Finance and Accounting Assignment - Math Problem Example

Since in this case, the payment is done at the beginning of the period every time, hence it is a case of an immediate annuity as each yearly payment is allowable to compound for an additional year as compared to the normal annuity case. In this context, Future Value of Annuity = A [{(1+i) ^n -1} / i] (Finance Formulas., n.d.) Where, A= Annual payment, i= interest rate per year, n= number of periods As in this case, each annual payment is completed at the start of each period, the same is allowed to compound for one extra period and hence its future value would be the product of value of a matching normal annuity and (1+ interest rate).

Future Value of Annuity Due = (1+i) * A [{(1+i)^n -1} / i ] (Finance Formulas., n.d.) The 65th birthday is the day the person wants to have $2 million in the savings account. It should also be kept in mind that a payment is made even on the last day i.e. on the 65th birthday. This last payment does not get a chance to be compounded and has to be simply added to the compounded value of the earlier made 35 payments. In the Future Value of Annuity Due formulae, it has to be noted that the last cash payment is made one year prior to the end of the 35th year.

Keeping in mind that a payment will be made even on the last day of 35 year period, the formulae for calculating the required annual payment would be, Future Value, FV = (1+i) * A [{(1+i)^n -1} / i ] + A A = F/ [ {((1+i)^n-1)/i} * (1+i) +1] It is decided that the person needs $2 million at the end of 35 years period, so in this scenario the Future Value would be $2 million. In this case, FV= $2000000, i= 5%, n= 35 years. Putting these values in the above equation, Annual Payment, A = 20,868.91 = $ 20,870 (approx) Thus, the person has to put aside $ 20,869 (approx) each year to make sure that he has $ 2 million in the savings account on the 65th birthday.

Problem 36 The person realizes that since the income would increase over the years it would be advisable to save less now and more in the later years. Thus, instead of putting the same amount aside, the person has altered his plans to let the amount to be set aside grow by 3% per year. This is a case of growing annuity which is similar to annuity as both ends after a certain period, however, growing annuity payments increase at a fixed constant rate unlike the annuity. It should be noted that since the first annual payment to the savings account is made today and continuing to do so on each birthday up to as well as including the 65th birthday, the number of periods would be 36.

The formula for Future Value of Growing Annuity is, FV = A [{(1+i)^n – (1+g)^n } / (i-g) ] (Finance Formulas., n.d.) Where A= First payment, i= interest rate, g= growth rate, n= number of periods Hence, The First Payment, A = FV * [(i-g)/ {(1+i)^n – (1+g)^n }] Here, FV= $2000000, i = 5%, g = 3%, n = 36. Putting these values in the above equation, First Payment = 13,823.91 = $ 13,824 (approx) Thus, the person will have to put $ 13,824 (approx) into the savings account today and keep on increasing the succeeding payments at a growth rate of 3% per year in order to get $ 2 million in the savings account on the 65th birthday.

References Finance Formulas. (n.d.). Future Value of Annuity. Retrieved July 14, 2011, from