Lesson Problems on discretely compound accounts

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Problems on discretely compound accounts


Discrete compounding refers to the method by which interest is calculated and added to the principal at certain set points in time.
For example, interest may be compounded yearly,  quarterly,  monthly,  weekly,  or even daily.

If the initial deposit at the account is  A  and annual interest is  p%,  then the formulas to calculate the Future Value of the compounded account are

        - for yearly compounding the Future Value after  t  years is   FV = A%2A%281%2Br%29%5Et;

        - for quartely compounding the future value after  n  quarters is   FV = A%2A%281%2Br%2F4%29%5En;

        - for monthly compounding the future value after  n  months is   FV = A%2A%281%2Br%2F12%29%5En,

where  r = p%2F100  is the decimal form of the percentage  "p".


In this lesson, you will find solutions to all major types of problems on discrete compounding.


Problem 1

An amount of  $10,000 is invested for a period of  5  years and compounded quarterly at  3.75%  per annum.
Find a future value of the investment.

Solution 

For this quarterly compounded account, use the formula


FV = 10000%2A%281+%2B+0.0375%2F4%29%5E%284%2A5%29 = 12051.77 dollars.    ANSWER

Problem 2

How much money would you have in  10  years if today you invested  $1000   a)  in an account compounded yearly at a rate of  4%?
in an account compounded quarterly at a rate of  4%?   in an account compounded monthly at a rate of  4%?

Solution 
    a)  FV = 1000%2A%281%2B0.04%29%5E10 = 1480.24 dollars;


    b)  FV = 1000%2A%281%2B0.04%2F4%29%5E%284%2A10%29 = 1488.86 dollars;


    c)  FV = 1000%2A%281%2B0.04%2F12%29%5E%2812%2A10%29 = 1490.84 dollars.

Problem 3

How long does it take for an investment to double in value if it is invested at  4​%  compounded quarterly?

Solution 

F = A%2A%281+%2B+r%2Fn%29%5E%28nt%29     (1)

where

F = future value
P = present value
r = annual interest rate expressed as a decimal
n = number of payments per year
t = number of years


This problem asks to find t for F= 2A,  r= 4% (0.04), and n=4 interest payments per year:

    2A = P%2A%281+%2B+0.04%2F4%29%5E%284t%29+ 


Cancel P from both sides, and simplify where possible:

    2 = %281.01%29%5E%284t%29+


Take log base 10 of both sides:

log(2) = 4t*log(1.01)

Solve for t:

t = log%28%282%29%29%2F%284%2Alog%28%281.01%29%29%29 = 0.30103%2F%284%2A0.004321%29 = 17.415 years.


In this problem, we must round the time to nearest greater integer number of quarters, which gives us the 


ANSWER.  t = 17.5 years.

Problem 4

How long does it take for an investment to double in value if it is invested at  16​%  compounded monthly?

Solution 

2A = A%2A%281%2B0.16%2F12%29%5En,


where "n" is the number of months.


Divide by A both sides


2 = %281%2B0.16%2F12%29%5En,   or, equivalently


2 = 1.013333%5En.


Take log base 10 from both sides


log(2) = n*log(1.013333)


n = log%28%282%29%29%2Flog%28%281.013333%29%29 = 52.3 months = 4 years and 5 months (rounded to the closest integer month).

Problem 5

Austin is going to invest  $470  and leave it in an account for  8  years.  Assuming the interest is compounded quarterly,
what interest rate,  to the nearest hundredth of a percent,  would be required in order for Austin to end up with  $600?

Solution 

Let "r" be the nominal annual percentage, i.e. the unknown value under the problem's question.

Then you have this equation


    600 = 470%2A%281%2Br%2F4%29%5E%284%2A8%29

    600%2F470 = %281%2Br%2F4%29%5E32

    1+%2B+r%2F4%29%5E32 = 1.276596


Take log base 10 from both sides


    32*log(1+r/4) = log(1.276596)

    log(1 + r/4) = log%28%281.276596%29%29%2F32 = 0.003314

    1+%2B+r%2F4 = 10%5E0.003314 = 1.00766

    r/4 = 1.00766 - 1 = 0.00766

    r = 0.00766*4 = 0.03064 = 3.064% = 3.06% rounded as requested.  ANSWER

Problem 6

What interest rate compounded monthly will yield an effective interest rate of  8​%?

