Question

A 15-year annuity of thirty $10,000 semiannual payments will begin 11 years from now, with the first payment coming 11.5 years from now.

Required : (a) If the discount rate is 10 percent compounded monthly, what is the value of this annuity 6 years from now?

(b) What is the current value of the annuity?

Answer 1

Interest rate is 10% per year compounded monthly, however we need to find the effective interest rate semi annually.

Annual interest = 10%

Semi annual interest rate = 5%

Effective interest rate = (1 + 0.05/6) ^6 - 1

= (1 + 0.0083333333) ^6 - 1

= (1.0083333333) ^6 - 1

= 1.05105 - 1

= 0.0511

So annual effective interest rate is 5.11% per year

PV of the annuity at the beginning of 11^th year.

PV of annuity = P* [1- (1+ r) ^-n]/ r

Where,

              Periodic deposit (P) = $10000

              Interest rate r = 5.11%

              Time (n) = 30

Let's put all the values in the formula to find PV o annuity

= 10000* [1- (1+ 0.0511)^-30]/ 0.0511

= 10000* [1- (1.0511)^-30]/ 0.0511

= 10000* [1- 0.224222359]/ 0.0511

= 10000* [0.775777641/ 0.0511]

= 10000* [15.1815585322896]

= 151815.59

So Present value of annuity is $151815.59

A.

We need to find the value of the annuity 6 years from now, first we need to calculate annual effective interest rate

Effective interest rate = (1 + i/m) ^m -1

Where,

Nominal interest rate (i) = 10% per year

Number of compounding in a year (m) = 12

Let's put all the values in the formula

Effective interest rate = (1 + 0.1/12) ^12 - 1

= (1 + 0.0083333333) ^12 - 1

= (1.0083333333) ^12 - 1

= 1.10471 - 1

= 0.1047

So annual effective interest rate is 10.47% per year

Value of annuity at the end of year = 151815.59/ (1 + 0.1047)^5

                                                             = 151815.59/ 1.6452

                                                             = 92277.89

B.

Current value of annuity = 151815.59/ (1 + 0.1047)^11

                                               = 151815.59/ 2.9901

                                                     = 50772.75

PLs note intermediate calculations are round off to 4 digits.

Final result may vary.