The Main Idea Behind . . . Example A Similar Idea Works . . . How Markowitz . . . Markowitz Portfolio Theory What If Side Effects . . . Helps Decrease Medicines’ Applications to . . . This Idea Also Works! Side Effect and Speed Up References to a . . . Reference to a Neural . . . Machine Learning Home Page Title Page Thongchai Dumrongpokaphan 1 and Vladik Kreinovich 2 ◭◭ ◮◮ 1 Faculty of Science, Chiang Mai University ◭ ◮ Chiang Mai, Thailand, tcd43@hotmail.com 2 University of Texas at El Paso, USA, vladik@utep.edu Page 1 of 32 Go Back Full Screen Close Quit
The Main Idea Behind . . . Example 1. The Main Idea Behind Markowitz Portfolio A Similar Idea Works . . . Theory How Markowitz . . . • In his Nobel-prize winning paper, H. M. Markowitz What If Side Effects . . . proposed a method for selecting an optimal portfolio. Applications to . . . This Idea Also Works! • To explain the main ideas behind his method, let us References to a . . . start with a simple case when: Reference to a Neural . . . – we have n independent financial instrument, each Home Page – with a known expected return-on-investment µ i Title Page – and with a known standard deviation σ i . ◭◭ ◮◮ • We can combine these instruments, by allocating the ◭ ◮ part w i of our investment to the i -th instrument. Page 2 of 32 � n • Here, we have w i ≥ 0 and w i = 1. Go Back i =1 Full Screen Close Quit
The Main Idea Behind . . . Example 2. Markowitz Portfolio Theory (cont-d) A Similar Idea Works . . . • For each portfolio, we can determine the expected re- How Markowitz . . . turn on investment µ and the standard deviation σ : What If Side Effects . . . Applications to . . . n n � � w i · µ i and σ 2 = w 2 i · σ 2 µ = i . This Idea Also Works! i =1 i =1 References to a . . . Reference to a Neural . . . • Some of such portfolios are less risky – i.e., have smaller Home Page σ – but have a smaller µ . Title Page • Other portfolios have a larger expected return on in- ◭◭ ◮◮ vestment but are more risky. ◭ ◮ • We can therefore formulate two possible problems. Page 3 of 32 Go Back Full Screen Close Quit
The Main Idea Behind . . . Example 3. Markowitz Portfolio Theory (cont-d) A Similar Idea Works . . . • The first problem is when we want to achieve a certain How Markowitz . . . expected return on investment µ ; What If Side Effects . . . Applications to . . . – out of all possible portfolios that provide such ex- This Idea Also Works! pected return on investment, References to a . . . – we want to find the portfolio for which the risk σ Reference to a Neural . . . is the smallest possible. Home Page • The second problem is when we know the maximum Title Page amount of risk σ that we can tolerate. ◭◭ ◮◮ • There are several different portfolios that provide the ◭ ◮ allowed amount of risk; Page 4 of 32 – out of all such portfolios, Go Back – we would like to select the one that provides the largest possible return on investment. Full Screen Close Quit
The Main Idea Behind . . . Example 4. Example A Similar Idea Works . . . • Let us consider the simplest case, when all n instru- How Markowitz . . . ments have the same µ and σ : What If Side Effects . . . Applications to . . . µ 1 = . . . = µ n , σ 1 = . . . = σ n . This Idea Also Works! • In this case, the problem is completely symmetric with References to a . . . respect to permutations. Reference to a Neural . . . Home Page • Thus, the optimal portfolio should be symmetric too. Title Page • So, all the parts must be the same: w 1 = . . . = w n . ◭◭ ◮◮ � n w i = 1, this implies that w 1 = . . . = w n = 1 • Since n . ◭ ◮ i =1 • For these values w i , the expected return on investment Page 5 of 32 is the same µ = µ i , but the risk decreases: Go Back n � i = n · 1 1 = 1 1 , hence σ = σ 1 σ 2 = w 2 i · σ 2 n 2 · σ 2 n · σ 2 √ n. Full Screen i =1 Close Quit
The Main Idea Behind . . . Example 5. What We Can Conclude From This Example A Similar Idea Works . . . A natural conclusion is that: How Markowitz . . . What If Side Effects . . . • if we diversify our portfolio, i.e., Applications to . . . • if we divide our investment amount between different This Idea Also Works! independent financial instruments, References to a . . . • then we can drastically decrease the corresponding risk. Reference to a Neural . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 32 Go Back Full Screen Close Quit
