You are given an annuity-immediate with 11 annual payments of 100 and a final balloon payment at the end of 12 years. At an annual effective interest rate of 3.5%, the present value at time 0 of all the payments is 1,000. Using an annual effective interest rate of 1%, calculate the present value at the beginning of the ninth year of all remaining payments.

The present value at the beginning of the ninth year of all remaining payments is 316.09

Explanation:

It is assumed that it is an ordinary annuity.

First, we need to calculate the final payment.

Use the following formula to calculate the final payment

Final Payment When an annuity is ordinary

Final payment = fv(rate,nper,pmt,pv) * (1+3.5%)

Where

rate = 3.5%

nper = 11

pmt = 100

pv = -1,000

Placing values in the formula

Final payment = fv(3.5%,11,100,-1000)*1.035

Final payment = $150.87

Now calculate the present value at the beginning of the ninth year of all remaining payments.

Present Value = ( Ninth payment / ( 1 + interest rate )^9 ) + ( Tenth payment / ( 1 + interest rate )^10 ) + ( Eleventh payment / ( 1 + interest rate )^11 ) + ( Final payment / ( 1 + interest rate )^11 )

Present Value = ( 100 / ( 1 + 3.5% )^9 ) + ( 100 / ( 1 + 3.5% )^10 ) + ( 100 / ( 1 + 3.5% )^11 ) + ( 150.87 / ( 1 + 3.5% )^11 )

Present Value = 73.37 + 70.89 + 68.49 + 103.34

Present Value = 316.09