equations on ti 86

Time Value of Money NPV and IRR

Equation Solving with the TI-86 (may work with TI-85)

(similar process works with TI-83, TI-83 Plus and may work with TI-82)

Time Value of Money, NPV and IRR equation solving with the TI-86............................. 2

Other TI-Calculators ....................................................................................................... 2

Manuals....................................................................................................................... 2

Transfer of Formulas Using Cable.............................................................................. 2

NPV and IRR .................................................................................................................. 2

The formula................................................................................................................. 2

Expression of the NPV Formula in TI-86 (and possibly TI-85)................................. 3

Expression of the Formula in TI-83, TI-83Plus (and possibly TI-82)........................ 3

Using the Equation Solver in TI-86 ................................................................................ 3

Solving for NPV ......................................................................................................... 4

Solving for IRR........................................................................................................... 5

Time Value of Money Calculations................................................................................ 6

The present value of an annuity formula .................................................................... 6

The future value of an annuity formula ...................................................................... 7

Future Value formula.................................................................................................. 7

Future Value formula with continuous compounding ................................................ 8

Effective Annual Rate................................................................................................. 8

Perpetuities and Constant Growth Perpetuities........................................................... 9

EquationsOnTI86 1 of 9 © Andrew Hall 2004

Time Value of Money, NPV and IRR equation solving with the TI-86

Other TI-Calculators This approach may work for the TI-85. You can use a similar process with the TI-83 and TI83Plus, so it may help you with the TI-82. On the TI-82 and TI-83 variables in equations appear to be limited to one character; so you may need to adjust the names you use accordingly.

Manuals Remember that pdf’s of the relevant sections of the manuals can be downloaded from the Course Web Page at http://www.people.umass.edu/adhall/ .

Transfer of Formulas Using Cable Come to regular office hours with your calculator and I will transfer these equations onto your TI-86 (and if it works your TI-85!)

NPV and IRR

The formula The formula which applies to both NPV and IRR is

( ){ } ( ){ } ( ){ } ( ){ }1 2 3

1 2 30...1 1 1NPV C CF CF CF CF

n

nr r r+ + + + += + + + 1 r+

Where C 0 1 2 3 n,CF ,CF ,CF ...CF can be positive or negative.

For the IRR the formula assumes that NPV=0 and looks as follows:

( ){ } ( ){ } ( ){ } ( ){ }1 2 3

1 2 30... 01 IRR 1 IRR 1 IRR 1 IRRC CF CF CF CF

n

n+ + + + + =+ + + +

For convenience, and assuming problems no larger than this on exams, let us work with the maximum number of periods = 7 so n=6:

( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5C1 C2 C3 C4 C5 C6C0+ + + + + +npv

1+r 1+r 1+r 1+r 1+r 1+r= 6

where r is either the Required Rate of Return or the IRR (Internal Rate of Return); C0 is the initial investment; C1 is the cashflow at the end of the first period; C2 is the cashflow at the end of the second period; NPV is either the value to be calculated for the net present value or NPV is set to zero to calculate, r, the IRR.


http://www.people.umass.edu/adhall/

Expression of the NPV Formula in TI-86 (and possibly TI-85) npv=C0+(C1/(1+r)^1)+(C2/(1+r)^2) +(C3/(1+r)^3) +(C4/(1+r)^4) +(C5/(1+r)^5) +(C6/(1+r)^6)

On the main screen:

• use 2nd and ALPHA to enter the letter “n”

• use 2nd and ALPHA to enter the letters “pv”

• use ALPHA and STO=> to enter an equals sign

• ALPHA C0 +

• Repeat (ALPHA C1 ÷ ( 1 + 2nd ALPHA r)^1 ) six times.

• use the ENTER KEY to store expression to expression variable.

Expression of the Formula in TI-83, TI-83Plus (and possibly TI-82) n=i+(a/(1+r)^1)+(b/(1+r)^2) +(c/(1+r)^3) +(d/(1+r)^4) +(e/(1+r)^5) +(f/(1+r)^6)

I have not worked out how to use variable names longer than a single character in these calculators. It will only really be relevant to the TI-82 users because TI-83 and TI-83Plus already have the TVM_Solver functionality. Refer to the section above this one on the “Expression of the NPV Formula in TI-86”. For “npv” substitute “n”; for “C0” substitute “i”; for “C1” – “a’; for “C2” – “b” ,…, for “C6” – “f”.

Using the Equation Solver in TI-86 Once the equation has been entered as an expression in the TI-86, typing 2nd GRAPH will take you into the equation solver.

Use CLEAR to clear any text to the right of “eqn:”.

At the bottom of your screen are 5 function names each associated with one of the function keys. The MORE key brings up the next five functions and another MORE will bring up the next five functions until it loops back to the first 5 functions. Use MORE to navigate until “npv” is one of the functions offered.

Use the function key below “npv” to select the equation you entered as npv=…

The line should now read “eqn:npv”

Hit the ENTER key.


The screen should contain something like

Exp=npv

exp=

C0=

C1=

r=

C2=

C3=

C4=

C5=

C6=

Bound=(-1E99,1E99)

Up to C3 should be visible. Use the cursor up and down keys to look at the full set of values. If you cursor onto the top line, the variables will disappear. Cursoring down will make them reappear.