The Main Idea Behind . . . Example 6. A Similar Idea Works Well in Measurement A Similar Idea Works . . . • Suppose that we have n results x 1 , . . . , x n of measuring How Markowitz . . . the same quantity x . What If Side Effects . . . Applications to . . . • Suppose that the measurement error x i − x has mean This Idea Also Works! 0 and standard deviation σ i . References to a . . . • Suppose that measurement errors corresponding to dif- Reference to a Neural . . . ferent measurements are independent. Home Page • Then we can decrease the estimation error if, Title Page – instead of the original estimates x i for x , ◭◭ ◮◮ � n – we use their weighted average � x = w i · x i , for ◭ ◮ i =1 � n Page 7 of 32 some weights w i ≥ 0 for which w i = 1 . i =1 Go Back • In this case, the standard deviation of the estimate � x � n Full Screen is equal to σ 2 = w 2 i · σ 2 i . Close i =1 Quit
The Main Idea Behind . . . Example 7. Measurement (cont-d) A Similar Idea Works . . . • We want to find the weights w i that minimize σ 2 under How Markowitz . . . � n What If Side Effects . . . the given constraint w i = 1. i =1 Applications to . . . • By using the Lagrange multiplier method, we get the This Idea Also Works! following unconstrained optimization problem: References to a . . . � n � n � � Reference to a Neural . . . w 2 i · σ 2 i + λ · w i − 1 → min . Home Page i i =1 i =1 Title Page • Differentiating with respect to w i and equating the ◭◭ ◮◮ derivative to 0, we get = − λ ◭ ◮ def 2 w i · σ 2 i + λ = 0 , so w i = c · σ − 1 i , where c 2 . Page 8 of 32 • This constant c can be found from the condition that Go Back σ − 2 � n 1 i w i = 1: we get c = ; thus, w i = . � � n n Full Screen i =1 σ − 2 σ − 2 j j j =1 j =1 Close Quit
The Main Idea Behind . . . Example 8. Measurement (final) A Similar Idea Works . . . � n 1 • For these weights, σ 2 = How Markowitz . . . w 2 i · σ 2 i = . � n What If Side Effects . . . σ − 2 i =1 j Applications to . . . j =1 � n This Idea Also Works! σ − 2 is larger than each of its terms σ − 2 • The sum j . j References to a . . . j =1 Reference to a Neural . . . • Thus, the inverse σ 2 of this sum is smaller than each Home Page of the inverses σ 2 j . Title Page • So, combining measurement results indeed decreases ◭◭ ◮◮ the approximation error. ◭ ◮ • In particular, when all measurements are equally accu- rate, i.e., when σ 1 = . . . = σ n , we get σ = σ Page 9 of 32 √ n. Go Back Full Screen Close Quit
The Main Idea Behind . . . Example 9. Optimal Portfolio When Different Instruments A Similar Idea Works . . . Are Independent How Markowitz . . . • So far, we considered the case when different financial What If Side Effects . . . instruments are independent and identical. Applications to . . . This Idea Also Works! • Let us now consider a more general case, when we still References to a . . . assume that: Reference to a Neural . . . – the financial instruments are independent, but Home Page – we take into account that these instrument may Title Page have individual values µ i and σ i . ◭◭ ◮◮ • In this case, the first portfolio optimization problem ◭ ◮ takes the following form: Page 10 of 32 � n w 2 i · σ 2 – minimize i Go Back i =1 � � n n Full Screen – under the constraints w i · µ i = µ and w i = 1 . i =1 i =1 Close Quit
The Main Idea Behind . . . Example 10. When Different Are Independent (cont-d) A Similar Idea Works . . . • Lagrange multiplier methods leads to: How Markowitz . . . � n � � n � What If Side Effects . . . � n � � w 2 i · σ 2 + λ ′ · i + λ · w i · µ i − µ w i − 1 → min . Applications to . . . i =1 i =1 i =1 This Idea Also Works! • Differentiating this expression with respect to w i and References to a . . . equating the derivative to 0, we conclude that Reference to a Neural . . . i + λ · µ i + λ ′ = 0 , i.e., w i = a · ( µ i · σ − 2 2 w i · σ 2 i ) + b · σ − 2 i , Home Page = − λ ′ = − λ Title Page def def where a 2 and b 2 . ◭◭ ◮◮ � n • For these w i , w i · µ i = µ and w i = 1 are: ◭ ◮ i =1 n Page 11 of 32 � def ( µ i ) k · σ − 2 a · Σ 2 + b · Σ 1 = µ, a · Σ 1 + b · Σ 0 = 1 , where Σ k = i . Go Back i =1 Full Screen • Thus, a = Σ 1 − µ · Σ 0 and b = µ · Σ 1 − Σ 2 . Σ 2 Σ 2 1 − Σ 0 · Σ 2 1 − Σ 0 · Σ 2 Close Quit
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