Financial Applications Portfolio selection Financial planning Designing optimal tax packages (government) Yield (revenue) management - Setting prices for goods to maximize revenue - Airline, hotel, catering industries Problem: Portfolio Planning Given a sum of money to invest, Decide how to invest the money to form a portfolio of shares and stocks Objectives: Maximize expected rate of return from the investment Minimize risk, that is variance of the return Multi-period planning A Financial Planning Problem You have $250,000 available now with the following Investment alternatives: A: Invest $1 at the beginning of any year and get $1.40 at the end of the second year. B: Invest $1 at the beginning of any year and get $1.70 at the end of the third year. C: Invest $1 at the beginning of year 2 and get $1.90 at the end of year 5. D: Invest $1 at the beginning of year 5 and get $1.30 at the end of year 5. Maximize the funds available at the end of year 5. Financial Planning (Constraints) 250,000 M1 M2 M3 M4 MS 1.9C2 M6 1.781 1.783 1.305 1.4A1 +11.782 1.4A2 1.4A3 14A4 C2 A181 A2 82 A3 83 A4. DS Y1 Y2 Y3 72 Y4 YS Financial Planning (var.) Decision variables: A1: $ invested in alternative A at the beginning of year 1. B1: $ invested in alternative B at the beginning of year 1. A2: $ invested in alternative A at the beginning of year 2. B2: $ invested in alternative B at the beginning of year 2. C2: $ invested in alternative C at the beginning of year 2. A3: $ invested in alternative A at the beginning of year 3. B3: $ invested in alternative B at the beginning of year 3. A4: $ invested in alternative A at the beginning of year 4. D5: $ invested in alternative B at the beginning of year 5. M: $ available in period i, (after the funds that should be mature at the beginning of that period become mature and after the investment of that period is made). i=1..6 Financial Planning (Constraints) 250,000 M1 M2 M3 M4 M5 1.9C2 M6 1.781 1.7831.305 1.4A1 +11.782 1.4A2 $1.4A3 14A4 A1 81 82 A3 83 A4 D5. Y1 Y2 Y3 Y4 YS Max M6 Y. MI-250000-AL-BI A1+BK 25000 2: MI-A2-B2-C2=M 3. M2-AB-B3 H4A =M Y6: M6=M5+1.4A2+ 196 All var 70 +1-78311.305 1875 Period 1 Balance Period 2 Balance 7 Period 3 Balance Period 4 Balance Period 5 Balance 10 Period 6 Balance 11 14 1.7 1.9 1.3 O Cash at the 12 end of Periods 13 14 15 16 17 18 25 MI=250000-A1-B1 MITA+B=250000 PHS 2500