J. Marín-Solano (UB), M. Bosch-Príncep (UB), J. Dhaene (KUL), C. Ribas (UB), O. Roch (UB), S. Vanduffel (KUL)

Transcription

1 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)

2 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.

3 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.

4 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.

5 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

6 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.

7 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.

8 3. UPPER AND LOWER BOUNDS FOR THE TERMINAL WEALTH Let X = (X 1, X 2,..., X n ) and let S = X 1 + X 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

9 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.

10 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 ].

11 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.

12 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

13 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 = 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.

14 p MC LB MV LB T UB % +1.46% % % +0.79% % % +0.57% % , % -0.00% % % +0.09% -3.92% % -0.34% +3.32% % +0.10% % % -0.27% % % -0.93% %

15 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).

16 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).

17 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,.

18 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)].

19 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 % MC LB MV LB T UB π % 33.83% 34.36% 57.14% π % 40.79% 39.87% 0% π % 25.38% 25.77% 42.86% K

20 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)].

21 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 % MC LB MV LB T UB π % 37.36% 38.44% 57.14% π % 34.62% 32.74% 0% π % 28.02% 28.82% 42.86% K