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
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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

NPV  C 0  CF1 1 r 
 CF 1 r   CF 1 r   CF 1 r  
1
2

2
3

3
n
 ... 
n
Where C0 ,CF1 ,CF2 ,CF3 ...CFn can be positive or negative.
For the IRR the formula assumes that NPV=0 and looks as follows:

C 0  CF1 1 IRR 
 CF 1 IRR   CF 1 IRR   CF 1 IRR    0
1

2
2

3
3
 ... 
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:
npv  C0+
C1
+
C2
+
C3
+
C4
+
C5
+
C6
1+r  1+r  1+r  1+r  1+r  1+r 
1
2
3
4
5
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.
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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.
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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.
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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.
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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

 
 
PVA A 1 r  1 r1 r   
n

PVA
DUE

A 1 r  1 r1 r    *(1 r)
n



 
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.
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The future value of an annuity formula

FVA A 1 r 

FVA
DUE

n


1 r 

A 1 r 
n



 1 r  *(1  r )

can be expressed to the calculator as:
fva=(pmt*((1+(r/m))^(yrs*m))-1)/(r/m)))*(1+(r/m)*beg)
Where
“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 “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
FV n  PV
1r m
mn
can be expressed to the calculator as:
fv=pv*((1/(r/m)) ^(yrs*m)
where
“fv” is the future value
“pv” is the present value
“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
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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 e
r *n
n


can be expressed to the calculator as:
fvc=pvc*e^(r*yrs)
where
“fvc” is the future value under continuous compounding
“pv” is the present value under continuous compounding
“yrs” is the number of years
“r” is the annual interest rate
“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
r m
Effective Annual Rate  1

m
1
can be expressed to the calculator as:
ear=((1+(r/m))^m) –1
where
“ear” is the effective annual rate
“yrs” is the number of years
“m” is the number of periods in a year
“r” is the annual interest 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.
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Perpetuities and Constant Growth Perpetuities
PVP  A r is equivalent to
PVCGP  CF  r  g 
if A=CF and g=0
1
1
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
“m” is the number of periods in a year
“r” is the annual interest rate
“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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