求名義利率 (數學英文題) 請詳解，please?

[匿名]

2015-12-25 3:15 am

At what nominal rate of interest would an annuity of $ 900 per quarter, paid at the end of each quarter, grow to a total of $ 60662.30 over a 10 year period?

更新1:

本書的答案是0.025，但我點都計唔到！

回答 (1)

2015-12-25 4:18 am

Let
nominal interest rate (per quarter) = i (in decimal),
R=(1+i)
P=quarterly payment=900
F=future value = 60662.30
n=number of periods
we have
F=P(R^n-1)/(R-1)
=>
60662.30=900(R^40-1)/(R-1)
We will denote
f(R)=900(R^40-1)/(R-1)-60662.30=0
=>
f(R)=9000R^40-606623R+597623
f'(R)=360000R^39-606623

and try to solve for R when f(R)=0
Solve by Newton's method:
R1=R0-f(R0)/f'(R0)...
Estimate R0=1.05 (5%)
R1=1.05-2401.8748/180708.741=1.0367
R2=1.0367-678.797/86153.685=1.028823
R3=1.028823-156.1187/48378.347=1.025596
R4=1.025596-20.69282/35804.731=1.0250181
R5=1.0250181-0.60947/33707.35=1.025000019
Say R=1.0250, or quarterly interest = 0.02500

Check:
FV=900*(1.025^40-1)/0.025=60662.30 ok.

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