Find the present value of a series of payments of R100 each, payable for 1 year at the beginning of each week, assuming a nominal interest rate of 10.4% compounded weekly and 52 weeks in a year.
To find the present value of the series of payments, we need to calculate the present value of each payment individually and then sum them up.
The formula to calculate the present value of a single payment is:
PV = Payment / (1 + r)^n
Where:
PV = Present value of the payment
Payment = Amount of the payment (R100)
r = Nominal interest rate per period (10.4% / 52 = 0.2% or 0.002)
n = Number of periods until payment is received (in weeks)
Since the payments are made at the beginning of each week for 1 year, there are 52 payments to consider.
Now, we calculate the present value of each payment:
PV1 = R100 / (1 + 0.002)^1 = R100 / 1.002 = R99.60
PV2 = R100 / (1 + 0.002)^2 = R100 / 1.004 = R99.60
...
PV52 = R100 / (1 + 0.002)^52 = R100 / 1.108 = R90.23
Now, sum up all the present values:
PV_total = PV1 + PV2 + ... + PV52 = R99.60 + R99.60 + ... + R90.23 = R4998.14
Therefore, the present value of the series of payments of R100 each payable for 1 year at the beginning of each week, assuming a nominal interest rate of 10.4% compounded weekly, is R4998.14.