Solving for NPV If solving for NPV enter the values of the variables C0, C1, r, C2, C3, C4, C5, C6.

Use the (-) for negative values. Fill in something for each variable i.e. use zeros for values for which there is no data, for example

Exp=npv

exp=

C0=-100

C1=0

r= 0.1

C2=0

C3=150

C4=0

C5=0

C6=0

Bound=(-1E99,1E99)

Cursor to the second line “exp=” and use the F5 key to invoke “SOLVE” from the functions list.


The Net Present Value will be calculated and entered into the second line as

Exp=npv

exp=12.68722013524

C0=-100

C1=0

r= 0.1

C2=0

C3=150

Solving for IRR If solving for IRR enter the values of the variables exp (equals zero for the IRR), C0, C1, C2, C3, C4, C5, C6.

Use the (-) for negative values. Fill in something for each variable i.e. use zeros for values for which there is no data, for example

Exp=npv

exp=0

C0=-100

C1=0

r=

C2=0

C3=150

C4=0

C5=0

C6=0

Bound=(-1E99,1E99)

Cursor to the fifth line “r=” and use the F5 key to invoke “SOLVE” from the functions list.


The Internal Rate of Return will be calculated and entered into the fifth line as

Exp=npv

exp=0

C0=-100

C1=0

r= 0.14471424255334

C2=0

C3=150

Time Value of Money Calculations

The present value of an annuity formula

{ } ( )

{ } ( )

1

1 *

1 1PVA

1 1PVA

A

A

n

nDUE r

r r r

r r r

= − = −

(1 )+

+

+

can be expressed to the calculator as:

pva=(pmt*((1/(r/m)-(1/((r/m)*(1+(r/m))^(yrs*m))))*(1+(r/m)*beg)

“m” is the number of periods in a year

“yrs” is the number of years

“r” is the annual interest rate

where “(r/m)” is the periodic interest rate

“(yrs*m)” is the number of periods

“beg” = 1 if the question asks for an annuity due

“beg” = 0 if the question asks for an annuity

In the solver, as with the NPV and IRR, “exp” refers to the left hand side of the equation or the PVA in this case. As before you can enter a value for exp and calculate one of the other variables: so if you want to calculate the payment given the PVA, yrs, m, r and knowing the mode (beg/end) you fill in exp=PVA, yrs =number, m= number, r = decimal expression of rate, and beg =1 or beg=0. Then go to the pmt line and hit F5 to solve.


The future value of an annuity formula

( ){ }( ){ }

1

1 *(1 )

1FVA

1FVA

A

A

n

nDUE

r

r r

r

r

= − = −

+

+

+


fva=(pmt*((1+(r/m))^(yrs*m))-1)/(r/m)))*(1+(r/m)*beg)

Where






“beg” = 1 if the question asks for an annuity due

“beg” = 0 if the question asks for an annuity

In the solver “exp” refers to the left hand side of the equation or the FVA in this case. As before you can enter a value for exp and calculate one of the other variables: so if you want to calculate the payment given the FVA, yrs, m, r and knowing the mode (beg/end) you fill in exp=FVA, yrs =number, m= number, r = decimal expression of rate, and beg =1 or beg=0. Then go to the pmt line and hit F5 to solve.

Future Value formula

{ }

1FV PVm n

n r m= +


fv=pv*((1/(r/m)) ^(yrs*m)

where

“fv” is the future value

“pv” is the present value







In the solver “exp” refers to the left hand side of the equation or the FV in this case. As before you can enter a value for the FV beside “exp=” and calculate one of the other variables. Enter values for the known variables (if annual then let m=1). Then go to the line with the variable for which you want to solve and hit F5 to solve.

Future Value formula with continuous compounding * FV PV r n

n e =


fvc=pvc*e^(r*yrs)

where

“fvc” is the future value under continuous compounding

“pv” is the present value under continuous compounding



“e^” is obtained using 2nd and LN keys.

In the solver “exp” refers to the left hand side of the equation or the FVC (under continous compounding) in this case. As before you can enter a value for the FVC beside “exp=” and calculate one of the other variables. Enter values for the known variables. Then go to the line with the variable for which you want to solve and hit F5 to solve.

Effective Annual Rate

{ } 1Effective Annual Rate 1m

r m= −+


ear=((1+(r/m))^m) –1

where

“ear” is the effective annual rate




In the solver “exp” refers to the left hand side of the equation or the EAR in this case. Enter values for the known variables. Then go to the “exp=” line and hit F5 to solve.


Perpetuities and Constant Growth Perpetuities

( ){ }1

1

A is equivalent to

if A= and g=0

PVPPVCGP CF

r

r gCF

=

= −

IF we express the Present Value Constant Growth Perpetuity to the calculator as

cgp=pmt/((r/m)-(grw/m))

where

“pmt” is either A or CF1



“grw” is the annual growth rate

In the solver “exp” refers to the left hand side of the equation or the PVCGP in this case. Enter values for the known variables. Then go to the “exp=” line and hit F5 to solve.

To solve for the growth rate or the interest rate or the payment enter the know variables go to the line for the variable you want to calculate and hit F5 to solve.

m should be equal to one unless the payment is occurring more frequently than annually.

grw should equal zero if the calculation is for the simple perpetuity.


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