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54

Month Principal Interest Balance

119.75

120.75

121.75

122.77

123.79

124.82

125.86

126.91

127.97

129.04

130.11

131.20

132.29

133,39

134.50

135.62

136.75

137,89

139.04

140.20

141.37

142.55

143.74

144.93

146.14

147.36

148.59

149.83

151.07

152.33

153.60

154.88

156.17

157.48

158.79

160.11

161.44

162.79

164.15

165.51

166.89

168.28

169.69

171.10

172.53

173.96

175.41

176.88

178.35

179.84

181.34

182.85

184.37

185.91

187.46

189.02

190.59

192.18

193.78

195.40

757.82

756.82

755.82

754.80

753.78

752.75

751.71

750.66

749.60

748.53

747.46

746.37

745.28

744.18

743.07

741.95

740.82

739.68

738.53

737.37

736.20

735.02

733.83

732.64

731.43

730.21

728.98

727.74

726.50

725.24

723.97

722.69

721.40

720.09

718.78

717.46

716.13

714.78

713.42

712.06

710.68

709.29

707.88

706.47

705.04

703.61

702.16

700.69

699.22

697.73

696.23

694.72

693.20

691.66

690.11

688.55

686.98

685.39

683.79

682.17

90,818.60

90,697.85

90,576.10

90,453.33

90,329.54

90,204.72

90,078.86

89,951.95

89,823.98

89,694.94

89,564.83

89,433.63

89,301.34

89,167.95

89,033.45

88,897.83

88,761.08

88,623.19

88,484.15

88,343.95

88,202.58

88,060.03

87,916.29

87,771.36

87,625.22

87,477.86

87,329.27

87,179.44

87,028.37

86,876.04

86,722.44

86,567.56

86,411.39

86,253.91

86,095.12

85,935.01

85,773.57

85,610.78

85,446.63

85,281.12

85,114.23

84,945.95

84,776.26

84,605.16

84,432.63

84,258.67

84,083.26

83,906.38

83,728.03

83,548.19

83,366.85

83,184.00

82,999.63

82,813.72

82,626.26

82,437.24

82,246.65

82,054.47

81,860.69

81,665.29

Month

Principal

Interest

Balance

197.03

680.54

81,468.26

198.67

678.90

81,269.59

200.32

677.25

81,069.27

201.99

675.58

80,867.28

203.68

673.89

80,663.60

205.37

672.20

80,458.23

207.08

670.49

80,251.15

208.81

668.76

80,042.34

210.55

667.02

79,831.79

212.31

665.26

79,619.48

214.07

663.50

79,405.41

215.86

661.71

79,189.55

217.66

659.91

78,971.89

219.47

658.10

221.30

656.27

78,531.12

223.14

654.43

78,307.98

225.00

652.57

78,082.98

226.88

650.69

77,856.10

228.77

648.80

77,627.33

230.68

646.89

77,396.65

232.60

644.97

77,164.05

234.54

643.03

76,929.51

236.49

641.08

76,693.02

238.46

639.11

76,454.56

240.45

637.12

76,214.11

242.45

635.12

75,971.66

244.47

633.10

75,727.19

246.51

631.06

75,480.68

248.56

629.01

75,232.12

250.64

626.93

74,981.48

252.72

624.85

74,728.76

254.83

622.74

74,473.93

256.95

620.62

74,216.98

259.10

618.47

73,957.88

261.25

616.32

73,696.63

263.43

614.14

73,433.20

265.63

611.94

73,167.57

267.84

609.73

72,899.73

270.07

607.50

72,629.66

272.32

605.25

72,357.34

274.59

602.98

72,082.75

276.88

600.69

71,805.87

279.19

598.38

71,526.68

281.51

5%.06

71,245.17

283.86

593.71

70,961.31

286.23

591.34

70,675.08

288.61

588.96

70,386.47

291.02

586.55

70,095.45

293.44

584.13

69,802.01

295.89

581.68

69,506.12

298.35

579.22

69,207.77

300.84

576.73

68,906.93

303.35

574.22

68,603.58

305.87

571.70

68,297.71

308.42

569.15

67,989.29

310.99

566.58

67,678.30

313.58

563.99

67,364.72

316.20

561.37

67,048.52

318.83

558.74

66,729.69

321.49

556.08

66,408.20

Month

Principal

Interest

Balance

Month

Principal

Interest

Balance

.

