Need to know:
Formula:
PV＝R × [ 1-(1+i)^-n ] /i
R＝PV× i÷[ 1－ (1+i) ^-n]
PV=the present value
R=payments(annuity)
i=interest rate per compounding period
n=the number of compounding periods
Example 3 & solution
Len borrowed $200 000 from the bank to purchase a yacht. If the
bank charges 6.6%/a compounded monthly, he will take 20 years
to pay off the loan.
a) How much will each monthly payment be?
Solution:
a) I=0.066/12=0.0055
n=20 × 12=240
PV=$200 000
PV=R ×(1-(1+i)^-n ) /i
200 000=R ×(1-(1+0.0055)^-240 ) /0.0055
200 000≈R ×133.072
200 000/133.072=R ×133.072/133.072
R≈1502.94
Len will have to pay $1502.94 per month for 20 years to pay off
the loan.
b) How much interest will he have paid over the term of the loan?
Solution:
b) A= 1502.94 ×240
= $360706.60
I= A-PV
= $ 360 706.60 - $ 200 000
= $ 160 706.60
Over the 20-year term of the loan, Len will have paid $ 160 706.60
in interest.
Book P521 (7)—Regular withdrawal
Emily is investing $128,000 at 7.8% a compounded monthly. She
wants to withdraw an equal amount from this investment each
month for the next 25 years as spending money. What is the most
she can take out each month?
Solution:
i＝0.078÷12= 0.0065
n=25×12= 300
PV=$128,000
R= PV× i ÷ [ 1－ (1+i) ^-n]
R= (128000 × 0.0065 )÷ [ 1- (1+0.0065)^-300]
R ≈ 971.03
So,the most she can take out is $971.03 each month.
Book P.522 (18)—Finding time
Kyla must repay student loans that total $ 17,000. She can afford to
make $325 monthly payments. The bank is charging an interest
rate of 7.2% a compounded monthly. How long will it take Kyla to
repay her loans?
Solution:
-n ≈ -62.964
i = 0.072/12= 0.006
n ≈ 62.964
R=$ 325 per month
∵ n=years × 12
PV= $ 17,000
∴years= n/12= 62.964/12
PV=R × (1－(1+ i)^-n ) / i
= 5.247
17000= 325 × (1-(1+0.006)^ -n ) /0.006
0.247×12 =2.964
≈3months
( 1-( 1.006)^-n )/0.006 ≈ 52.308
1- (1.006)^-n= 0.313848
∴ It take Kyla to repay her
(1.006)^-n=0.686152
loans about 5 years and
-n㏒1.006= ㏒ 0.686152
3 months .
Thank you for listening!
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