Question

Q7)

Lin has $100000.00 invested in the stock market and $50000.00 in his savings account. At the end of each year, he looks at the total of his investments and transfers money from one to the other so that his stock market investment is again twice as much as his savings account. His savings account always pays 3% compounded quarterly.

a) In the first year, his stock market investment earns 4.00% (effective annual rate). How much money does he transfer from his stocks to his savings account (or vice versa)?

b) In the second year, his stock market investment loses 16.00% (effective annual rate). How much money does he transfer from his stocks to his savings account (or vice versa)?

c) In the third year, his stock market investment again makes 3.00% (effective annual rate). How much money does he have in total?

d) If he had just taken the total amount of money he started with ($150000.00) and invested it for 3 years ending up with the same amount of money as your answer for part c), what would his effective annual interest rate have been?

Answer 1

Step 1: Computation of Amount in Saving Bank Account after 1 year

Deposit amount =$ 50,000

Rate of interest = 3% Compounded Quarterly

We know that Future Value = Present value ( 1+i/4) ^4n

Here I = Rate of interest and n = No.of years

Here I = 3% and n= 1 year

Future value = $ 50000( 1+3/400)^4(1)

= $ 50000( 1.0075) ^4

= $ 50,000*1.030339

= $ 51516.95

Saving Bank Account after 1 year is $ 51,516.95

Computation of total value of the investment at the end of year 1

Value of the investment = Amount invested * ( 1+ i)

Here I = Rate of interest i.e 4%

= $ 100000*( 1.04)

= $ 104000

Given investment in the stock market is again twice as much as his saving account

Total balance = Stock market investment + Saving Bank Account balance

= $ 104000+$ 51516.95

= $ 155516.95

Divide the Total amount into 3 parts such that 2 parts belong to stock market and 1 part belong to the saving bank account

Stock Market investment for next year = $ 155516.95/3*2 = $ 103677.96

Saving bank account balance = $ 155516.95/3 =$ 51838.99

The amount to be wthdrawn from stock market and invested in saving bank account = $ 104000- $ 103677.96

= $ 322.04

Hence Saving Account balance = $ 51516.95 + $ 322.04 = $ 51838.99

Step 2: Computation of Amount in Saving Bank Account for 2nd year

Deposit amount =$ 51838.99

Rate of interest = 3% Compounded Quarterly

We know that Future Value = Present value ( 1+i/4) ^4n

Here I = Rate of interest and n = No.of years

Here I = 3% and n= 1 year

Future value = $ 51838.99( 1+3/400)^4(1)

= $ 51838.99( 1.0075) ^4

= $ 51838.99*1.030339

= $ 53411.73

Saving Bank Account after 2 year is $ 53411.73

Computation of total value of the investment at the end of year 2

Value of the investment = Amount invested * ( 1+ i)

Here I = Rate of interest i.e -16%

= $ 103677.96*( 1-0.16)

= $ 103677.96*0.84

= $ 87089.49

Given investment in the stock market is again twice as much as his saving account

Total balance = Stock market investment + Saving Bank Account balance

= $ 87089.49+$ 53411.73

= $ 140501.22

Divide the Total amount into 3 parts such that 2 parts belong to stock market and 1 part belong to the saving bank account

Stock Market investment for next year = $ 140501.22/3*2 = $ 93667.48

Saving bank account balance = $ 140501.22/3 =$ 46833.74

The amount to be wthdrawn from saving bank account and invested in stock Market = $ 53411.73-$ 46833.74= $ 6577.99

Hence Saving Account balance = $ 53411.73 - $ 6577.99 = $ 46833.74

Stock Market investment after transfer = $ 87089.49+$ 6577.99 = $ 93667.48

Step 3: Computation of Amount in Saving Bank Account for 3rd year

Deposit amount =$ 46833.74

Rate of interest = 3% Compounded Quarterly

We know that Future Value = Present value ( 1+i/4) ^4n

Here I = Rate of interest and n = No.of years

Here I = 3% and n= 1 year

Future value = $ 46833.74( 1+3/400)^4(1)

= $ 46833.74( 1.0075) ^4

= $ 46833.74*1.030339

= $ 48254.63

Saving Bank Account after 3rd year is $ 48524.63

Computation of total value of the investment at the end of year 3

Value of the investment = Amount invested * ( 1+ i)

Here I = Rate of interest i.e 3%

Value of the Investment = $ 93667.48*1.03

= $ 93667.48*1.03

= $ 96477.50

Given investment in the stock market is again twice as much as his saving account

Totalbalance = Stock market investment + Saving Bank Account balance

Total Balance = $ 96477.50+$ 48254.63

= $ 144732.13

So after 3 years the total balance is $ 144732.13

Step 4:If we invest $ 150000 and after 3 years the balance is $ 144732.13

  Rate of retiurn ( Closing Balance - Opening Balance ) / Opening Balance * 100

( $ 144732.13-$ 150000) / $ 150000 *100

-3.5119%

Effective Annual interest rate = -3.5119/3 = -1.170633