how do i find the present value and evaluate the formula i h

how do i find the present value and evaluate the formula? i have the answer but cannot figure out how. thanks

Solution

Present value formula :

   The present value of an annuity is the amount that would have to be deposited in one lump sum today in order to produce exactly the same balance at the annuity’s maturity

Future value of an ordinary annuity is S = R( (1+i)ⁿ - 1 )/i ) and if P dollars are deposited today at the same compound interest rate i, then at the end of n payments, the amount in the account is P(1+i)ⁿ.

This amount must be the same as the future value of the annuity.

That is ,

  P(1+i)ⁿ = R( (1+i)ⁿ - 1 )/i )

P = [ R( (1+i)ⁿ - 1 )/i ) ] / ( 1+i )ⁿ

P = [ R( ( 1+i )ⁿ - 1 ) ] / i( 1+i )ⁿ

P = R [ ( ( 1+i )ⁿ - 1 ) / i( 1+i )ⁿ

   P = R [ 1 - ( (1+i)^-n / i ]

Therefore,

The present value P of an annuity of n payments of R dollars at the end of the consecutive interest periods with interest compounded at a rate of i per period is ,

P = R [ 1 - ( (1+i)^-n / i ]

From the given information

R = $150 , i = 8% compounded semiannually , n = 7 years compounded semiannually

i = 0.08/2 = 0.04 i.e n = 7x2 = 14

Hence,

Present value P = R [ 1 - ( (1 + i)^-n / i ) ]

= 150 [ 1 - ( ( 1 + 0.04)^-14 ) / (0.04) )

= 150 x 10.56312293

= $1584.468439

Therefore,

Present value = $1584.468439