Solution 

The problem asks which NOMINAL annual interest compounded monthly is equivalent to the annual interest of 8% compounded annually.


Let x be the nominal interest rate  under the question.


Then the account grows from month to month with the effective growing coefficient %281%2Bx%2F12%29.


Thus the equation to find x is THIS


    %281%2Bx%2F12%29%5E12 = 1 + 0.08,   or

    %281%2Bx%2F12%29%5E12 = 1.08.


From the equation


    1+%2B+x%2F12 = 1.08^(1/12) = 1.006434.


and finally you get


    x%2F12 = 1.006434 - 1 = 0.006434.


Therefore,  x = 12*0.006434 = 0.0772.


Thus the equivalent (or effective) nominal annual interest rate is 7.72%.


ANSWER.  NOMINAL annual interest compounded monthly, equivalent to the annual interest of 8% compounded annually, is 7.72%.


         Or, in more compact form,  8% annual interest compound annually, is equivalent to 7.72% annual interest compound monthly.

Problem 7

At what rate,  compounded monthly,  should  $25,000  be deposited in a bank to gain an interest of  $4,500  in  3  years?

Solution

            It is one of the typical problems on periodically compounded account,
            but slightly changed/modified comparing with traditional formulation.

            So,  my first step is  to convert  the problem to its traditional form.

            To convert it,  notice that having the principal of  $25,000  and gaining the interest of  $4,500
            MEANS  that the  Future  Value  is   $25,000 + $4,500 = $29,500.

            So,  now we have this traditional formulation 

           +----------------------------------------------------------------+
           |   Find the nominal annual rate at which the principal $25,000  |
           |   compounded monthly becomes $29,500 in 3 years ?              |
           +----------------------------------------------------------------+


Now this standard problem is  EASY  to  SOLVE. 


    29500 = 25000%2A%281%2Br%2F12%29%5E%283%2A12%29


    29500%2F25000 = %281%2Br%2F12%29%5E36


    %281%2Br%2F12%29%5E36 = 1.18


    log%2810%2C%281+%2B+r%2F12%29%29 = %281%2F36%29%2Alog%2810%2C%281.18%29%29 = 0.001997

    1+%2B+r%2F12  = 10%5E0.001997 = 1.004609.

      r%2F12    = 1.004609 - 1 = 0.004609

        r    = 12*0.004609 = 0.055308 = 0.055 (rounded) = 5.5%.    ANSWER


CHECK.  25000%2A%281%2B0.055308%2F12%29%5E%283%2A12%29 = 29500.84.   Good enough.


My other lessons in this site on logarithms,  logarithmic equations and relevant word problems are
    - WHAT IS the logarithm,
    - Properties of the logarithm,
    - Change of Base Formula for logarithms,
    - Evaluate logarithms without using a calculator
    - Simplifying expressions with logarithms
    - Solving logarithmic equations,
    - Solving advanced logarithmic equations
    - Solving really interesting and educative problem on logarithmic equation containing a HUGE underwater stone
    - Proving equalities with logarithms
    - Solving logarithmic inequalities
    - Using logarithms to solve real world problems,  and
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    - Population growth problems
    - Radioactive decay problems
    - Carbon dating problems
    - Bacteria growth problems
    - A medication decay in a human's body
    - Problems on appreciated/depreciated values
    - Inflation and Salary problems
    - Miscellaneous problems on exponential growth/decay
    - Problems on continuously compound accounts
    - Tricky problem on solving a logarithmic system of equations
    - Entertainment problem: Uninterrupted withdrawing money from a retirement fund
    - Entertainment problems on logarithms
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    - Upper level problems on solving logarithmic equations
    - OVERVIEW of lessons on logarithms, logarithmic equations and relevant word problems

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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