var [Yk i ] = σi 2, and Cov [Yk i, Y j l ] = 0 if k l, σ ij if k = l.
Let Π(t) = (Π 0 (t), Π 1 (t),..., Π m (t)) denote the vector describing the proportions of wealth invested in each asset at time t. In general, a vector Π(t) will define an investment strategy. If one unit of a security is constructed according to the investment strategy Π(t), let P (t) be the price of that unit at time t. Then, m [ dp (t) P (t) = Π i (t) dp i m ] (t) P i (t) = m Π i (t)(µ i r) + r dt+ Π i (t)σ i db i (t). i=0 i=1 If Π(t) is prefixed, P (t) can be obtained by solving the stochastic differential equation above (constantly rebalanced portfolio). i=1
2. BUY AND HOLD STRATEGY. TERMINAL WEALTH The new amounts of money α(t) are invested at time t = 0, 1,..., n 1 in some prefixed proportions π(t) = ( π 0 (t), π 1 (t),..., π m (t)). Fractions π i (t) are always the same. Denoting π i (0) = π i, then ( π 0 (t),..., π m (t)) = (π 0,..., π m ), for every t = 0, 1,..., n 1. New quantities are invested once in a period of time (typically, once in a year), i.e., α i if t = i, for i = 0, 1,..., n 1, α(t) = 0 otherwise. The decision maker follows a buy and hold strategy, i.e., no securities are sold.
Objective: To compute the terminal wealth W n (π) for a given buy and hold strategy π = (π 0, π 1,..., π m ). Let Zj i be the sum of returns of 1 unit of capital invested at time t = j of asset i from time t = j to the final time t = n, Z i j = n k=j+1 Y i k. The terminal wealth invested in asset i is W i n(π) = n 1 j=0 π i α j e Zi j, whereas the terminal wealth will be given by W (π) = m W i (π) = i=0 m i=0 n 1 j=0 π i α j e Zi j.
3. UPPER AND LOWER BOUNDS FOR THE TERMINAL WEALTH Let X = (X 1, X 2,..., X n ) and let S = X 1 + X 2 + + X n. It can be shown that where S c = n i=1 S l cx n i=1 F 1 X i (U) and S l = X i cx S c, n E [X i Λ]. i=1
If S = n ᾱ i e Z i with ᾱ i 0, i=1 S c = n i=1 F 1 ᾱ i e Z i (U) = n ᾱ i e E [ Z i ]+σ Zi Φ 1 (U). i=1 For a given Λ = n γ j Zj, j=1 S l = n ᾱ i E [e Z i Λ] = i=1 n ᾱ i e E [ Z i ]+ 1 2(1 r 2 i )σ 2 Zi +r i σ Zi Φ 1 (U). i=1 We need values of γ j that minimize of the distance between S and S l.
Maximal Variance lower bound approach. As we have that Var[S] = Var[S l ] + E[Var[S Λ]], it seems reasonable to choose the coefficients γ j such that the variance of S l is maximized: γ k = ᾱ k e E [ Z k ]+ 1 2 σ2 Zk. Taylor-based lower bound approach. Λ is a linear transformation of a first order approximation to S: γ k = ᾱ k e E [ Z k ].
Comonotonic Upper Bound B&H strategy: W c (π) = m i=0 n 1 j=0 π i α j e (n j)(µ i 1 2 σ2 i )+ n jσ i Φ 1 (U). Note that W c (π) is a linear combination of fractions π i, i = 0,..., m. Comonotonic Lower Bound B&H strategy: W l (π) = m i=0 n 1 j=0 π i α j e (n j) (µ i 1 2 r2 ij σ2 i )+r ij n jσi Φ 1 (U) where the correlation coefficients r ij are given by r MV ij = σ i [ (n j) m s,k=0 m k=0 n 1 l=0 π kα l (n max(j, l))σ ik e (n l)µ k n 1 t,l=0 π sπ k α t α l (n max(t, l))σ sk e (n t)µ s+(n l)µ k ] 1/2.
