RELEASE DATE:  5/1/06       Latest Revision: 3/23/15 
© 2003 Frank Kizer    NOTE:  See copy restrictions and printing hints at the end of this document.  

General: 
*  TMV Solver - Unless otherwise indicated,  all calculations will be with the TMV Solver.  To access
   this, press APPS, ENTER, ENTER. 
*  Most of these instructions will be carried out using a problem as an example.  Note that some of the  
   problems could be solved, possibly even easier, without the Finance APP, but this document deals with
   that APP only. 
*  Minus Signs - Note that some answers will have a minus sign before them.  These are there because 
   the calculator  follows the cash-flow sign convention in which cash outflows (investments for example)
   are negative and inflows are positive.  For many problems, you can ignore this sign.  When it's
   important, that will be indicated.
*  Setting N, P/Y, and C/Y - As a general rule, when there are no periodic payments, such as in  
   interest calculations, "N" is set equal to the number of years and P/Y is set at 1.  C/Y will be set to
   the number of compounding periods a year.  Notice that for daily compounding, C/Y will be set at 360
   or possibly 365 for some problems.  For loans, annuities, and other such things with periodic payments,
   P/Y will be set for the number of payments a year, "N"  will be the number of payments, and C/Y will be
   set for the number of compoundings per year.
* Note to Students:  The presence of this document does not imply that students should rely solely on the calculator
  to find the answers to their problems.  Doing the problems by hand definitely helps to understand the problem and
  gives good practice in manipulating the formulas. I encourage students to use the calculator to check their answers
  and to obtain answers only when allowed by the professor.

I.  Simple and Compound Interest.

    1. Simple Interest:
        Comment: Simple interest can be easily obtained from the formula I=PRT, but for non-students and others who may rely solely on the
        calculator, I will include simple interest. Suppose a student had $1000 which she did not need for 3 years. If she invested it for 3 years at 5%
        annual interest, how much did she have at the end of the 3 years?
      a) Enter values so that the display appears as follows: N=1; I%=5*3; PV = 1000; PMT=0; P/Y =1; C/Y=1; END.
      b) Set the cursor on FV and press ALPHA; SOLVE. Note that SOLVE is the third function of the ENTER key. The answer will be -1150. 
          Ignore the negative sign.
     c) Note that if you want the interest accumulated, then just subtract $1000 from the answer obtained in the above operation.
     Note that this works fine in all cases except when you want to find the interest. Suppose that instead of the above
     problem, the problem was the same except that the student wanted to know what interest she would need to have
     $1150 in three years.
     a) Enter values so that the display appears as follows: N=1; PV = 1000; PMT=0; FV=-1150; P/Y =1; C/Y=1; END.
     b) Set the cursor opposite I% and press ALPHA; SOLVE. The answer will be 15.
     c) Move the cursor over past 15 and enter ÷, 3; then press ENTER. The correct answer, 5, will be displayed.

2. Compound Interest:
        Ex 1:  Suppose that you invest $5000 for 6.5 years at 5.25% interest compounded quarterly,  
        how much money will you have at the end of the period? 
         a) Press APPS, ENTER, ENTER to display the TMV Solver.
         b)  Enter values so that the following display is completed:  N=6.5; I%=5.25; PV = -5000; PMT=0;
              P/Y =1; C/Y=4; END.
        c)  Set the cursor on FV and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 7017.93.
        d)  Note that if you want the interest accumulated, then just subtract $5000 from the answer
             obtained in the above operation. 

       Ex 2:  Suppose that you have $1200 and you need $1800 in 7 years,  at what interest compounded
       quarterly,  will you need to invest the money to earn this amount?
        a)  Enter values so that the following display is completed:  N=7; I%=0; PV = -1200; PMT=0; FV=1800,
             P/Y =1; C/Y=4; END.
        b)  Set the cursor on I%, and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 5.835 rounded to 3 decimal places.

        EX 3:  Interest Compounded Continuously:
           Although the formula A=Pe^rt is just about as easy to work with as using the Finance APP, some users have difficulty
           working with  the formula to obtain time or rate.  So, I will include this example of continuous compounding.
            Let's take the information in Ex 2 above except that we have interest compounded continuously.
            a)  Enter the information exactly as in Ex 2 except that for C/Y, enter 1E10.  Do that by pressing 2, 2ND
                 EE (the comma key), 9, ENTER. 
            b)  Set the cursor on I%, and press ALPHA; SOLVE.  Note that SOLVE is the third function of the
              ENTER key. Your answer should be 5.792 rounded to 3 decimal places.

       EX 4:  Time to Double Investment at Continuous Compounding:
          Suppose you want to know how long would be required to double your investment at an interest rate of 5 percent with continuous
          compounding.
          a) Take any convenient value such as $100 and double it to get $200.
          b) Enter values so that the following display is completed: N=; I%=5; PV = -100; PMT=0; FV=200, P/Y =1; C/Y=1E10; END
          c) Set the cursor opposite N  and press ALPHA; SOLVE. .  The answer would be 13.86… or about 13 years and 315 days..

