Financial Management Question?

Having passed the driving test, Suki is planning to save HK$600,000 to


buy a sports car. Her plan is to put aside a fixed amount of money at the


end of each month into her saving account for five years. The annual


interest rate is 3%, and interest is paid to her savings account monthly.





Q1 How much does Suki have to save each month to realize her plan?





Q2 After having the twenty-fourth month鈥檚 savings paid into the saving


account today, Suki then decides to scrap the plan by stopping


making any further savings deposits. She decides that at the end of


each of the coming six months she would take out an equal amount of


money until all savings, including all interest earned, are withdrawn


from the bank. What is the fixed sum Suki has to withdraw each


month to ensure she gets a zero balance in her savings account in six


months鈥?time?





PLEASE HELP THANKS ALOT|||When they say that the annual interest rate is 3% but they also say that the interest is paid to the account monthly, they are usually saying that the annual interest rate is really 3.0416% because for some reason in the distant past, people thought the annual interest rate could be calculated from the monthly interest rate by just multiplying the monthly by 12 which is what you would do with simple interest but not with compound interest and instead of admitting to doing the math wrong, they decided to make up terms to describe it. Such an erroneous annual interest rate is often called the nominal annual interest rate and the actual annual interest rate is called the effective, the compounding interval is specified so that the effective interest rate can be calculated from the nominal. This means that the 3% annual interest rate mentioned is 0.25% per month which equates to 3.0416% per year.





If x is the amount of money that needs to save at the end of each month over 5 years then it needs to satisfy the equation:





F = x * (1+r)^59 + x * (1+r)^58 + x * (1+r)^57 ... + x * (1+r)^0





Where F = 600000


r = 0.0025


and the exponents are ranging from 59 to 0 because it's 60 deposits, each at the end of the month. We can generalize this to an arbitrary number of months n to get:





F(n) = x * (1+r)^0 + x * (1+r)^1) + x * (1+r)^2 ... + x * (1+r)^n





Using the summation of a geometric sequence equation, we get:





F(n) = x * (1 - (1+r)^(n+1))/(1 - (1+r))





therefore





x = F(n) * (1 - (1+r)) / (1 - (1+r)^(n+1))


x = 600,000 * (1 - 1.0025)/(1 - 1.0025^60)


x = 9,281.22 (rounded up to the nearest 100th to make sure she saves up enough)





After 24 months, she would've saved:





F(23) = 9,281.22 * ( 1 - 1.0025^24 ) / ( 1 - 1.0025 )


F(23) = 229,272.28 (rounded down to be conservative)





She decides to withdraw this over six months in six equal monthly withdrawals. This would be an annuity and the equation is also a summation of a geometric sequence but in a different form so to derive the equation let P be the value, x be the monthly withdrawal and r be the monthly interest rate. Then think about how much would be left after one month, then two, three etc.





After one month:





F(1) = P * (1+r) - x





After two months:





F(2) = F(1) * (1+r) - x


.:


F(2) = P * (1+r)^2 - x * (1+r) - x





After three months:





F(3) = P * (1+r)^3 - x * (1+r)^2 - x * (1+r) - x





As you can see, it can be generalized as:





F(n) = P * (1+r)^n - x * (summation of a geometric sequence for k from 0 to n-1 of the term (1+r)^k )


.:


F(n) = P * (1+r)^n - x * (1 - (1+r)^n) / (1 - (1+r))





Therefore:





F(6) = 0 = P * (1+r)^6 - x * (1 - (1+r)^6 ) / (1 - (1+r))


.:


x = P * (1+r)^6 * (1 - (1+r)) / (1 - (1+r)^6)


x = 229,272.28 * 1.0025^6 * (1 - 1.0025) / (1 - 1.0025^6)


x = 38,547.09 (rounded down to be conservative)





So for Q1, she can save 9,281.22 per month and for Q2 she can withdraw 38,547.09 per month. Note the the savings and annuity equations are actually both summation of a geometric sequence so you don't actually have to memorize different equations so long as you're willing to derive the form that you need.