553.40

66,084.03

533.36

344.21

40,772.20

326.87

550.70

65,757.16

537.80

339.77

40,234.40

329.59

547.98

65,427.57

542.28

335.29

39,692.12

332.34

545.23

65,095.23

546.80

330.77

39,145.32

335.11

542.46

64,760.12

551.36

326.21

38,593.96

337.90

539.67

64,422.22

555.95

321.62

38,038.01

340.72

536.85

64,081.50

560.59

316.98

37,477.42

343.56

534.01

63,737.94

565.26

312.31

36,912.16

346.42

531.15

63,391.52

569.97

307.60

36,342.19

349.31

528.26

63,042.21

574.72

302.85

35,767.47

352.22

525.35

62,689.99

579.51

298.06

35,187.96

355.15

522.42

62,334.84

584.34

293.23

34,603.62

358,11

519.46

61,976.73

589.21

288.36

34,014.41

361.10

516.47

61,615.63

594.12

283.45

33,420.29

364.11

513.46

61,251.52

599.07

278.50

32,821.22

367.14

510.43

60,884.38

604.06

273.51

32,217.16

370.20

507.37

60,514.18

609.09

268.48

31,608.07

373.29

504.28

60,140.89

614.17

263.40

30,993.90

376.40

501.17

59,764.49

619.29

258.28

30,374.61

379.53

498.04

59,384.96

624.45

253.12

29,750.16

382.70

494.87

59,002.26

629.65

247.92

29,120.51

385.88

491.69

58,616.38

634.90

242.67

28,485.61

389.10

488.47

58,227.28

640.19

237.38

27,845.42

392.34

485.23

57,834.94

645.52

232.05

27,199.90

395,61

481.96

57,439.33

650.90

226.67

26,549.00

398.91

478.66

57,040.42

656.33

221.24

25,892.67

402.23

475.34

56,638.19

661.80

215.77

25,230.87

405.59

471.98

56,232.60

667.31

210.26

24,563.56

408.96

468.61

55,823.64

672.87

204.70

23,890.69

412.37

465.20

55,411.27

678.48

199.09

23,212.21

415.81

461.76

54,995.46

684.13

193.44

22,528.08

419.27

458.30

54,576.19

689.84

187.73

21,838.24

422.77

454.80

54,153.42

695.58

181.99

21,142.66

426.29

451.28

53,727.13

701.38

176.19

20,441.28

429.84

447.73

53,297.29

707.23

170.34

19,734.05

433.43

444.14

52,863.86

713.12

164.45

19,020.93

437.04

440.53

52,426.82

719.06

158.51

18,301.87

440.68

436.89

51,986.14

725.05

152.52

17,576.82

444.35

433.22

51,541.79

731.10

146.47

16,845.72

448.06

429.51

51,093.73

737.19

140.38

16,108.53

451.79

425.78

50,641.94

743.33

134.24

15,365.20

455.55

422.02

50,186.39

749.53

128.04

14,615.67

459.35

418.22

49,727.04

755.77

121.80

13,859.90

463.18

414.39

49,263.86

762.07

115.50

13,097.83

467.04

410.53

48,796.82

768.42

109.15

12,329.41

470.93

406.64

48,325.89

774.82

102.75

11,554.59

474.85

402.72

47,851.04

781.28

96.29

10,773.31

478.81

398.76

47,372.23

787.79

89.78

9,985.52

482.80

394.77

46,889.43

794.36

83.21

9,191.16

486.82

390,75

46,402.61

800.98

76.59

8,390.18

490.88

386.69

45,911.73

807.65

69.92

7,582.53

494.97

382.60

45,416.76

814.38

63.19

6,768.15

499.10

378.47

44,917.66

821.17

56.40

5,946.98

503.26

374.31

44,414.40

828.01

49.56

5,118.97

507.45

370.12

43,906.95

834.91

42.66

4,284.06

511.68

365.89

43,395.27

841.87

35.70

3,442.19

515.94

361.63

42,879.33

848.89

28.68

2,593.30

520.24

357.33

42,359.09

855.96

21.61

1,737.34

524.58

352.99

41,834.51

863.09

14.48

874.25

528.95

348.62

41,305.56

870.28

7.29

3.97



It is desired to find the variance of the sum

J = X] + XiX2 -!-+ X1X2 . . x

where x,, Xj , . . . , x are independent and identically distributed random variables. Bodi formula (3.34) for - and formula (3.36) for are of diis form. Let nil " second moments about the origin, i.e.

£[x] = m, and E

= m2

for it = 1, 2, . .., n. From independence we have

Els] = y:

k = i

which we will denote by sinii) .

The formula which we wish to prove is

var[5] = -i. 5( 2) - -- s{mi) - \s{mi)Y

ni2 - nil "2 ~ ""l

The proof is by madiematical induction. Let n = 1. We know that

var[j] = var[X(] = - m,.

The right-hand side of die formula to be proven is

m2+ nil

2 ~ nil ~

Thus, the formula holds for n = 1.

/Wj - nti - ~ i .

Now assume that the formula holds for - 1. Define

I = Xi + X1X2 +

+ Xfy . ..x.

n-l-

We have

varM = var[/ + XjXj . . . x]

= var[/] + varxiX2 ...x] + 2cov[/, x,X2...x]

We need to evaluate diese three terms. From the induction assumption

varM = -i s i{m2) - -s-iimO - K-i("i)l

m2 - m2 - nti

"h + mi 2 , , "-n

m2 -(- /«2 +-)- 1

(m, + nu m2 -m, *

- \i + m\ +

From independence

var[x,X2

For the covariance term we have

+ m.

