1. A perpetuity is an annuity whose term is infinite (i.e., an annuity whose payments continue forever). The present value of a perpetuity-immediate is a ––  | = v + v 2 + v 3 + … = 1 v —— = 1 – v v — = iv 1 — i Similarly, the present value of a perpetuity-due is.. a ––  | = 1 + v + v 2 + … = 1 —— = 1 – v 1 — = iv 1 — d Note that since perpetuities continue forever, there is no such thing as an accumulated value for a perpetuity. a – n| Note that can be interpreted as the difference between payments for two perpetuities each paying 1 at the end of each period; the first one has present value, and the second one is deferred n periods so its present value is 1 – v n = —— i Sections 3.5, 3.9 1/i vn/i.vn/i.

2. An investment of $1,000,000 is to be made where a 5% effective interest rate is assumed. Payments of $50000 each year will begin in exactly one year. The first 10 payments go to Abernathy, the next 20 payments go to Barnabas, and continuing payments thereafter go to Charity. Find the lump sums (adding to $1,000,000 of course) that could be immediately paid to Abernathy, Barnabas, and Charity, which would be equivalent to the perpetuity. The present value of the payments to Abernathy is a –– 10 | 0.05 50000 =50000(7.7217) =$386,085 The present value of the payments to Barnabas is a –– 30 | 0.05 50000 =– a –– 10 | 0.05 50000(15.3725 – 7.7217) =$382,540 The present value of the payments to Charity is a ––  | 0.05 50000 =– a –– 30 | 0.05 50000(1/0.05 – 15.3725) =$231,375

3. We shall next consider situations in which interest can vary each period, but compound interest is still in effect. Let i k denote the rate of interest applicable from time k – 1 to time k. If i k is applicable only for period k regardless of when the payment is made, then a – n| =(1 + i 1 ) –1 +(1 + i 1 ) –1 (1 + i 2 ) –1 + …+ (1 + i 1 ) –1 (1 + i 2 ) –1 …(1 + i n ) –1 If i k is applicable for the payment made at time k over all k periods, then a – n| =(1 + i 1 ) –1 +(1 + i 2 ) –2 + …+ (1 + i n ) –n If i k is applicable only for period k regardless of when the payment is made, then.. s – n| =(1 + i 1 )(1 + i 2 )…(1 + i n ) +(1 + i 2 )(1 + i 3 )…(1 + i n ) + …+ (1 + i n ) If i k is applicable for the payment made at time k over all k periods, then.. s – n| =(1 + i 1 ) n +(1 + i 2 ) n–1 + …+ (1 + i n )

4. Find the accumulated value of a 12-year annuity-immediate of $500 per year, if the effective rate of interest (for all money) is 8% for the first 3 years, 6% for the following 5 years, and 4% for the last 4 years. The accumulated value of the first 3 payments to the end of year 3 is 500 s – 3 | 0.08 =500(3.2464) =$1623.20 The accumulated value of the first 3 payments to the end of year 8 at 6% and then to the end of year 12 at 4% is 1623.20(1.06) 5 (1.04) 4 =$2541.18 The accumulated value of payments 4, 5, 6, 7, and 8 to the end of year 8 is 500 s – 5 | 0.06 =500(5.6371) =$2818.55 The accumulated value of payments 4, 5, 6, 7, and 8 to the end of year 12 at 4% is 2818.55(1.04) 4 =$3297.30

5. Find the accumulated value of a 12-year annuity-immediate of $500 per year, if the effective rate of interest (for all money) is 8% for the first 3 years, 6% for the following 5 years, and 4% for the last 4 years. The accumulated value of payments 9, 10, 11, and 12 to the end of year 12 is 500 s – 4 | 0.04 =500(4.2465) =$2123.25 The accumulated value of the 12-year annuity immediate is $2541.18 +$3297.30 +$2123.25 =$7961.73

6. Find the accumulated value of a 12-year annuity-immediate of $500 per year, where the first 3 payments are invested at an effective rate of interest of 8%, the following 5 payments are invested at an effective rate of interest of 6%, and the last 4 payments are invested at an effective rate of interest of 4%. The accumulated value of the first 3 payments to the end of 3 years is 500 s – 3 | 0.08 =500(3.2464) =$1623.20 The accumulated value of the first 3 payments to the end of year 12 at 8% is 1623.20(1.08) 9 =$ 3244.78 The accumulated value of payments 4, 5, 6, 7, and 8 to the end of year 8 is 500 s – 5 | 0.06 =500(5.6371) =$2818.55 The accumulated value of payments 4, 5, 6, 7, and 8 to the end of year 12 at 6% is 2818.55(1.06) 4 =$3557.66

7. Find the accumulated value of a 12-year annuity-immediate of $500 per year, where the first 3 payments are invested at an effective rate of interest of 8%, the following 5 payments are invested at an effective rate of interest of 6%, and the last 4 payments are invested at an effective rate of interest of 4%. The accumulated value of payments 9, 10, 11, and 12 to the end of year 12 is 500 s – 4 | 0.04 =500(4.2465) =$2123.25 The accumulated value of the 12-year annuity immediate is $3244.78 +$ 3557.66 +$2123.25 =$8925.69