Future Value and Present Value

Using financial computing

Lesson Overview

Unit 10 - Future Value and Present Value: Using Financial Computing

The set of activities contained within follow those presented in John Webber, Math for Business and Life, Fourth Edition, Olympus Publishing Company, 2009  in Unit 10 of the book.  Similar activities can be found in other books.

Unit Objectives

Unit 10.1  	Time-Value-of-Money terminology

Unit 10.2	Financial Computing

Unit 10.3	Future Value

Unit 10.4	Present Value

Goals

Understand Time-Value-of-Money Terminology

Finding the Future Value for a single deposit

Finding the Present Value when given a Future Value

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.1  Time-value-of-money terminology

With compounding interest rates, we not only earn interest on the initial deposit, but was also earn interest on the previously accumulated interest.  Due to this fact, we must understand a principle call the Time-value-of-money.

The variables needed to calculate any Time-value-of-money problem are:

({{Variable, Definition}, {n, Total number of periods}, {i, Interest rate per period}, {PV, Present Value}, {PMT, Payment}, {FV, Future Value}})

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.1  Time-Value-of-Money terminology

Time-value-of-money problems are quite simple and normally straight-forward.  Some attention to determining the correct period and interest rate will serve you well for these types of problems.  The values needed for the calculations may not be the numbers that are used in common discussions.  For example, when asking about a car loan last week, I was told the rate was 7.05% per year.  One might first believe n=1 and i=7.05%.  Instead we find out that Banks charge interest on your car loans on a daily basis, even though they quote you an annual rate.  In this case, the correct number of n (periods) is 365 and the i (interest rate) is (7.05/365) .

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.1  You Give It A Try

You start a savings plan, earning 6% compounded quarterly, by depositing $300.  You then deposit $100 at the end of each quarter.  You want to know your account balance at the end of 10 years.  Identify n, i, PV, PMT and FV.

n = 10 4; 
Print["The number of periods(n) is :",n, " periods."  ]
Print["Careful: Many people would say 10 periods, based on the identified 10 years, but interest is computed quarterly."]
i = 6/4;
Print["The interest rate (i): ",N[ i], " %."]
Print["Since the period in question is quarterly,we take((6%)/4)and we get 1.5%.  1.5% is equal to .015"]
pv=300;    
Print["The Present Value (PV) is: $", pv,". The amount we started with."]
pmt = 100;
Print["The Payments (PMT) are: $",pmt,". The amount of deposits."]

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.1  You Give It A Try

You want to own your own home.  You have worked hard and saved up for a nice down payment.  Your banker states that a thirty-year mortgage is currently available at rate of 5.6%.  Having studied Math for Business and Life, you quickly ask your banker what the APR for the loan is.  Highlight the values for  n and i.  And calculate the APR.

n = 10,950  -> 30 years times 365 days per year.   30 * 365 = 10,950
i = 0.01534%  ->  Since the period in question is daily, we take ((5.6%)/365) and we get .01534%.

Check

5.6/365

Calculating APR

100*(((10000*(((1+.000153425))^365)-10000))/10000)

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.2  Future Value Computing

The following demonstration allows you to change values for PV,n,i in a single deposit future value calculation.  Move the sliders and select the appropriate compounding period.  The results are automatically calculated below.

Manipulate[
Pane[
Text@Style[
Grid[{
{Row[{"Your initial deposit of $",Style[p,Blue,Bold]}]},
{"\n"},
{Row[{"compounded ",
Style[Switch[compound,1,"annually",2,"semi-annually",3,"quarterly",4,"monthly",5,"weekly",6,"daily",7,"continuously"],Red,Bold],
" at an annual interest of ",
,Style[i,Purple,Bold],"%"}]},
{"\n"},
{Row[{ "for ",y," year",If[y==1,"","s"]," will be worth "}]},
{"\n"},
{Row[{"$",
Style[Switch[compound,
1,NumberForm[N[(1+(i/100))^(y)*p],{7,2}],
2,NumberForm[N[(1+(i/200))^(2*y)*p],{7,2}],
3,NumberForm[N[(1+(i/400))^(4*y)*p],{7,2}],
4,NumberForm[N[(1+(i/1200))^(12*y)*p],{7,2}],
5,NumberForm[N[(1+(i/5200))^(52*y)*p],{7,2}],
6,NumberForm[N[(1+(i/36500))^(365*y)*p],{7,2}],
7,NumberForm[N[p*E^((i*y)/100)],{6,2}]
],Bold,Darker[Green]]}]},
{"\n"},
{Style[Row[{
"equivalent annual interest rate is : ",
Switch[compound,
1,NumberForm[N[((1+((i/100)/1))^1-1)*100],{5,2}],
2,NumberForm[N[((1+((i/100)/2))^2-1)*100],{5,2}],
3,NumberForm[N[((1+((i/100)/4))^4-1)*100],{5,2}],
4,NumberForm[N[((1+((i/100)/12))^12-1)*100],{5,2}],
5,NumberForm[N[((1+((i/100)/52))^52-1)*100],{5,2}],
6,NumberForm[N[((1+((i/100)/365))^365-1)*100],{5,2}],
7,NumberForm[N[(E^(i/100)-1)*100],{5,2}]
],
"%"
}],14,Bold]
}
},Alignment->Center],
17],
{550,300},Alignment->{Center,Center}],
{{p,100,"initial deposit"},0,10000,1,Appearance->"Labeled"},
{{compound,3,"compounded"},{1->"annually",2->"semi-annually",3->"quarterly",4->"monthly",5->"weekly",6->"daily",7->"continuously"},ControlType->Setter},
{{i,3,"% annual interest"},0,25,.01,Appearance->"Labeled"},
{{y,1,"years"},1,50,1,Appearance->"Labeled"}
]

