PRE-CALCULUS. STUDENT ID:. [Pick the date]. Question 1.

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PRE-CALCULUS
STUDENT ID:
[Pick the date]

Question 1
PART A
Principal = R
Rate of interest = R% per period
Let the amount be deposited after every 1 year. Then as per the question, compounding
would be done after every 4 months.
Present value of investment would be the discounted value of the future cashflows expected
at the end of N periods.
Hence, present value = R*(1+(R/300))3N/(1+(R/300))3N + R*(1+(R/300))3(N-1)/(1+(R/300))3N +
R*(1+(R/300))3(N-2)/(1+(R/300))3N + ................................................ +
R*(1+(R/300))3/(1+(R/300))3N + R/(1+(R/300))3N
The above is an example of a finite geometric progression where first term =
(R*(1+(R/300))3N/(1+(R/300)3N = R and common ratio = 1/(1+(R/300)
Hence, based on the sum of finite GP, the prevent value of investment =
R(1-(1/(1+(R/300))N)/(1- (1+(R/300)) = 300*(1-(1/(1+(R/300))N) = This is the new
formula based on finite sum of GP
PART B
Face value of bond =$1,000
Maturity period = 1 year
APR = 1.3% compounding monthly
Present value of the annuity (Using Method 1) = 50/(1+ (1.3/1200))3 + 50/(1+ (1.3/1200))6 +
50/(1+ (1.3/1200))9 + 1050/(1+ (1.3/1200))12 = $1,185.48 (Computation with using
annuity formula)

Present value of the annuity (Using Method 2) = R(1-(1/(1+(R/300))N)/(1- (1+(R/300)) =
50(1-(1/(1+(1.3/4)/300))4)/(1+(1.3/4)/300)) + 1000/(1+ (1.3/1200))12 = $1,185.48
(Computation with using the formula in question 1)
It is noteworthy that in the annuity formula, it was assumed that the same amount would be
obtained after every period which is true for the coupon payment. Hence, an additional term
for present value of the face value has been added.