Question

A bank offers a nominal annual interest rate of 8% for a new savings account.

(i) What is the effective rate when compounded quarterly? Draw the cash flow diagram.

(ii) If you want to build $500,000 in savings in 2 years, what quarterly annuity must be made? Draw the cash flow diagram.

Answer 1

Interest rate = 8%

1) Quarterly interest rate = 8 / 4 = 2%

With annual interest rate of 8% annually, $100 invested today would become $108 after one year.

With interest rate of 2% quarterly, $100 invested today would become $108.24 after one year.

2) First payment made of $X will be invested for 7 quarters, then $X after 7 quarters = X (1 + 0.02) ^ 7

Second payment made of $X will be invested for 6 quarters, then $X after 6 quarters = X (1 + 0.02) ^ 6

Third payment made of $X will be invested for 5 quarters, then $X after 5 quarters = X (1 + 0.02) ^ 5

Forth payment made of $X will be invested for 4 quarters, then $X after 4 quarters = X (1 + 0.02) ^ 4

Fifth payment made of $X will be invested for 3 quarters, then $X after 3 quarters = X (1 + 0.02) ^ 3

Sixth payment made of $X will be invested for 2 quarters, then $X after 2 quarters = X (1 + 0.02) ^ 2

Seventh payment made of $X will be invested for 1 quarter, then $X after 1 quarter = X (1 + 0.02) ^ 1

Eighth payment made of $X will be invested for 0 quarters, then $X after 0 quarter = X (1 + 0.02) ^ 0 = X

Thus sum of all payment become $500,000.

X (1.02) ^ 7 + X (1.02) ^ 6 + X (1.02) ^ 5 + X (1.02) ^ 4 + X (1.02) ^ 3 + X (1.02) ^ 2 + X (1.02) ^ 1 + X = 500,000

X = 58411.21 (approx.)