THIS NOTEBOOK IS THE SOURCE CODE FROM

Future Value Calculator from The Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/FutureValueCalculator/

Contributed by: Sarah Lichtblau

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.3  Future Value

The formula for finding the Future Value, given we know the Present Value is:

Future Value = Present Value *  (((1+interest ))^(number of periods))

or

FV=PV (((i+1))^n)

And the formula for finding the Present Value, given we know the Future Value is:

Present Value = ((Future Value)/(((1+interest))^(number of periods)))

or

PV=(FV/(((i+1))^n))

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.3  Future Value

Lets give it a try!     What is the value of a savings account in 2015, if we have $5,000 in the account and we are paid 2.5% annually.

What do we know?

Fv=Pv (((i+1))^n)

Pv = 5000;
n=6 ;     (* years : 2015 - 2009 *)
i =  .025;  (* 2.5% *)

How do we set this up?

Fv = Pv*(((1+i))^n)

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.4  Present Value

The formula for finding the Present Value, given we know the Future Value is:

Present Value = ((Future Value)/(((1+interest))^(number of periods)))

or

PV = (FV/(((1+i))^n))

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.4  Present Value Computing

The following demonstration allows you to change values for FV, n and i in a present value calculation.  Move the sliders and select the appropriate compounding period.  The results are automatically calculated below.

Manipulate[
Pane[
Text@Style[
Grid[{
{Row[{"If the money is compounded ",
Style[Switch[compound,1,"annually",2,"semi-annually",3,"quarterly",4,"monthly",5,"weekly",6,"daily",7,"continuously"],Red,Bold]}]},
{"\n"},
{Row[
{" at an annual interest of ",
,Style[i,Purple,Bold],"%"}]},
{"\n"},{Row[{"In order to have $",Style[fut,Blue,Bold]," in ",y," year",If[y==1,"","s"]}]},
{"\n"},{Row[{"You will need to invest "}]},
{Row[{"$",
Style[Switch[compound,
1,NumberForm[N[(fut/((1+(i/100))^(y)))],{7,2}],
2,NumberForm[N[(fut/((1+(i/200))^(2y)))],{7,2}],
3,NumberForm[N[(fut/((1+(i/400))^(4y)))],{7,2}],
4,NumberForm[N[(fut/((1+(i/1200))^(12y)))],{7,2}],
5,NumberForm[N[(fut/((1+(i/5200))^(52y)))],{7,2}],
6,NumberForm[N[(fut/((1+(i/36500))^(365y)))],{7,2}],
7,NumberForm[N[(fut/(E^((i*y)/100)))],{6,2}]
],Bold,Darker[Green]]}]},
{"\n"},

{Style[Row[{
"equivalent annual interest rate is : ",
Switch[compound,
1,NumberForm[N[((1+((i/100)/1))^1-1)*100],{5,2}],
2,NumberForm[N[((1+((i/100)/2))^2-1)*100],{5,2}],
3,NumberForm[N[((1+((i/100)/4))^4-1)*100],{5,2}],
4,NumberForm[N[((1+((i/100)/12))^12-1)*100],{5,2}],
5,NumberForm[N[((1+((i/100)/52))^52-1)*100],{5,2}],
6,NumberForm[N[((1+((i/100)/365))^365-1)*100],{5,2}],
7,NumberForm[N[(E^(i/100)-1)*100],{5,2}]
],
"%"
}],14,Bold]
}
},Alignment->Center],
17],
{550,300},Alignment->{Center,Center}],
{{compound,3,"compounded"},{1->"annually",2->"semi-annually",3->"quarterly",4->"monthly",5->"weekly",6->"daily",7->"continuously"},ControlType->Setter},
{{i,3,"% annual interest"},0,25,.01,Appearance->"Labeled"},
{{y,1,"years"},1,50,1,Appearance->"Labeled"},
{{fut,100,"Future Value"},0,100000,1,Appearance->"Labeled"}
]

Unit 10 - Future Value and Present Value: Using Financial Computing

Unit 10.4  Present Value

Lets give it a try!     How much money will you need to save today, if we want to have $5,000 in the account in 3 years.  Banks are currently paying 1.5% annually, but paying interest semi-annually.

What do we know?

Pv = (Fv/(((1+i))^n))

Fv = 5000;
n=6 ;     (*  3 years * 2 (semi-annually) *)
i =  ((.015)/2);

How do we set this up?

Pv = (Fv/(((1+i))^n))