1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved DSCI 3870 Chapter 8 NON-LINEAR OPTIMIZATION Additional Reading Material

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 8 – Non-Linear Optimization Additional Reading Material n Markowitz Portfolio Model n Forecasting Adoption of a New Product

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Markowitz Portfolio Model n There is a key tradeoff in most portfolio optimization models between risk and return. n The index fund model (Lymann Brothers example) presented earlier managed the tradeoff passively. n The Markowitz mean-variance portfolio model provides a very convenient way for an investor to actively trade-off risk versus return. n We will now demonstrate the Markowitz portfolio model by extending the Lymann Brothers example.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Markowitz Portfolio Model n In the Lymann Brothers example there were four scenarios and the return under each scenario was defined by the variables R 1, R 2, R 3, and R 4. n If p s is the probability of scenario s, and there are n scenarios, then the expected return for the portfolio R is n If we assume that the four scenarios in the Lymann Brothers model are equally likely, then

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Markowitz Portfolio Model n The measure of risk most often associated with the Markowitz model is the variance of the portfolio. n For our example, the portfolio variance is n For our example, the four planning scenarios are equally likely. Thus,

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Markowitz Portfolio Model n The portfolio variance is the average of the sum of the squares of the deviations from the mean value under each scenario. n The larger the variance value, the more widely dispersed the scenario returns are about the average return value. n If the portfolio variance were equal to zero, then every scenario return R i would be equal.

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved n There are two basic ways to formulate the Markowitz model: (1) Minimize the variance of the portfolio subject to constraints on the expected return, and(1) Minimize the variance of the portfolio subject to constraints on the expected return, and (2) Maximize the expected return of the portfolio subject to a constraint on risk.(2) Maximize the expected return of the portfolio subject to a constraint on risk. We will now demonstrate the first (1) formulation, assuming that Lymann Brothers’ client requires the expected portfolio return to be at least 9 percent. We will now demonstrate the first (1) formulation, assuming that Lymann Brothers’ client requires the expected portfolio return to be at least 9 percent. Example: Markowitz Portfolio Model

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Objective Function Minimize the portfolio variance: Minimize the portfolio variance: n Define the Constraints Define the return for each scenario: 25.64 IS + 15.31 LC + 18.74 MC + 14.19 SC + 7.88 IB = R 1 27.62 IS + 18.77 LC + 18.43 MC + 12.37 SC + 9.45 IB = R 2 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3 5.80 IS + 11.06 LC + 6.28 MC  1.92 SC + 10.56 IB = R 3  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4  3.13 IS + 4.75 LC  1.04 MC + 7.32 SC + 3.31 IB = R 4 Example: Markowitz Portfolio Model

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Constraints (continued) All the money must be invested in the portfolio: IS + LC + MC + SC + IB = 1 IS + LC + MC + SC + IB = 1 Define the expected return for the portfolio: The portfolio return must be at least 9 percent: Non-negativity: IS, LC, MC, SC, IB > 0 Example: Markowitz Portfolio Model

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Markowitz Portfolio Model n Optimal Solution R 1 = 10.63(10.63% portfolio return for scenario 1) R 1 = 10.63(10.63% portfolio return for scenario 1) R 2 = 12.20 (12.20% portfolio return for scenario 2) R 2 = 12.20 (12.20% portfolio return for scenario 2) R 3 = 8.93 ( 8.93% portfolio return for scenario 3) R 3 = 8.93 ( 8.93% portfolio return for scenario 3) R 4 = 4.24 ( 4.24% portfolio return for scenario 4) R 4 = 4.24 ( 4.24% portfolio return for scenario 4) R bar = 9.00( 9.00% expected portfolio return)R bar = 9.00( 9.00% expected portfolio return) IS = 0 ( 0.0% of portfolio in international stock) IS = 0 ( 0.0% of portfolio in international stock) LC =.251 (25.1% of portfolio in large-cap blend) LC =.251 (25.1% of portfolio in large-cap blend) MC = 0 ( 0.0% of portfolio in mid-cap blend) MC = 0 ( 0.0% of portfolio in mid-cap blend) SC =.141 (14.1% of portfolio in small-cap blend) SC =.141 (14.1% of portfolio in small-cap blend) IB =.608 (60.8% of portfolio in intermediate bond) IB =.608 (60.8% of portfolio in intermediate bond) 100.0% of portfolio

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Forecasting Adoption of a New Product n Forecasting new adoptions (purchases) after a product introduction is an important marketing problem. n We introduce here a forecasting model developed by Frank Bass. n Nonlinear programming is used to estimate the parameters of the Bass forecasting model.

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Forecasting Adoption of a New Product n The Bass model has three parameters that must be estimated. m is the number of people estimated to eventually adopt a new product m is the number of people estimated to eventually adopt a new product q is the coefficient of imitation which measures the likelihood of adoption due to a potential adopter influenced by someone who has already adopted the product q is the coefficient of imitation which measures the likelihood of adoption due to a potential adopter influenced by someone who has already adopted the product p is the coefficient of imitation which measures the likelihood of adoption assuming no influence from someone who has already adopted the product. p is the coefficient of imitation which measures the likelihood of adoption assuming no influence from someone who has already adopted the product.