 3. Effective Interest Rate:
     Suppose that a one bank tells you that it pays 3.9% compounded monthly and another tells you
     that it pays 4% compounded semi-annually.  Which one is the best investment?
     a)  Press APPS, ENTER, move the cursor down  to C:EFF( and press ENTER.  (Alternatively, you
          may press ALPHA C.)  "EFF (" will be pasted to the screen.
     b)  Enter 3.9, 12) and press ENTER.  The effective interest rate will be 3.97%.
     c)  Press 2nd, ENTRY (the second function of ENTER); then edit the entry so that you have
          EFF(4, 2); then press ENTER.  Your answer will be 4.04.  So, this is the best investment.

II. Annuities and Mortgages:

     1. Ordinary Annuities:
         For our purposes, an ordinary annuity will be one in which equal payments are made at equal
         periods of time, the compounding period is the same as the payment period, and the payments
         are made at the end of the period. Note Well:  Because there are payments in an annuity, "N" in
         the TMV Solver must set equal to the number of payment periods.
         Ex. 1:  Suppose that you pay $20,000 each year into an annuity for 7 years.  If the interest is 6%
         compounded annually, how much will you have at the end of the period?
         a) Press APPS, ENTER, ENTER to display the TMV Solver.
         b)  Enter values so that the following display is completed:  N=7; I%=6; PV = 0;PMT=-20000;
              P/Y =1; C/Y=1; END.
         c)  Set the cursor on FV and press ALPHA, SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 167876.75.

     2. Annuities Due:
         Annuities Due have the same setup as ordinary annuities, except that BEGIN is highlighted
         instead of END.
         Ex. 1:  Suppose that you pay $500 each year into an annuity due for 7 years.  If the interest is
         6% compounded annually, how much will you have at the end of the year?
         a) Press APPS, ENTER, ENTER to display the TMV Solver.
         b)  Enter values so that the following display is completed:  N=7; I%=6; PV = 0;PMT=-500;
              P/Y =1; C/Y=1; BEGIN
         c)  Set the cursor on FV and press ALPHA, SOLVE.  Note that SOLVE is the third function of the
             ENTER key. Your answer should be 4448.73..., rounded to 2 decimal places.

     3. Sinking Funds:
         Sinking funds have the same characteristics as annuities,  but they are for purposes other than an
         annuity. They may be to accumulate enough money to buy a car, pay off a loan, or any other purpose.
         Follow the same procedure for these as for annuities.

     4.  Mortgages:
          Suppose a family buys a home for $200000 and makes a down payment of $20000.  They take  
          out a $180000 mortgage at 7.5% for 30 years.  What is the monthly payment required to
          amortize this loan? How much interest is paid on this loan?
          Payment:
            a) Press APPS, ENTER, ENTER to display the TMV Solver.
            b)  Enter values so that the following display is completed:  N=360; I%=7.5; PV =
                180000; FV=0; PMT=0; P/Y =12; C/Y=12; END.
            c)  Set the cursor on PMT and press ALPHA, SOLVE.  Note that SOLVE is the third function of 
                 the ENTER key. Your answer should be 1258.59, rounded to 2 decimal places.
          Interest:
            You can calculate the interest either using the calculator or by hand. 
              Calculator
                a) After calculating the paymet, set the cursor on FV and press APP, ENTER, and movethe cursor
                    down to ƩInt( and press ENTER. The expression ƩInt( will be displayed opposite FV.
                b) Enter the months spread, for example 1 to 360 so that you have ƩInt(1, 360).
                c) Press ENTER and the value 273091.00 will be displayed opposite FV.
               By Hand:
                 a) To calculate by hand, use the following formula:
                     Total Interest = Monthly Payment * Number of Months - Original Amount of Loan.
                      You may get a small difference between these two methods because of rounding.
                NOTE: You can also use ƩPrn( and bal( in the same way. For example, ƩPrn(5, 9) will give you the
                    amount of principal paid in that period and bal(3) will give you the balance after payment 3.
                             

