A 4-year annuity of eight $11,400 semiannual payments will begin 9 years from no
ID: 2382899 • Letter: A
Question
A 4-year annuity of eight $11,400 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now.
If the discount rate is 11 percent compounded monthly, what is the current value of the annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
If the discount rate is 11 percent compounded monthly, what is the current value of the annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
Explanation / Answer
The problem is to be divided into two parts.
First, the present value of the amount of annuities payable at the end of each half year for 4 years (8 half years) 9 years from now. Also, as it is mentioned that the first annuity is payable 0.5 years after the completion of the 9 year period. Hence the formula applicable for this would be the calculation of Present Value Annuity Factor which is as follows
PVAF = {1-(1/(1+r)^n)}/r
This is derived from the following
An annuity consists of payment of a constant payment received at the end of each year. If P()) is the present value of the investment to receive the investment and r is the interest rate applicable for the period and C is the amount of annuity received at the end of each period, the flows can be calculated using the formula
P(0) = (C/(1+r))+(C/(1+r)^2)+(C/(1+r)^3)+...+(C/(1+r)^n)
This is reduced to P(0) = C*[ (1/1+r)+(1/1+r^2)+...+(1/1+r)^n)]
This can be written in the following notation
P(0) = C * PVAF(r,n) where PVAF(r,n) is equal to the second term in above equation
This can be reduced to PVAF(r,n) = {1-(1/(1+r)^n)}/r
The same factor was taken into account while calculating the present value in the below solution.
Part II of the solution involves taking this value and arriving at the present value at the end of 8th year as the amount needs to be invested at the beginning of the second period for receiving the annuity.
Part I of solution
In the given problem
Discount rate r = 11% compounded monthly
Effective monthly discounting factor r = (11/100)/12 = 0.00916667
Semi annual payment amount A = 11400
period of annuity n = 4 years = 8 semi annual payments of the annuity amount
Present value of the Annuity payments P(9) can be calculated using the formula
P(9) = A * PVAF(r,n)
PVAF factor is calculated as follows
PVFA(r,n) = {1-(1/(1+r)^n)}/r
substituting r, n values from above
PVAF(r,n) = {1-(1/(1+0.00916667)^8}/0.00916667
= {1-(1/(1.00916667)^8}/0.00916667
Value of 1.00916667^8 = 1.075729772
Substituting the above value
PVAF(r,n) = {1-(1/1.075729772)}/0.00916667 = {1- 0.92960149}/0.00916667
= 0.07039851 / 0.00916667
= 7.6798346618
Using this PVAF factor the present value P(9) can be calculated as follows
P(9) = A * PVAF(r,n) = 11400*7.6798346618
= 87550.115145
Part II of the solution
P(9) is the amount required at the beginning of 9th year to create an annuity which starts yielding an amount of USD 11400 semi-annually for the next 4 years at the given discount rate of 11%.
To get the value of P(9) at the beginning of 9th year (or end of 8th year) we need to find the present value P(0) to be invested now.
The Values for this calculation are as follows
Amount required at the beginning of 9th year = 87550.115145
monthly discount rate r = 0.00916667
Period = 8 years = 8*12 = 96 months
Present Value can be calculated using the formula
P(0) = P(9) / (1+r)^n
= 87550.115145 / (1+0.00916667)^96
= 87550.115145 / (1.00916667)^96
= 87550.115145 / 2.40125487
= 36,460.150998 or 36460.15 (rounded off)
Hence an amount of USD 36460.15 need to be invested now to receive an annuity of USD 11400 Semi-annually 9 years from now with the first semi-annual annuity payment starting 9.5 years