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

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

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