       5.  Mortgage Loan Calculations:
            Calculate Individual values:
                  Suppose you have an 10-year loan of $80,000.00 at 8.5 percent with payments each month. 
                   Make an amortization table for the first three payments.  You might first want to make a table  
                   such as that in Figure 1 below to enter your data.  The calculated data has already been entered in
                   this table.
             To Calculate the Monthly Payment::
                   a)  Press APPS,  ENTER, ENTER
                   b)  Put the following information in the display that appears:  N=10*12; I% = 8.5; PV=-80000;
                        FV=0; P/Y=12;C/Y = 12; END.
                  c)  Put the cursor at PMT, press ALPHA, SOLVE  and the payment of 991.885 will be displayed
                       opposite PMT.
             To Calculate a Specific Principal Balance:
                  a)  From the Home screen, press APPS, ENTER, move the cursor to display bal( on the Home screen.
              b)  Enter values so that your display looks like this:  bal(3) . The numbers inside the parentheses  indicate the  
                      balance will be calculated after the third payment.
                 c)  Press ENTER and the value indicated in the Table 1 below for the third payment will be displayed. You can
                       calculate additional balances by pressing 2nd, ENTER, and editing the payment number.
              To Calculate a Specific  Principal Payment: 
             a)  From the Home screen, press APPS, ENTER, move the cursor to display ∑Prn(.   Press ENTER and the term
                      ∑Prn( will be pasted to the Home screen. 
                b)  Enter values so that your display looks like this: ∑Prn(3, 3) . The numbers inside the parentheses 
                       indicate the principal payment will be calculated for  the third payment.
                 c)  Press ENTER and the value indicated in the Table 1 below for the third payment will be displayed.
              To calculate a Specific  Interest Payments.
                 a)  From the Home screen, press APPS,  ENTER; move then cursor down ∑Int.   Press ENTER and the term ∑Int
                  will be pasted to the Home screen. 
                b)  Enter values so that your display looks like this: ∑Int(3, 3) . The numbers inside the parentheses 
                       indicate the principal payment will be calculated for  the third payment.
                 c)  Press ENTER and the value indicated in the table below for the third payment will be displayed.
                 Of course you could fill out a few lines of a table such as that below using this method, but there's a better method
                 which I've included in the amortization method below.

 5.  Amortization Table for a Loan:
           General:  The manual procedure, which I will explain first, takes a lot of time if you have to
           calculate  several loans or several lines on a table. Therefore, I have added a little program
           that I wrote to save you some work.  The program follows  this explanation. 

         Manual Procedure:
          General:  The manual procedure, which I will explain first, takes quite a lot of time if you have to
           calculate  several loans. Therefore, I have added a little program that I wrote to save you some work. 
          The program follows  this explanation. 

          Details of Manual Procedure:
              Suppose you have a 10-year loan of $80,000.00 at 8.5 percent with payments each month. 
              Make an amortization table for the first three payments.  You might first want to make a table  
              such as the following to enter your data.  The calculated data has already been entered in
              this table.                
               

                                           Table 1   

              Semi-automated Method Using TVM Solver, Graph, and Table:
                      In previous versions I had not included this method because I thought that the program would be
                      used by those who have considerable work of this type to do.  The programs seems not to have been
                      used much, so I am including this,  somewhat tedious, I'm afraid, method to add more flexibility.
                      a)  First, I recommend that you make a table such as the one immediately
                            above.  You might wanat to name the columns X, Y₇, Y_8,Y_9,Y_0. 
                      b)  Press APPS, ENTER, ENTER to display the TMV Solver.
                      c)  Put the following information in the display that appears:  N=10*12; I% = 8.5; PV=80000;
                           FV=0; P/Y=12;C/Y = 12; END.
                      d)  Put the cursor at PMT, press ALPHA, ENTER, and the payment of 991.885 will be displayed
                            opposite PMT.
                       e)  Press Y= to go to the graphing screen.  We are going to enter some financial functions in positions
                            Y₇, Y_8,Y_9,Y_0. (You could enter them in Y₁, etc. if you prefer, but I am entering them so that the upper variables
                            can be used for other functions.)
                       f)  Place the cursor opposite Y₇ and press APPS, ENTER. Now move the cursor down to tmv_Pmt
                           and press ENTER to paste tmv_Pmt opposite Y₇.
                       g)  Place the cursor opposite Y₈ and press APPS, ENTER.  Now move the cursor down to ΣPrn(
                           and press ENTER to paste ΣPrn(  opposite Y₈.
                       h) Enter characters so that you have ΣPrn(X,X) opposite Y₈.
                       i)  Place the cursor opposite Y₉ and press APPS, ENTER. Now, move the cursor down to bal( and press ENTER.
                       j) Enter characters so that you have bal(X) opposite Y₀.
                       k)  Press 2ND, TBLSET, and set TblStart = 0 and ΔTbl=1 and Indpnt to Ask
                 l)  Press 2ND, TABLE and enter the payment number or numbers that you want information for.. When
                     you enter a number in the X-column the values for theY-variables will be entered. You will need to
                     scroll right to see the columns on the right since only three columns take up the whole screen.
                 Obviously, if you want to calculate a table for a different mortgage, just do the calculation for the
                  payment again and then use the table to get the values for the second mortgage without having
                  to make new entries in the Y= positions. Be sure to deselect the Y-variables before graphing a
                  function or you'll tie your calculator up for some time graphing unwanted stuff.

                      Using the Program:  This is a simple program that should take only a few minutes to enter if you  
                      have some rudimentary knowledge of how to enter programs.  You can find information on entering
                      programs in your TI User Manual  or in the programs section on this  Website.  (Click on TI  
                      Programming Keystrokes near the bottom of the navigation panel to the left.)   After one student has the
                      program stored in a calculator,  it takes less than three minutes, including setup to transfer the program to
                      in another student's calculator.  NOTE:  The colons to the left on the lines of code are automatically entered
                     when you enter the program  by hand.

