5. Suppose we have a bank account with a nominal APR of 12% what would bethe eff

Question

5. Suppose we have a bank account with a nominal APR of 12% what would bethe effective rate ifwe daily? After 12 monthly deposits of S200 at 5% compounded monthl one year. 6. y, find the amount of the annuity at the end of 7. What is the value of an annuity at the end of 20 years if $2,000 is deposited each year into an accoust earning 8.5% compounded annually? How much interest was earned? 8. Our company decides that we will need to buy a new computer system in 4 years at an estimated cost of 12,000. How much would we have to deposit each month into an accont earning 3.5%s compounded monthly to reach our goal? 9. what is the present value of an annuity with payments of S200 per month for 5 years earning interest of 6% 10. Bill would like to set up a college fund for his son which will help him pay for living expenses while in son to be able to withdraw $250 per month for lour years. This amount will be held in an aunt eaming 4% interest per year son starts college to cover all these withdrawals? compounded monthly. ow much needs to be in Bill's account when his

Explanation / Answer

Solution

Back-up Theory

A sum P at rate of interest r% compounded annually for T years becomes:

A = P{1 + (r/100)}T ……………………………………………………………………..(1)

If compounded quarterly, the amount is: AQ = P{1 + (r/400)}4T ………………………..(2)

If compounded monthly, the amount is: AM = P{1 + (r/1200)}12T ………………………..(3)

If compounded daily, the amount is: AD = P{1 + (r/36500)}365T ………………………..(4)

Now, to work out the solutions,

Q5

Here r = 12% and compounding is daily and T = 1. Hence, vide (4),

A = P{1 + (12/36500)}365

= 1.000329365

= 1.12757

=> effective interest rate is 12.757% ANSWER

Q6

Here, P = 200, r = 5% and compounding is monthly and T = 12 months for the first deposit, 11 months for the second deposit, ………., 1 month for the last (12th) deposit.

So, the total amount,

A = 200{1 + (5/1200)}12 + 200{1 + (5/1200)}11 + …….. + 200{1 + (5/1200)}1

= 200{(1205/1200)12 + (1205/1200)11 + ……… + (1205/1200)1}

= 200(1205/1200){(1205/1200)12 – 1}/{(1205/1200) – 1} [employing the formula for the sum of a geometric progression]

= (200 x 1.004167 x 0.051166)/0.004167

= 2466.205

Thus, the amount after 12 months would be $2466.21 ANSWER

Q7

Here, P = 2000, r = 8.5% and compounding is annual and T = 20 years for the first deposit, 19 years months for the second deposit, ………., 1 year for the last (20th) deposit.

So, the total amount,

A = 2000{1 + (8.5/100)}20 + 2000{1 + (8.5/100)}19 + …….. + 2000{1 + (8.5/100)}1

= 2000{(108.5/100)20 + (108.5/100)19 + ……… + (108.5/100)1}

= 2000(1.085){(1.085)20 – 1}/{(1.085 – 1} [employing the formula for the sum of a geometric progression]

= (2000 x 1.085 x 4.112046)/0.085

= 104978.1

Thus, the amount after 20 yearss would be $104978.1 ANSWER

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