+ m 2

+1

"2 ~ 21

2 - nil (D)

- ,

(B) (C)

2cov[r, x,X2.. .x] = 2E[UiX2 . . .x] - 2£W£[x,X2 . . . x]

-1 , 2 -2 , 2 1 - OT2OT1 -

OT2OT1 - OTj 2 W2 - OTj

- 2 m,

, -t- -(-

-2(ot,

-I- , -I-

-I- .

2 - /Wj (F)

OT2 - OT[

- 2 "7 , + +

+

-I-

We need to show diat the sum of the diree terms gives us our induction hypothesis. The reader should verify diat (A) + (D) -(- (F) gives die fust term in die formula to be proven; (B) + (G) gives die second term; and (C) -I- (E) + (H) gives die diird term.

Thus, the proof by mathematical induction is complete.

Derivation of the variance of an annuity



Bibliography

1. Black, F. and Scholes, M. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy Vol. 81 (1973), 637-654.

2. Box, G.E.P. and Jenkins, G.M. Times Series Analysis (Second Edition), San Francisco: Holden-Day, 1976.

3. Boyle, P.P. "Rates of Return as Random Variables," Journal of Risk and Insurance Vol. 43 (1976), 693-713.

4. Boyle, P.P. "Immunization Under Stochastic Models of die Term Structure," Journal of the Institute of Actuaries Vol. 105 (1978), 177-187.

5. Brealey, R.A. and Myers, S.C. Principles of Corporate Finance (Third Edition), New York: McGraw-Hill, Inc. 1988.

6. Butcher, M.V. and Nesbitt, C.I. Mathematics of Compound Interest, Ann Arbor: Uhichs Books, Inc., 1971.

7. Butsic, R.P. "Determining die Proper Interest Rate for Loss Reserve Discounting: An Economic Approach," Casualty Actuarial Society Call Papers (1988), 147-188.

8. Clancy, R.P. "Options on Bonds and Applications to Produce Pricing," Transactions of the Society of Actuaries Vol. 37 (1985), 97-130.

9. Cox, J.C.; IngersoU, I.E.; and Ross, S.A. "A Theory of die Term Sdiicture of Interest Rates," Econometrica Vol. 53.2 (1985), 385-407.

10. Cox, J.C.; Ross, S.A.; and Rubinstein, M. "Option Pricing: A Simplified Approach," Journal of Financial Economics Vol. 7 (1979), 229-263.

11. DArcy, S.P. "Use of die CAPM to Discount Property-Liability Loss Reserves," Journal of Risk and Insurance Vol. 55 (1988), 481-491.

21. McCutcheon, J.J. and Scott, W.F. An Introduction to the Mathematics of Finance, Heinemann: London, 1986.

22. Milgrom, P.R. "Measuring die Interest Rate Risk, Transactions of the Society of Actuaries Vol. 37 (1985), 241-257.

23. Miller, R.B. and Wichem, D.W. Intermediate Business Statistics, New York: Holt, Rinehart and Winston, 1977.

24. Panjer, H.H. and Bellhouse, D.R. "Stochastic Modelling of Interest Rates Widi Applications to Life Contingencies," Journal ofRik and Insurance Vol. 47 (1980), 91-110.

25. Panjer, H.H. and Bellhouse, D.R. "Stochastic Nodelling of Interest Rates Widi Applications to Life Contingencies - Part ," Jounal of Risk and Insurance Vol. 48 (1981), 628-637.

12. Fen, A.M. "Interest Rate Futures: An Alternative to Traditional Immunization in die Financial Management of Guaranteed Investment Contracts," Transactions of the Society of Actuaries Vol. 37 (1985), 153 184.

13. Ferguson, R.E. "Duration," Proceedings of the Casualty Actuarial Society Vol. 70 (1983), 265-288.

14. Fisher, I. TJie Theory of Interest, 1930. Reprint. New York: Augustus M. Kelley, PubUshers, 1986.

15. Ibbotson, R.G. and Sinquefield, R.A. "Stocks, Bonds, Bills and Inflation," Ibbotson Associates Yearbook, 1986.

16. Jean, W.H. "On Multiple Rates of Return," Journal of Finance Vol. 23 (1968), 187-191.

17. Jetton, M.F. "Interest Rate Scenarios," Transactions of the Society of Actuaries Vol. 40 (1988), 423-437.

18. Kocherlakota, R.; Rosenbloom, E.S.; and Shiu, E.S.W. "Algoridims for Cash-Flow Matching," Transactions of the Society of Actuaries Vol. 40 (1988), 477-484.

19. Kolb, R.W. Options: An Introduction Miami: Kolb Publishing Co., 1991.

20. Macaulay, F.R. Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States Since 1856, New York: National Bureau of Economic Research, 1938.



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