                     :PROGRAM: LAONAMRT
                     :"FKIZER  V:050106"
                     : Disp "ENTR DATA IN APPS"
                     :Input "1ST PMT NO. ", B
                     :Input "LAST PMT NO. ", E
                     :1→X
                     :ClrList L₁, L₂, L₃, L₄, L₅
                     :For(P,B,E
                     :P→L₁(X)
                     :tmv_Pmt→L₂(X)
                     :∑Int( P,P→L₃(X)
                     :∑Prn( P,P→L₄(X)
                     :bal(P→L₅(X)
                     :X+1→X
                     :End
                     :Stop

                     Using the Program:  Here's how to use this program, assuming you already have it entered.
                      1)  Follow the first three steps in the manual method described above; then press 2nd, QUIT.
                      2)  Pres, PRGM; move the cursor down to the name of the program you want to use and press ENTER.
                      3) The statement 1ST PMT NO. will appear.  Enter the number of the first payment you want to 
                          calculate data for and press ENTER. 
                      4)  LAST PMT NO. will then appear.  Enter the number for the last payment you want to calculate
                           and  press ENTER.  Obviously, if you want only one payment, that number will be entered for
                           both the first and last payment number.
                      5)  The calculator will store the amounts for Payment, Interest, Principal Payment, and Principal
                           Balance in that order in lists L_1, L₂, L₃, L_4, and L₅ .
                      6)  To access the data tables, press STAT, ENTER.  
                      7)  You will notice that the data has only five characters (Numbers plus decimal and negative sign, if
                            any.).  If you want a more accurate answer, scroll to the number of interst and a more accurate value
                             will be displayed below the tables containing the lists.
                       
III.  Loans:
      Loans, car loans for example, have the same structure as ordinary annuities.  Let's do an example
      to demonstrate that.
      Ex 1:  Suppose that a car costs $26,000 and your down payment is $4000.  The balance will be paid off in
      36 monthly payments with a interest of 10% per year on the unpaid balance. Find the monthly
      payment.
      a)  Press APPS, ENTER, ENTER to display the TMV Solver.
      b)  Enter values so that the following display is completed:  N=36; I%=10; PV = 22000;PMT=0;
           FV=0; P/Y =12; C/Y=12; END.
      c)  Set the cursor on PMT and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be 709.88, rounded to 2 decimal places.

IV.  Investments:

      1. Bonds:
          Ex 1:  Suppose that a $1000, 10-year, 8% bond is issued when the market rate is 7.5%. 
          Interest is paid semiannually.  What can you expect to pay for the bond?
          a) Press APPS, ENTER, ENTER to display the TMV Solver.
          b)  Enter values so that the following display is completed:  N=20; I%=7.5; PV =0;PMT=40;
           FV=1000; P/Y =2; C/Y=2; END.  It's important to realize that the cost is based on the interest
           to maturity.
          c)  Set the cursor on PV and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be -1034.74, rounded to 2 decimal places.

           Ex 2:  Suppose that you have to pay $1034.74 for a $1000, 10-year, 8% bond with interest paid
           twice a year.  What is the interest to maturity for the bond?
           a)  Enter values so that the following display is completed:  N=20; I%=0; PV =-1034.74;PMT=40;
                FV=1000; P/Y =2; C/Y=2; END.
           b)  Set the cursor on I% and press ALPHA, SOLVE.  Note that SOLVE is the third function of
           the ENTER key. Your answer should be 7.5%.

       2.  Present value:
             The syntax for Net Present Value (NPV) is:  npv(interest rate, CFO, CFList[CFFreq]).  Now,
             let's define what these mean: 
             Interest Rate = the rate by which to discount the cash flows over one period.
             CFO = the initial cash flow at time zero.
             CFOList = A list of cash flow amounts AFTER the initial cash flow, CFO.
             CFFreq = How many there are of each amount.  The default is 1.
             Ex. 1:  Suppose you are offered an investment that will pay the cash flows in the table below at
             the end of each year for the next 5 years.  How much would you be willing to pay for it if you
             wanted 10 percent interest per year?
             

               a) Press STAT, ENTER to go to the lists.  It there are numbers in the list you choose to use,
                   you can erase those numbers by highlighting the list name, for example L₁, pressing CLEAR;
                   then ENTER.  Do not use DEL.
               b) Enter the numbers starting with 100 in list L_1.   To enter a number, just enter it and press ENTER. 
               c)  Press 2nd, QUIT to leave the list.
               d)  Press APPS, ENTER, 7. "npv(" will be pasted to the home screen.
               e)  Make entries so that you have the following: npv(10, 0, L₁.  To enter L₁, press 2nd L₁.  (L₁
                    is the second function of the number 1 key.)
               f)  Press ENTER.  Your answer should be 1065.26 rounded to two decimal places.
               NOTE 1:  Instead of using the lists, you could enter the following:
               npv(10, 0, {100, 200, 300, 400, 500}). Then press ENTER.  I frankly prefer to use lists because
               of the increased flexibility.
               NOTE:  If you have several CONSECUTIVE cash flows, you can create a frequency table in
              another list, L_2, for example.  You will need to enter the frequency for each of the CFO values,
              even if it is 1.  Your entry then would be npv(10, 0 L₁, L₂ .
              Ex. 2: Suppose that we wanted to find the future value.  Rather than using the TMV solver for
              each cash flow and adding them up, just multiply the answer from Ex. 1 by (1+.10)^5.  To do
              that, press 2nd, Ans, x (multiply), (1+.10)^5.  Your answer should be 1715.61.
              Ex. 3: Suppose that you were offered the above investment for $800.  What is the NPV?
              CFO is now -800.  The cash outflow is negative.  So, we would enter, npv(10, -800, L₁.  Your
              answer should be 265.26 rounded to 2 decimal places.  
     
