Math 418.B                                               Lecture 3  

 

 

 

Arithmetic Payments

 

 

                                      P                            P+Q                    P+2Q   . . . .            P+(n-1)Q 

       |________________|_______________|________________|_________________|

       0                              1                             2                               3                     . .  .     n        

                                                                                                                           
                                    Special cases:                                                                                                      

 

 

Geometric Payments

 

                                      1                             1+k                       . . . .               

       |________________|_______________|________________|_________________|

       0                              1                             2                               3                     . .  .     n        

                                                                                                                           
                                                                                                                        

 

 

Continuous Payment

 

                  Rate of Payment at time t 

v  = 1/(1+ i )      Discounting factor

 

         Present Value of the annuity paid from time 0 to time n with constant  

                          interest rate

 

 

        Present Value of the annuity paid from time 0 to time n with

                                    variable force of interest

Examples:

 

  1. Find the present value of a perpetuity-immediate whose successive payments are 1, 2, 3, 4, . . . , at an effective interest rate of 6%.

 

 

 

 

 

 

            Ans. 294.44

  1. Find the present value of an annuity-immediate such that payments start at 1, increase by annual amounts of 1 to a payment of 10, and then decrease by annual amounts of 1 to a final payment of 1. Annual effective rate is 6%.

 

 

Ans. 57.42

  1. Find the expression in terms of annuities for your answer in 2.

 

 

 

 

 

 

 

 

 

 

 

  1. Find the present value of an annuity-immediate such that payments start at 1, each payment thereafter increases by 1 until reaching 10, and then remain at that level until 25 payments in total are made. Annual effective rate is 6%.

 

 

 

 

 

 

 

 

 

            Ans. 91.20

  1. An immediate annuity provides for 20 annual payments. The first payment is $1000. The payments increase in such a way that each payment is 4% greater than the preceding payment. Find the present value of this annuity if the annual effective interest rate is 6%.

 

 

 

 

 

 

 

            Ans. 15839.87

  1. Find the present value of a perpetuity which pays 1 at the end of the third year, 2 at the end of the sixth year, 3 at the end of the ninth year, …, etc., if the annual effective interest rate is 6%.

 

 

 

 

 

            Ans. 32.64

 

 

  1. Find the accumulated value at the end of 10 years of an annuity in which payments are made at the beginning of each half-year for five years. The first payment is $2000, and each payment is 90% of the prior payment. Interest is credited at 10% convertible quarterly.

 

 

 

 

 

 

 

 

 

 

      Ans. 29474.59

 

 

 

  1. Express (with an integral) the present value of a continuously increasing annuity with a term of n years if the force of interest is and if the rate of payment at time t is per annum.