108156.pdf

270 Mathematics for Finance

The conditiong<rmust be satisfied or otherwise the series will be divergent.

Using this formula and the tail-cutting procedure, we can find that for a stream

ofnpayments

C

r−g−

C(1 +g)n

r−g

1

(1 +r)n=C

1 −

(1+g

1+r

)n

r−g.

2.18The timetit will take to earn $1 in interest satisfies

e^0.^1 t× 1 , 000 , 000 ∼= 1 , 000 , 001.

This givest∼= 0 .00001 years, that is, 315.36 seconds.

2.19a) The value in year 2000 of the sum of $24 for which Manhattan was bought

in 1626 would be

24e(2000−1626)×^0.^05 ∼= 3 , 173 , 350 , 575

dollars, assuming continuous compounding at 5%.

b) The same amount compounded at 5% annually would be worth

24(1 + 0.05)^2000 −^1626 ∼= 2 , 018 , 408 , 628

dollars in year 2000.

2.20$100 deposited at 10% compounded continuously will become

100e^0.^1 ∼= 110. 52

dollars after one year. The same amount deposited at 10% compounded

monthly will become

100

(

1+^012.^1

) 12

∼= 110. 47

dollars. The difference is about $0. 05.

If the difference is to be less than $0.01, the compounding frequencym

should satisfy

100

(

1+^0 m.^1

)m

> 110. 51.

This means thatmshould be greater than about 55.19.

2.21The present value is

1 , 000 ,000e−^20 ×^0.^06 ∼= 301 , 194

dollars.

2.22The ratersatisfies

950e^0.^5 r=1, 000.

It follows that

r= 01. 5 ln^1 , 950000 ∼= 0. 1026 ,

that is, about 10.26%.

2.23The interest rate is

r= 20 /.^0312 =0. 18 ,

that is, 18%.