       3.  Internal Rate of Return (Irr):  
            Suppose you wanted to find the Irr for the npv example above.
            a)  First enter all of the cash flows except the first in list L₁.
           b)  Press APPS, ENTER, 8.  The term "irr(" will be displayed on the home screen.
           c)  Make entries so that you have the following:  irr(-800, L₁.Your answer should be 19.538.  This
                assumes that the numbers in the table of cash flows above have been entered in list L₁.
           Comments:  If you get an error message using this procedure and don't understand why, go to
           the home page, click on "More Detalied P2" under TI FAQs, and read FAQ 56.

         4.  Modified Internal Rate of Return (MIrr):  
           Step 1:  First we'll find the Future Value:
           a) Press STAT, ENTER to go to the lists.  It there are numbers in the list you choose to use,
                   you can erase those numbers by highlighting the list name, for example L₁, pressing CLEAR;
                   then ENTER.  Do not use DEL.
           b) Enter the numbers starting with 100 in list L_1.   To enter a number, just enter it and press ENTER. 
           c)  Press 2nd, QUIT to leave the list.
           d) Press APPS, ENTER, ENTER to display the TMV Solver.
                e)  Enter values in the display as follows::  N=5; I%=0; PV =-800; PMT=0;
                 FV=1715.61; P/Y =1; C/Y=; END.
            Now, we want to enter a calculated value into FV. To do that, place the cursor opposite FV, press
            CLEAR to clear the value there; the do the following:
           f)  Press APPS, ENTER, 7. "npv(" will be pasted to the home screen.
           g)  Make entries so that you have the following: npv(10, 0, L₁).  To enter L₁, press 2nd L₁.  (L₁ is the
                second function of the number 1 key.)
           h) Now, we want to multiply this by (1.1)^5.  To do that enter 1.1^5.  You should now have
              the this expression:  npv(10, 0, L₁)1.1^5.  When you move the cursor away from FV you
              should have 1715.61
           i) Set the cursor on I% and press ALPHA, SOLVE.  Note that SOLVE is the third function of
                 the ENTER key. Your answer should be 16.48 rounded to two decimal places. 

V. A Few Portrolio Calculations:

    1.  Alpha, Beta, CorrCoef, and RSquared:
         Although they may seem quite complex so far as their uses in a portfolio, in concept, α and β are quite simple mathematically.
         The α term is just the y-intercept of the line y=mx +b that you learned about somewhere around seventh grade and β is
         the slope of that line. Note that these calculations are grouped together only because they result from the same calculator
         operation and not for any financial reason.
         Comments about alpha and beta:
         But the question my be, "Where does the line come from?" Well, the x-values are usually a benchmark; we'll use the S&P 500,
         and the y-values are a fund or stock that you want to investigate. The line is the least-squres best linear fit to the data points.
         Alpha, as it applies to a portfolio is the non-systematic value that remains when the market value (the S&P 550 data) is zero.
         Although all of these values can be calculated by hand using the list arithmetic, we'll use the function of the calculator that
         does all of the arithmetic.
          Comments about r and r²:
         Although the terms r, and r² are somewhat more complex because of the arithmetic calculations involved, the calculator
         takes care of all of the arithmetic.  The correlation coefficient, r, is an indication of how strongly the two data
         groups are correlated. In our case, how strongly is a fund correlated to the S&P 500. The term r² (also called R Squared
         and R² ) in statistics called the coefficient of determination, is as follows: r² = (explained variation)/(total variation), where
         explained variation is that  predicted by the best-fit line. For example, if r²  is 90%, then 90% of the variation in the best-fit
        line is explained by the variation of the S&P 500.
        So, let's calculate these values for two different international funds.

While we're at it let's find the averages. We'll need them later.
For the S&P 500, press STAT, move the cursor to Math and press ENTER, ENTER. In the display the
value for xbar =9.209.
For Fund A do the same except when you get the display 1-Var Stats, press 2nd, L₂ to get 1-Var Stats L₂.
Then press ENTER to get xbar = 10.217
Find the Various Relationships:

  Relationship between S*P 500 and Fund A:
  a) Press STAT, ENTER and enter the S&P 500 returns in list L₁ and the Fund A returns in list L₂.
  b) Press STAT, go to CALC and press 8 for LinReg(a+bX).
  c) Place the cursor opposite Xlist, press 2nd L₁; then 2nd L₂ opposite Ylist.^.
   d) On the screen that appears, move the cursor to "Calculate" and press ENTER. The values listed in the table will appear.
      Note that if you don't have the Stats Wizard, press 8 and then ENTER.

