St 226 Week 1 Homework Solution

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1. Force of interest 0.05 100 exp Effective rate of return 0.05 100 (1.05) 2 = 102.47 2. The value is 6 6
1

1 × 0.05 2

= 102.53

exp
0.5

δ (s) ds

= exp
0.5

(0.01s + 0.04) ds
t=6

=

exp

0.01

t2 + 0.04t 2

= 1.49.
t=0.5

3. 150 in 3 years accumulates to
3

150 exp
0

0.05 + 0.02s2 ds

= 150 exp

0.02

t3 + 0.05t 3

t=3

= 208.65
t=0

200 in 5 years accumulates to
3 5

200 exp
0

0.05 + 0.02s2 ds +
3 t=3

0.05 + 0.01s2 ds t3 + 0.05t 3
t=5

=

200 exp

0.02

t3 + 0.05t 3

exp
t=0

0.01

= 426.23
t=3

250 in 8 years accumulates to
3 8

250 exp
0

0.05 + 0.02s2 ds +
3 t=3

0.05 + 0.01s2 ds t3 0.01 + 0.05t 3
t=8

=

250 exp

t3 0.02 + 0.05t 3

exp
t=0

= 2248.73
t=3

To find the effective rate earned we have to solve 600 = 208.65 (1 + i) −3

+ 426.23 (1 + i)

−5

+ 2248.73 (1 + i)

−8

Trying various values such as i = 0.1 and i = 0.2, we have LHS¡RHS. With i = 0.25, RHS=620 and with i = 0.26, RHS=596, so it is between 0.25 and 0.26 and closer to 0.26. Further investigation shows it is 0.258, i.e. 25.8% 4. Same exercise

1

5. We start with the present value
6 s 2 s

ρ (s) exp −
0 5 0

δ (u) du ds =
1 s

12 exp −
0 6

δ (u) du ds +
s

(10 + s) exp −
2 0 2

δ (u) du ds +
5 s

15 exp −
0

δ (u) du ds =

12 exp −
1 5 2 0

0.08 du ds +
s

(10 + s) exp −
2 6 2 0

0.08 du −
2 5

du 10 + u
s

ds +

15 exp −
5 2 0

0.08 du −
2 5

du − 10 + u

0.06 du ds =
5

12 exp (−0.08s) ds +
1 6 2

(10 + s) exp (−0.08 × 2) exp − ln 15 12
5

10 + s 12

ds +

15 exp (−0.08 × 2) exp − ln
5

exp (−0.06 (s − 5)) ds = 12 ds + 10 + s

12

exp (−0.08) − exp (−0.16) + 0.08
6

(10 + s) exp (−0.16)
2

12 exp (−0.16) exp (−0.06 (s − 5)) ds =
5

1 − exp (−0.06) exp (−0.08) − exp (−0.16) +12 (5 − 2) exp (−0.16)+12 exp (−0.16) = 51.25 0.08 0.06 The accumulated value is 12 2 5

51.25 exp
0

0.08 du +
2

du + 10 + u
10

6

0.06 du
5

= 51.25 exp (0.16)

15 exp (0.06) = 79.83 12

6. 1000

(1.0381)
t=1 10

t−1

(1.09)
−t

−t

=

1000 1.0381 1000 1.0381
10

t=1

1.09 1.0381

=

(1.05)
t=1

−t

=

1000 at 0.05 1000 a = 7.72173 = 7438.33. 1.0381 10| 1.0381

2

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