Question

Eric deposits 1000 into a bank account. The bank credits interest at a nominal annual rate of i convertible semiannually for the first 7 years and a nominal annual rate of 2i convertible quarterly for all years thereafter. The accumulated amount in the account at the end of 5 years is X. The accumulated amount in the account at the end of 10.5 years is 1980. Calculate X

Answer 1

Answer:

1276.5

Explanation:

Today, Eric deposited an amount(PO) that is = 1000

The annual nominal Interest rate for the first 7 years = i  

This is convertible semiannually.

As such,

The semi-annual interest rate for first 7 years =$\frac{i}{2}$  

The number of semi-annual periods in 7 years (k) = 2*7 = 14

The magnitude of the deposit at end of year 7( P7),

= P7 ==  P0 * (1 + i/2)14 ------ equation 1

On the other hand,

The nominal interest rate after 7 years = 2i    which is convertible quarterly

The effective quarterly interest rate = $\frac{2i}{4}$=$\frac{i}{2}$

The number of quarters between the end of year 7 and year 10.5 = 4* (10.5 - 7) = 14

The amount of deposit at the end of year 10.5 year = P10.5  

$P_{10.5}$= $P_{7}$ * (1 +$\frac{i}{2}$$_{14}$  ---equation 2

We are given the following:

$P_{10.5}$ =1980

We shall thus substitute this value into the equation as

1980 = $P_{7}$ *( 1 + i/2)14

Further, we have worked out that  

P7 ==  P0 * (1- i/2)14 *(1

       

We shall therefore substitute equation 1 into equation 2 as follows

1980 = P0 * (1+ i/2)14 *( 1 + i/2)14  

1980 = 1000* (1+ i/2)14 *( 1 + i/2)14  

This can also be rewritten as

1000* (1+ i/2)14 *( 1 + i/2)14  =1980

(1+ i/2)14 *( 1 + i/2)14  =1980/1000

(1+ i/2)28 =1.98

(1+ i/2) = 1.981/28

(1+ i/2) = 1.0247

i/2 = 1.0247 -1

i/2 = 0.0247

1 = 0.0247 *2 =0.0494

At the end of year 5, the amount has accumulated to an amount = X

X =  Po * ( 1 + i/2 )10

  = 1000 * ( 1 + 0.0494/2)10

   = 1000 * ( 1 + 0.0247)10

    = 1000 + ( 1.0247)10

    =   1000* 1.276467

  =  1276.5