Relationship between S&P 500 and Fund B:
  a) Press STAT, ENTER and enter the data MSCI data in list L₁ and the Fund B data in list L₃.
  b) Press STAT, go to CALC and press 8 for LinReg(a+bx).
  c) Place the cursor opposite Xlist, press 2nd L₁; then 2nd L₃ opposite Ylist.^.
   d) On the screen that appears, move the cursor to "Calculate" and press ENTER. The values listed in the table will appear.
      Note that if you don't have the Stats Wizard, press 8 and enter the lists so that you have the following: LinReg(a+bx) L₁, L₃; then
      press ENTER to display that appears on the same line as Fund B in the table above.

Comments:  Funds A and B are two different international funds. These are usally compare to the MSCI, but for this exercise, it
doesn't make any difference.You'll notice the two funds have about the same correlation asindicated by the values of r. The values
for β indicate how much a fund return moves when the benchmark moves a unit value. You'll notice that Fund A moves somewhat
more than Fund B. The r² values indicate that slightly more of the volatility of Fund A is explained by the volatility of the S&P 500
than is Fund B. Finally, alpha indicates the difference between the value as prdicted by the slope, β, and the market return, in this
case the S&P 500. For the market return, α is zero. You'll notice that Fund B is
somewhat higher.

2. Sharpe Ratio:
     The Sharpe is a means of comparing returns of investments over a broad range of assets. The ratio measures the excess return
     of an asset relative to it's standard deviation. Remember that alpha measures the excess return relative to it's beta. The excess
     return is usually that above the 6-months T-bill over the same period. Suppose that an asset has a return over a certain period of
     arithmetic mean of 12% and return of the 6-months T-bill arithmetic mean of 3.0%. Further suppose that the standard deviation is
     15%. Then the Sharpe ratio is (12-3)/15 =0.6

3. Geometric Mean:
     Let's do the geometric mean of the S&P 500 as listed in the table at the beginning of this section.
     a) Press STAT, ENTER and enter the returns data in list L₁ or whatever list is convenient.
     b) Place the cursor on the list name of list L₂ or whatever list you choose, and enter 2nd, L₁/100 +1 and press ENTER.
         Now, we're going to use the formula (products of L₂)^1/k , where "k" is the number of return values. In out case that's
          10.
     c) Press 2nd, STAT, move the cursor to MATH on the menu that appears and press 6 for prod(.
     d) Press 2nd, L₂ so that you have prod(L₂ on the display. Press ENTER.
      e) Now enter the following: (Ans)^1/10 -1 . You may need to enter the entire exponent in parentheses if you have
           an older calculator. If you did the S&P returns, you should get 7.4.

4. Mean-Variance Optimization:
     I'm going to do a very limited mean-variance optimization. I realize that Excel or a portfolio optimizer is the preferred method
     of doing optimizations, but I'm thinking that maybe doing it with a calculator will give some insight as to what is being done
     in Excel. Caveat: I'm aware that mean-variance optimization assumes that returns are normally distributed, that it often uses
     historical rather than future data, and other shortcomings. I'm not going to deal with those issues. This section if for helping
     students better understand what mean-variance optimization is all about, not evaluation of the effectiveness of procedures.
     So, with that, let's write down the formulas for the portfolio return and standard deviation.
     Portfolio Expected Return:
     E(R_p) = w^T µ  Where w^T is the transpose of the vector of the weights, w.
     Standard Deviation of Portfolio:
     σ_p =(w^T V w)^(1/2) Where w^T is as explained immediately above, V is the variance-covariance matrix and w is just the
     vector of weights.
     So, let's take a highly unrealistic portfolio of  S&P 500  and Fund A with the returns as given above and a weighting of .25 for
      the S&P and .75 for Fund A. So, that would give us a weighting vector as follows:
    [[.25]
    [.75]]
    Let's put that in matrix [H] for heavy
     The terms used for the return matrix would be as follows:
      So, µ =[[7.4]  
             [6.79]]
   These values were obtained by taking the geometric means of the values in the tables above where r, r² etc. are listed.
   Let's put them in matrix [I]. Item 3, above, is a procedure for doing geometric mean.
   So, the formula for the return is as follows:
  E(R_p) = w^T µ
  So, in terms of our matrices, that would be as follows:
  E(R_p) =[H]^T [I]

a) To get [H]^T , we put the numbers in matrix [H]. Then press 2nd, QUIT to quit the matrix editor.
b) Press 2nd, MATRIX, move the cursor to highlight [H] and press ENTER.
c) Press 2nd, MATRIX, move the cursor to MATH, and press 2 for ^T . You then should have [H]^T .
d) If you have the µ matrix in [I], then press x, the multiply sign; then press 2nd, MATRIX, and press 2
     for matrix [I].
e) Press ENTER to get the answer of 6.94.

