## Answers

a) This can be done using below formula which can be derived as we do derivation of annuity formula.

A, accumulated amount = 8000

i, interest rate for the period = 5%/2 = 2.5% because interest rate calculated at semi annual rate

n, period = 8x2 = 16 periods

Therefore P = (8000x 2.5%)/((1+0.025)^{16} -1) = 412.79

Hence **X = 412.79**

b) Amount in Sunk fund at the end of 6th Year is

Using the same formula,

we can rearrange to

Now, we have to find A

P = X = 412.79

i = 2.5%

n = 6X2 = 12

Putting values in formula

A = 412.79(1.025^{12} -1)/.025 = $5694.67

Therefore loss to the bank = 8000- 5694.67 = $2305.33

c) Interest paid by Carlos after every 6 months = i X Principal = 6%/2 X 8000 = 3% of 8000 = $240

Therefore Total Money got from Interest = 240x12 = 2880

total money bank collected = Interest + Sunk fund = 2880 + 5694.67 = 8574.67

This is more than $8000 by $574.67

d) Interest after 6 years =

i = 6%/2 = 3%

n = 6x2 = 12

Therefore Interest = 8000(1.03^{12} -1) = $3406.087

Principal received = $5694.67

Total Money bank has = 9100.757

To calculate semi annual yield

9100.757 = 8000(1+i_{annual}/2)^{12}

^{=> iannual = 2X ((9100.76/8000)}^{1/12} - 1 ) = 2.16%