and m r T ij = m k=0 n 1 s,k=0 t,l=0 n 1 l=0 π kα l (n max(j, l))σ ik e (n l) [µ k 1 2 σ2 k] σ i (n j) 1/2 π s π k α t α l (n max(t, l))σ sk e (n t) [µ s 1 2 σ2 s]+(n l)[µ k 1 2 σ2 k] 1/2
Numerical illustration: 2 risky, 1 risk-free. µ 1 = 0.06, µ 2 = 0.1, σ 1 = 0.1, σ 2 = 0.2, Pearson s correlation: 0.5, r = 0.03. Every period α i = 1, invested in proportions: 19% risk-free asset, 45% first risky asset, 36% in the second risky asset. This amount is invested for i = 0,..., 19, whereas in i = 20 the invested amount is α 20 = 0. The simulated results were obtained with 500,000 random paths.
p MC LB MV LB T UB 0.01 20.018 +2.39% +1.46% -20.01% 0.025 23.072 +1.49% +0.79% -18.54% 0.05 25.056 +1.07% +0.57% -16.80% 0.25 33,488 +0.00% -0.00% -10.14% 0.5 42.307-0.11% +0.09% -3.92% 0.75 55.161 +0.12% -0.34% +3.32% 0.95 86.258 +0.24% +0.10% +14.49% 0.975 101.5 +0.11% -0.27% +18.07% 0.99 123.99-0.23% -0.93% +22.08%
4. OPTIMAL PORTFOLIO Possible criteria: maximizing an expected utility, Yaari s dual theory of choice under risk: max π ρ f [W n (π)] = max π 0 f(pr(w n (π) > x))dx, risk measures (some of them correspond to distorted expectations ρ f [W n (π)] for appropriate choices of the distorsion function f).
Value at Risk at level p: Q p [X] = F 1 X (p) = inf{x R F X(x) p}. If F X is an strictly increasing function, then it coincides with the related risk measure Q + p [X] = sup{x R F X (x) p}, p (0, 1). Additive for sums of comonotonic risks. Conditional Left Tail Expectation at level p (CLT E p [X]): CLT E p [X] = E [ X X < Q + p [X] ], p (0, 1).
For the upper and lower bounds in B&H strategy: Q p [W c (π)] = Q p [W l (π)] = CLT E p [W c (π)] = CLT E p [W l (π)] = m i=0 m i=0 m i=0 m i=0 n 1 j=0 n 1 j=0 n 1 j=0 n 1 j=0 π i α j e (n j)(µ i 1 2 σ2 i )+ n jσ i Φ 1 (p), π i α j e (n j) (µ i 1 2 r2 ij σ2 i )+r ij n jσi Φ 1 (p), π i α j e µ i(n j) 1 Φ( n jσ i Φ 1 (p)) p π i α j e µ i(n j) 1 Φ( n jr ij σ i Φ 1 (p)) p,.
Maximizing the Value at Risk: for a given probability p and a given investment strategy, let K p (π) be the p-target capital defined as the (1 p)-th order + -quantile of terminal wealth, K p (π) = Q + 1 p [W (π)]. For the optimal case: Alternatives: Kp = max π Q+ 1 p [W (π)]. K c p = max π Q + 1 p [W c (π)] or K l p = max π Q + 1 p [W l (π)].
r MC LB MV LB T UB π 0 100% 100% 100% 100% π 1 0% 0% 0% 0% π 2 0% 0% 0% 0% K 27.817 27.817 27.817 27.817 6% MC LB MV LB T UB π 0 33.24% 33.83% 34.36% 57.14% π 1 41.82% 40.79% 39.87% 0% π 2 24.94% 25.38% 25.77% 42.86% K 25.718 25.914 25.805 22.78
Maximizing the CLTE: max π CLT E 1 p [W (π)]. This optimization problem describes decisions of risk averse investors. The CLT E 1 p has the following nice property (lacking with the VaR): CLT E 1 p [W c (π)] CLT E 1 p [W (π)] CLT E 1 p [W l (π)]. Alternative: max π CLT E 1 p [W l (π)].
r MC LB MV LB T UB π 0 100% 100% 100% 100% π 1 0% 0% 0% 0% π 2 0% 0% 0% 0% K 27.817 27.817 27.817 27.817 6% MC LB MV LB T UB π 0 37.70% 37.36% 38.44% 57.14% π 1 34.12% 34.62% 32.74% 0% π 2 28.18% 28.02% 28.82% 42.86% K 23.431 24.089 23.962 21.163