Question

1. How many years are required for P10,000.00 to increase to P20,000.00 if invested at 9% per year compounded annually, semi-annually, quarterly and monthly.

   
   

2.  If the sum of P12,000.00 is deposited in an account earning interest at the rate of 9% compounded quarterly, what will be the accumulated amount at the end of 8 years?

3. Annual deposits were made in a fund earning 10% per annum. The first deposit wasP2000.00 and each deposit thereafter was P200.00 less than the preceding one. Determine the amount in the fund after 

4. A debt of P 15,000.00 was paid for as follows: P4,000.00 at the end of three months, P5000.00 at the and of 12 months, P3000 at the end of 15 months and a final payment F at the end of 21 months.  If the rate of interest was 18% compounded quarterly, find the final payment F.

Answer 1

1. Future value = Present value(1+r)^t

Where r = rate of interest and t = No of periods

Compounded annually

r = 0.09

P20,000 = P10,000(1+0.09)^t

1.09^t = 2

log(1.09^t) = log 2

t * log(1.09) = log 2

t = log 2/log(1.09) = 0.301/0.03743 = 8.04 years

Compounded semiannually

r = 0.09/2 = 0.045

P20,000 = P10,000(1+0.045)^t

1.045^t = 2

log(1.045^t) = log 2

t * log(1.045) = log 2

t = log 2/log(1.045) = 0.301/0.019116= 15.75

No of years = 15.75/2 = 7.88 years

Compounded quarterly

r = 0.09/4 = 0.0225

P20,000 = P10,000(1+0.0225)^t

1.0225^t = 2

log(1.0225^t) = log 2

t * log(1.0225) = log 2

t = log 2/log(1.0225) = 0.301/0.009663 = 31.15

No of years = 31.15/4 = 7.79 years

Compounded monthly

r = 0.09/12 = 0.0075

P20,000 = P10,000(1+0.0075)^t

1.0075^t = 2

log(1.0075^t) = log 2

t * log(1.0075) = log 2

t = log 2/log(1.0075) = 0.301/0.003245 = 92.76

No of years = 92.76/12 = 7.73 years

Answer 2

SOLUTION :

1.

a.

Annual compounding :

r = 9% = 0.09

=> 1 + r = 1.09

PV = 10000.00 P

FV = 20000.00 P

Period = n years 

FV/PV = (1 + r)^n 

=> 20000.00 / 10000.00 = 1.09^n

=> 2.00 = 1.09^n

Taking log :

=> log(2.00) = n log(1.09)

=> n = log(2.00) /log(1.09)

=> n = 8.04 years  (ANSWER).

b.

Semi-annual compounding :

r = 9/2 % = 0.045

=> 1 + r = 1.045

PV = 10000.00 P

FV = 20000.00 P

Period = 2 n semi-annuals. (n is in years)

FV/PV = (1 + r)^(2n) 

=> 20000.00 / 10000.00 = 1.045(2^n)

=> 2.00 = 1.045^(2n)

Taking log :

=> log(2.00) = 2n log(1.045)

=> n = 0.5 * log(2.00) /log(1.045)

=> n = 7.87 years  (ANSWER).

c.

Quarterly compounding :

r = 9/4 % = 0.0225

=> 1 + r = 1.0225

PV = 10000.00 P

FV = 20000.00 P

Period = 4 n quarters. (n is in years)

FV/PV = (1 + r)^(4n )

=> 20000.00 / 10000.00 = 1.0225(4^n)

=> 2.00 = 1.0225^(2n)

Taking log :

=> log(2.00) = 4n log(1.0225)

=> n = 0.25 * log(2.00) /log(1.0225)

=> n = 7.79 years  (ANSWER).

d.

Monthly compounding :

r = 9/12 % = 0.0075

=> 1 + r = 1.0075

PV = 10000.00 P

FV = 20000.00 P

Period = 12 n quarters. (n is in years)

FV/PV = (1 + r)^(12n )

=> 20000.00 / 10000.00 = 1.0075(12^n)

=> 2.00 = 1.0075^(12n)

Taking log :

=> log(2.00) = 12n log(1.0075)

=> n = (1/12) * log(2.00) /log(1.0075)

=> n = 7.73 years  (ANSWER).