Question 84045
{{{A=p(1+r/n)^(n*t)}}} Start with the given equation


{{{A=20000(1+0.08/n)^(n*3)}}}  Plug in p=20000, r=0.08


a) 

Lets calculate the return if the bank compounds annually

Let n=1 and plug it into  {{{A=20000(1+0.08/n)^(n*3)}}}

{{{A=20000(1+0.08/1)^(1*3)}}}       Start with the given expression
{{{A=20000(1+0.08)^(1*3)}}}     Divide 0.08 by 1 to get 0.08
{{{A=20000(1+0.08)^(3)}}}     Multiply the exponents 1 and 3 to get 3
{{{A=20000(1.08)^(3)}}}       Add 1 and 0.08 to get 1.08
{{{A=20000(1.259712)}}}      Raise 1.08 to 3 to get 1.259712
{{{A=25194.24}}}       Multiply 20000 and 1.259712 to get 25194.24

So our return is $25194.24
b) 

Lets calculate the return if the bank compounds quarterly

Let n=4 and plug it into  {{{A=20000(1+0.08/n)^(n*3)}}}

{{{A=20000(1+0.08/4)^(4*3)}}}       Start with the given expression
{{{A=20000(1+0.02)^(4*3)}}}     Divide 0.08 by 4 to get 0.02
{{{A=20000(1+0.02)^(12)}}}     Multiply the exponents 4 and 3 to get 12
{{{A=20000(1.02)^(12)}}}       Add 1 and 0.02 to get 1.02
{{{A=20000(1.26824179456255)}}}      Raise 1.02 to 12 to get 1.26824179456255
{{{A=25364.8358912509}}}       Multiply 20000 and 1.26824179456255 to get 25364.8358912509

So our return is $25364.84
c) 

Lets calculate the return if the bank compounds monthly

Let n=12 and plug it into  {{{A=20000(1+0.08/n)^(n*3)}}}

{{{A=20000(1+0.08/12)^(12*3)}}}       Start with the given expression
{{{A=20000(1+0.00666666666666667)^(12*3)}}}     Divide 0.08 by 12 to get 0.00666666666666667
{{{A=20000(1+0.00666666666666667)^(36)}}}     Multiply the exponents 12 and 3 to get 36
{{{A=20000(1.00666666666667)^(36)}}}       Add 1 and 0.00666666666666667 to get 1.00666666666667
{{{A=20000(1.27023705162065)}}}      Raise 1.00666666666667 to 36 to get 1.27023705162065
{{{A=25404.741032413}}}       Multiply 20000 and 1.27023705162065 to get 25404.741032413

So our return is $25404.74
d) 

Lets calculate the return if the bank compounds daily

Let n=365 and plug it into  {{{A=20000(1+0.08/n)^(n*3)}}}

{{{A=20000(1+0.08/365)^(365*3)}}}       Start with the given expression
{{{A=20000(1+0.000219178082191781)^(365*3)}}}     Divide 0.08 by 365 to get 0.000219178082191781
{{{A=20000(1+0.000219178082191781)^(1095)}}}     Multiply the exponents 365 and 3 to get 1095
{{{A=20000(1.00021917808219)^(1095)}}}       Add 1 and 0.000219178082191781 to get 1.00021917808219
{{{A=20000(1.27121572005174)}}}      Raise 1.00021917808219 to 1095 to get 1.27121572005174
{{{A=25424.3144010349}}}       Multiply 20000 and 1.27121572005174 to get 25424.3144010349

So our return is $25424.31


e) What observation can you make about the size of the increase in your return as your compounding increases more frequently?


As the compounding frequency increases, the return slowly approaches some finite number (which in this case appears to be about $12213.69). Think about it, banks wouldn't be too fond of shelling out an infinite amount of cash.



f)Calculate A with continuous compounding

Using the contiuous compounding formula {{{A=Pe^(rt)}}} where e is the constant 2.7183 and letting r=0.1, P=10,000, and t=2 we get


{{{A=20000(2.7183)^(0.08*3)}}} Start with the given equation


{{{A=20000(2.7183)^(0.24)}}} Multiply 0.1 and 2


{{{A=20000(1.27125118988893)}}} Raise 2.7183 to 0.2


{{{A=25425.0237977787}}} Multiply


So using continuous compounding interest we get a return of $25425.02 (which is real close to what we got from a daily compounding frequency)



g)Now suppose, instead of knowing t, we know that the bank returned to us $15,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t)


{{{25000=20000e^(0.08t)}}} 


{{{25000/20000=e^(0.08t)}}} Divide both sides by 20,000


{{{1.25=e^(0.08t)}}} 

{{{ln(1.25)=0.08t}}} Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.


{{{ln(1.25)/0.08=t}}} Divide both sides by 0.08


So we get


{{{t=0.2231/0.08=2.78875}}}

{{{t=2.78875}}}


So it will take a little over 2 and a half years to generate $25,000



h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer?


Since we want to double our money, let A=2*20,000. So A=40,000. Now solve for t:

{{{40000=20000e^(0.08t)}}} 


{{{40000/20000=e^(0.08t)}}} Divide both sides by 10,000


{{{2=e^(0.08t)}}} 

{{{ln(2)=0.08t}}} Take the natural log of both sides. This eliminates "e".The natural log (pronounced "el" "n") is denoted "ln" on calculators.


{{{ln(2)/0.08=t}}} Divide both sides by 0.08


So we get


{{{t=0.69314/0.08=8.66425}}}

{{{t=8.66425}}}


So it will take about 8 and a half years to double your money.