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Forecasting Adoption of a New Product n Developing the Forecasting Model F t, the forecast of the number of new adopters during time period t, is F t, the forecast of the number of new adopters during time period t, is F t = (likelihood of a new adoption in time period t ) F t = (likelihood of a new adoption in time period t ) x (number of potential adopters remaining at x (number of potential adopters remaining at the end of time period t – 1) the end of time period t – 1)

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved n Developing the Forecasting Model Essentially, the likelihood of a new adoption is the likelihood of adoption due to innovation plus the likelihood of adoption due to imitation.Essentially, the likelihood of a new adoption is the likelihood of adoption due to innovation plus the likelihood of adoption due to imitation. Let C t  1 denote the number of people who have adopted the product up to time t  1.Let C t  1 denote the number of people who have adopted the product up to time t  1. Hence, C t  1 / m is the fraction of the number of people estimated to adopt the product by time t – 1.Hence, C t  1 / m is the fraction of the number of people estimated to adopt the product by time t – 1. The likelihood of adoption due to imitation is q ( C t  1 / m ).The likelihood of adoption due to imitation is q ( C t  1 / m ). The likelihood of adoption due to innovation and imitation is p + q ( C t  1 / m ).The likelihood of adoption due to innovation and imitation is p + q ( C t  1 / m ). Forecasting Adoption of a New Product

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved n Developing the Forecasting Model The number of potential adopters remaining at the end of time period t – 1 is m  C t  1.The number of potential adopters remaining at the end of time period t – 1 is m  C t  1. Hence, the complete forecast model is given byHence, the complete forecast model is given by F t = ( p + q ( C t  1 / m )) ( m  C t  1 ) F t = ( p + q ( C t  1 / m )) ( m  C t  1 ) Forecasting Adoption of a New Product

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved n Nonlinear Optimization Problem Formulation F t = ( p + q ( C t  1 / m )) ( m  C t  1 ), t = 1, …., N F t = ( p + q ( C t  1 / m )) ( m  C t  1 ), t = 1, …., N E t = F t  S t, t = 1, …., N E t = F t  S t, t = 1, …., N where N = number of time periods of data available E t = forecast error for time period t E t = forecast error for time period t S t = actual number of adopters (or a multiple of S t = actual number of adopters (or a multiple of that number such as sales) in time period t that number such as sales) in time period t Forecasting Adoption of a New Product

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Forecasting New-Product Adoption n Maid For You Maid For You is a residential cleaning service firm that has been quite successful developing a client base in the Chicago area. The firm plans to expand to other major The firm plans to expand to other major metropolitan areas during the next few years. Maid For You would like to use its Chicago subscription data (on the next slide) to develop a model for forecasting service subscriptions in regions where it might expand. The first step is to estimate values for p (coefficient of innovation) and q (coefficient of imitation).

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Forecasting New-Product Adoption n Subscribers and Cumulative Subscribers (1000s) Month Subscribers S t Cum. Subscribers C t Month Subscribers S t Cum. Subscribers C t 1 0.53 0.53 1 0.53 0.53 2 2.94 3.47 2 2.94 3.47 3 3.60 7.07 3 3.60 7.07 4 4.8511.92 4 4.8511.92 5 3.4415.36 5 3.4415.36 6 2.7618.12 6 2.7618.12 7 1.8219.94 7 1.8219.94 8 0.9320.87 8 0.9320.87 9 0.6121.48 9 0.6121.48

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Objective Function Minimize the sum of the squared forecast errors: Minimize the sum of the squared forecast errors: Example: Forecasting New-Product Adoption

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Forecasting New-Product Adoption n Define the Constraints Define the forecast for each time period: 1) F 1 = pm 2)F 2 = ( p + q ( 0.53/ m )) ( m – 0.53) 3) F 3 = ( p + q ( 3.47/ m )) ( m – 3.47) 4) F 4 = ( p + q ( 7.07/ m )) ( m – 7.07) 4) F 4 = ( p + q ( 7.07/ m )) ( m – 7.07) 5) F 5 = ( p + q (11.92/ m )) ( m – 11.92) 6) F 6 = ( p + q (15.36/ m )) ( m – 15.36) 7) F 7 = ( p + q (18.12/ m )) ( m – 18.12) 8) F 8 = ( p + q (19.94/ m )) ( m – 19.94) 9) F 9 = ( p + q (20.87/ m )) ( m – 20.87)

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Forecasting New-Product Adoption n Define the Constraints (continued) Define the forecast error for each time period: 10) E 1 = F 1 – 0.53 11) E 2 = F 2 – 2.94 12) E 3 = F 3 – 3.60 13) E 4 = F 4 – 4.85 13) E 4 = F 4 – 4.85 14) E 5 = F 5 – 3.44 15) E 6 = F 6 – 2.76 16) E 7 = F 7 – 1.82 17) E 8 = F 8 – 0.93 18) E 9 = F 9 – 0.61

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Forecasting New-Product Adoption n Optimal Forecast Parameter Values Parameter Value Parameter Value p 0.08 p 0.08 q 0.62 q 0.62 m 21.26 m 21.26 The value of the imitation parameter q =.62 is The value of the imitation parameter q =.62 is substantially larger than the value of the innovation parameter p =.08. Subscriptions gain momentum over time due mainly to very favorable word-of- mouth.

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved n Optimal Solution Example: Forecasting New-Product Adoption Month Forecast Subscribers Error 11.770.53 1.24 11.770.53 1.24 22.052.94-0.89 22.052.94-0.89 33.293.60-0.31 33.293.60-0.31 44.124.85-0.73 44.124.85-0.73 54.033.44 0.59 54.033.44 0.59 63.142.76 0.38 63.142.76 0.38 71.931.82 0.11 71.931.82 0.11 80.880.93-0.05 80.880.93-0.05 90.270.61-0.34 90.270.61-0.34

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved n Subscribers versus Forecasts Example: Forecasting New-Product Adoption Subscribers Month Subscribers (1000s) 54321 1 2 3 4 5 6 7 8 9 Forecast

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 8