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TI-83 Plus/TI-84 Financial Math

Contents:  Simple interest, compound interest, effective rate, annuities, sinking funds, loan amortization, rate of return, present value,
Interest accumulated, principal remaining, alpha, beta, correlation coeffficient, r Squared, mean-variance optimization, least squares
best fit, and much more.
Last Revised:
3/23/2015

INDEX:

To facilitate lookup, the instructions are divided into the following categories:

I.   Interest - Simple Interest, Compound Interest, Interest Compounded Continuously, Effective Interest Rate.

II.   Annuities and Mortgages - Ordinary Annuities,  Annuities Due, Sinking Funds,
I
II.   Loans -  Car Loans, Loan Amortization Table by hand, Loan Amortization Table Semi-Automated
method, Loan Amortization Table Calculator Program,
IV.  Investments – Price of a bond; Interest to Maturity of a Bond, Present Value, Internal Rate of
return (Irr), Modified Internal Rate of Return (mirr),

V. A few Portfolio Calculations - Alpha, Beta, CorrCoef, and R Squared, Geometric Return, A smidgen of mean-variance
optimization.

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
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=Pert 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

2. Annuities Due:
Annuities Due have the same setup as ordinary annuities, except that BEGIN is highlighted
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.

 Payment Number Amount of Payment Principal Payment Interest Payment Principal Balance 0 \$80,000.00 1 \$991.89 \$425.22 \$566.67 \$79574.80 2 \$991.89 \$428.23 \$563.65 79146.54 3 \$991.89 \$431.26 \$560.62 78715.285

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, Y8,Y9,Y0.

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, Y8,Y9,Y0. (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
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 L1, L2, L3, L4, 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

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?

 PERIOD CASH FLOWS 0 0 1 100 2 200 3 300 4 400 5 500

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 L1.   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, L2, 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

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 L1.   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.
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.
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.

 FUND OR BENCHMARK YEARLY RETUNS α β r2 r S& P 500 10.88, 4.91, 15.79, 5.49, -37.00, 26.46, 15.06, 2.11, 16.00, 32.39, Fund A 13.89,16.27,19.26,13.43, -48.02, 52.20, 14.48,-12.33, 18.72, 14.27 -0.73078 1.1888 0.7600 0.8718 Fund B 20.84, 15.57, 26.64, 15.52, -44.10, 36.73, 11.04, -14.52,18.21. 15.14 0.8707 1.0691 0.7582 0.8707

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(Rp) = wT µ  Where wT is the transpose of the vector of the weights, w.
Standard Deviation of Portfolio:
σp =(wT V w)^(1/2) Where wT 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(Rp) = wT µ
So, in terms of our matrices, that would be as follows:
E(Rp) =[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:
σp =(wT V w)^(1/2)
The sticky point is getting the variance-covariance matrix, V. Remember that the formula for that matrix is as follows:
σxy =1/n Ʃi=1,n (Xi-X̄)(Yi-
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:
σp =(wT 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.
p)2=([H]T *[D]* [H])
σp =
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 2nd,  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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