Portfolio Standard Deviation:
  The portfolio standard deviation requires considerably more work. Remember that the formula for the standard deviation
is as follows:
σ_p =(w^T V w)^(1/2)
The sticky point is getting the variance-covariance matrix, V. Remember that the formula for that matrix is as follows:
σ_xy =1/n Ʃ_i=1,n (X_i-X̄)(Y_i-ȳ) 
Where xbar and ybar are the arithmetic means.
So put the returns from the table above in matrix [A]. Put the S&P values in the first column and those for Fund A in
the second column. Then in matrix [B], enter a 10 x 10 matrix with all 1s. To save some time in entering all of the 1s,
you can go to the matrices and dimension [B] as a 10 x 10. Then go to the home screen. Press 2nd, MATRIX, move
the cursor to MATH, select Fill( and press ENTER. Then enter information so that you have Fill(1, [B]. Press ENTER
to fill the matrix. Note tha [B] must be entered from the list of matrices.
Let's do the variance-covariance matrix in two steps as follows:
Diff = [A]-1/10*[B]*[A]--> [C]  Eq 1
The symbol --> is obtained by pressing the STO button.
Covr = V =1/10*[C]^T [C] -->[D]
Now, we want to implement the following equation for the standard deviation:
σ_p =(w^T V w)^(1/2)
Since we can't take the square root of a matrix, we'll have to solve for the square; then take the square root of the number
inside the 1x1 matrix.
(σ_p)²=([H]^T *[D]* [H])
σ_{p =}√(σ_p² ) (NOTE: Don't take the square root  of Ans because it's a matrix. Take the square root of the number inside the
matrix.

Here's a little program to do the above calculations. You'll need to put the returns in matrix [A] and make a k x k matrix of 1s,
and the averages of the returns in ]I].

:MVOPT
:"FKIZER 06/09/14"
:Input "1ST. WEIGHT" , F
:Input "2ND WEIGHT , S
:Input "NO. RETURNS ",N
:[[F] [S]]--> [H]
:[H]^T *[I]-->[E]
:[A]-1/N*[B]*[A]-->[D]
:1/N*[D]^T*[D]-->[C]
:[H]^T *[C]*[H]-->[J]
:Matr►list([J]L₆)
:ClrHome
:Disp "RETN ",[E]
:Disp "STD DEV ", √(L₆)

I may be back later to add more detail as to what the matrices are doing for the variance-covariance matrix,
but for now, that's it. 

II. Solving Linear Programming Problems Graphically:

   1) Graphical Linear Programming without the Inequality Application:  .
       Find the maximum of the objective function z=2x +5y, subject to the following constraints:
            3x+2y≤6  (Eq 1a)
            -x+2y≤4   (Eq 2a)
             x≥0, y≥0
             a)  First put the two-variable equations in slope-intercept form.
                   y≤-3/2x+3  (Eq 1b)
                   y≤1/2x+2   (Eq 2b)
                   (Remember  that you may need to change the direction of the inequality sign if you have to
               multiply or divide by -1 during the rearranging of the equation.)
              b)  Enter the right side of those equations opposite Y1 and Y2 respectively. Opposite the equal sign
                    for Y3 enter 0 (zero).
              c)  Set the WINDOW at Xmin = -1, Ymin=-1, Ymax= the largest value of "b" plus a few units, say 5 in
                    this problem.
                    Xmax is a bit more difficult to anticipate.  If there is an equality with a negative slope, I usually
                    make Xmax = 4/3*b/m, round it off to the next largest whole number and add a few units. You can
                    enter the arithmetic opposite Xmax. You might want to press GRAPH and see if all of the
                    corner points of the bounded region are on the screen.
              Now we will enter the inequality signs:
           a)  Shading of the graph is determined by the symbol to the left of the "Y=" entry.  Using the left
                arrow, move the cursor all the way to the left of the Y= symbol.
           b)  Pressing ENTER in that position will display different symbols.  For < or <, press ENTER
                until the upright triangle is displayed.  For > or >, press ENTER until the upside down
                triangle is displayed.
           c) After you have the correct symbol displayed, press ENTER to graph the inequality.
           Finding  the x- and y-values of the corner points. 
                You may find it easier to locate the corner points if you change the inequalities back to equalities
                 and press GRAPH to graph only the lines.  Write down the values of each corner point.  They
                 will be used to evaluate the objective function
                a) Press 2^nd,  CALC.
            b) Press 5 to select intersect.
            c)  To get the corner point at the intersection of the two graphs, move the cursor  a little away from
                  the intersection and press ENTER to mark the first graph line and move the cursor to the next
                  graph line.
            d)  Move the cursor to a little away from the intersection and press ENTER to mark the second graph
                 line and set the cursor to guess the intersection.
            e) Move the cursor approximately to the intersection and press ENTER.
            f) The coordinates for the point of intersection, x=.5 and y=2.25, will appear at the bottom of the screen.
            g)  To obtain the value where the graph line intersects the x-axis, repeat the procedure
                 for the intersection of two lines except that for the second curve you will need to insure
                 that the cursor is on the x-axis.  If it is not, press the down arrow. The answer is x=2, y=0.
            h)  The corner point where the graph intersects the y-axis is obtained by pressing 2ND, CALC,
                 pressing ENTER to select Value,  entering 0 (zero) opposite X, and then pressing ENTER.
          Having the calculator evaluate the objective function.
               a)  First you need to enter the proper expressions.  From the home screen, press .5, STO, X, ALPHA, :
                (decimal point key), 2.25, STO, ALPHA, Y, ALPHA, :, 2, X, +, 5, ALPHA, Y.  You should now
                have the following on the home screen:  .5→X:2.25→Y:2X+5Y
               b)  Press ENTER and the corner point will be evaluated at 12.25.
               c)  To evaluate additional points, press 2nd, ENTER and the above expression will be displayed.
                     Enter the new corner point and press ENTER.  Repeat this as many times as needed.

