1. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-1 Special Situations Chapter 12 McGraw-Hill Ryerson©

2. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-2 Calculate the… After completing this chapter, you will be able to: Learning Objectives Present Value of a perpetuity or deferred perpetuity LO 2. LO 1. Present Value and Future Value of an annuity whose payment size grows at a constant rate

3. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-3 Calculation of… Perpetuity or Deferred Perpetuity

4. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-4 Ordinary Perpetuities A perpetuity is an annuity whose payments continue forever. A $100,000 bequest is made to Seneca College to establish a perpetual bursary fund. If the college invests the funds to earn 6% compounded annually, the maximum amount that can be paid out on each anniversary of the bequest is … $100,000 * 0.06 = $6,000 If more than this was to be paid out, a loss of principal would result.

5. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-5 Ordinary Perpetuities P resent V alue of: If the payment interval equals the compounding interval, the perpetuity is an ordinary simple perpetuity it is an ordinary general annuity Otherwise… PV = PMT / i Formula

6. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-6 What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at the end of each month if interest is 6.0% compounded monthly in perpetuity? = 6000 / (.06/12) PV = PMT / i Formula = $1,200,000

7. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-7 What monthly compounded nominal rate of return must an endowment of $1 million earn to fully fund a perpetuity with an ongoing cost of $4,000 at the end of each month? PV = PMT/ i Formula Reorganize to find i i = 4000 / 1 000 000 = 0.004 i = PMT / PV = 0.4% per month The required nominal rate of return is: 12 * 0.4% = 4.8% compounded monthly

8. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-8 What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at the end of each month if interest is 6.0% compounded annually in perpetuity? Since this is a general perpetuity, we need to determine c and i 2 C = number of compoundings per year number of payments per year =.0833 = 1 12 i 2 = (1+i) c - 1 = (1.06) 0.0833 -1 = 0.00486755 = 6000 / 0.00486755 = $ 1,232,652.83 PV

9. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-9 Calculating initial endowment for a General Perpetuity

10. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-10 Since this is a general perpetuity, we need to determine c and i 2 C = number of compoundings per year number of payments per year =.1667 = 2 12 i 2 = (1+i) c - 1 = (1.02) 0.1667 -1 = 0.00330589 = 700 / 0.00486755 = $ 211,743.26 What amount must be placed in a perpetual fund today if it earns 4.0% compounded semi-annually and monthly payments of $700 in perpetuity are to start 1 month from now? PV

11. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-11 What amount must be placed in a perpetual fund today if it earns 4.0% compounded semi-annually and monthly payments of $700 in perpetuity are to start 1 YEAR from now? We have already determined the value at the beginning of the payments PV = $ 211,743.26 PV = FV(1 + i) - n Formula PV = 211743.26 (1 + 0.00330589) - 11 = $204,193.83 This is the value now This is the value 11 months from now

12. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-12 Constant Growth A nnuities LO 2.

13. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-13 Constant Growth Annuities … Annuities in which the payments change by the same percentage from one payment to another Let g = rate of growth in payment size between successive payments

14. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-14 FV = PMT (1+ i) n - (1+g) n [ i - g ] Formula PV = PMT 1- (1+g) n (1+ i) - n [ i - g ] Formula Constant Growth Annuities The following formulae will be used:

15. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-15 You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2,000 in the first year and increase the contribution by 4% every year thereafter. a)How much will you have in your RRSP at the time of your 20th contribution if the plan earns 7.5% compounded annually ? b)What will be the amount of your last contribution? Constant Growth Annuities Extract necessary data...

16. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-16 PMT = i = 0.075 n = 20$2000 PV = 0FV = ? g = 4% You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2000 in the first year and increase the contribution by 4% every year thereafter. Solve … FV = PMT (1+ i) n - (1+g) n [ i - g ] [ FV = 2000 (1.075) 20 - (1.04) 20 0.075 - 0.04 ] Solve … a)How much will you have in your RRSP at the time of your 20th contribution if the plan earns 7.5% compounded annually? …

17. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-17 1.04 0.035 20 1.075 2000 20 2.1911 4.24792.0567117,527.31 Solve … Amount in the RRSP at the time of the 20th contribution PMT = i = 0.075 n = 20$2000 PV = 0FV = ? g = 4%

18. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-18 You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2000 in the first year and increase the contribution by 4% every year thereafter. (b) What will be the amount of your last contribution? The final payment will be the Future Value of $2000 after 19 compoundings at 4% = 2000( 1+ 0.04) 19 = $ 4,213.70 FV = PV(1 + i ) n Formula

19. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-19 Constant Growth Annuities How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually? The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually. Solution

20. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-20 Constant Growth Annuities The cost will be the PV of the payments. Extract necessary data... How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually? The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually. PMT = i = 0.05/2 = 0.025 n = 50 $10000 PV = ? g = 3%/2 = 0.015 Solve … PV = PMT [ 1- (1+ g) n (1+ i) -n i - g ] [ PV= 10000 1- (1.015) 50 (1.025) -50.025 –. 015 ] Solve …

21. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-21 [ PV= 10000 1- (1.015) 50 (1.025) -50.025 –. 015 ] 0.2909 2.10520.6125-0.3875 1.025 0.01 50 1.015 10000 50 1 387,496.12 Cost of the annuity How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually? The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually.

22. Special Annuities Special Annuities1212 McGraw-Hill Ryerson© 12-22 This completes Chapter 12