 2)  Graphical Linear Programming with the Inequalities Application:
         Find the maximum of the objective function z=2x +5y, subject to the following constraints:
            3x+2y≤6  (Eq 1a)
            -x+2y≤4   (Eq 2a)
             x≥0, y≥0
           a)  First put the two-variable equations in slope-intercept form.
                 y≤-3/2x+3  (Eq 1b)
                 y≤1/2x+2   (Eq 2b)
           b)  Enter the right side of those equations opposite Y1 and Y2 respectively and enter 0 opposite Y3.
           c)  Set the WINDOW at Xmin = -1, Ymin=-1, Ymax= the largest value of "b" plus a few units, say 5 in
                 this problem. The value for  Xmax is a bit more difficult to anticipate.  If there is an equality with
                 a negative slope, I usually make Xmax = 4/3*b/m, round it off to the next largest whole number
                 and add a few units. You can enter the arithmetic opposite Xmax. You might want to press GRAPH
                 and see if all of the corner points of the bounded region are on the screen.
           Now we will enter the inequality signs.
         a)  Move the cursor to the sign (either equal or inequality) after Y1.  If the inequality symbols do not
                   appear at the bottom of the screen, you will need to start the Inequality App.  Do that by pressing
                   APPS, move the cursor down to Inequal,  or Inequalz for the international version, and press
                   ENTER, ENTER.  The Y= editor screen should be displayed. 
         b)  Place the cursor on the equal sign opposite Y1 and press ALPHA, F3 (ZOOM).  The equal sign
              should have been replaced by the inequality ≤.
             c)  Do the same for Y2; then opposite Y3, press ALPHA, F5 (GRAPH).  The symbol ≥ should have
                   replaced the equal sign before the 0. 
             d)  Now, move the cursor up to the "X" in the upper left corner and press ENTER.  With the cursor on the
                   equal sign opposite X1, press ALPHA, F5 (GRAPH) to enter ≥;  then enter a zero after that symbol.
             e)  Press GRAPH  to draw and shade the graphs.  
             f)  Press ALPHA, F1, 1 to define the feasible region.
             If you only want to graph, you may stop here.
             At this time we will find the x- and y-values of the corner points.  We will use a method to have the
             calculator determine the corresponding values of the objective function.  If you prefer to calculate the
             value of the objective function by hand--I actually recommend that.-- just record the values as you find them
             without pressing STO in the following procedure so that you can substitute them into the objective function.
             a)  Press ALPHA, F3 (ZOOM).  If Y1∩Y2 appears in the upper left of the screen, the values  x=.5, y=2.25 will
                  be displayed.  Record these values if you choose to evaluate the objective function by hand.  Otherwise,
                  press STO, ENTER.
              b) If Y1∩Y2 appears in the upper part of the screen, press the right arrow to move to the point x=2, y=0 and press
                   STO, ENTER or record the values for hand calculation.
              c)  Press the right arrow; press the down arrow, and then the left arrow and the last point, x=2, y=0
                    should be displayed.  Press STO, ENTER if you want to store the point.
             Having the calculator evaluate the objective function.  Method I:
              a)  Press STAT, ENTER, to bring up the lists with the stored data.
              b)  First we will name the list after list INEQY.  We will name it OBJ.  Move the cursor to the name
                    block at the top of that list.  Press 2ND, ALPHA, O, B, J, ENTER.
              Now, we will define the list OBJ.
              c)  With the cursor on the list name, press ALPHA,  (the + key), 2, x(multiply), 2ND, LIST and scroll down to
                    INEQX and press ENTER.
             d)  Press +, 5, x (multiply), 2ND, LIST, scroll to INEQY and press ENTER.  Press ALPHA, ".  You should now have
                   "2*└INEQX+5*└INEQY" at the bottom left of the screen after "OBJ=".  Press ENTER and the values
                   of the objective function for each set of coordinates will be displayed in the OBJ list. 
             Method II:
              Actually, this requires a number of rather tedious strokes.  If you don't want to just do it using the calculation mode, then there's
               an easier method. 
               a)  Just write down the x- and y-values as you obtain them, and then enter them in list L₁ for x and L₂ for y.
               b) Now, place the cursor over the name for L₃ and enter the 2, 2nd, L₁, +,5, 2nd, _L2.  
               c)  Press ENTER and the objective values will be entered in list L₃.

III. Simplex Method:
       1.  The Simplex Method is described in the LinProg Document here. 

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