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8.5 Annuities

8.5 Annuities. (regular withdrawal and finding time). 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.

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8.5 Annuities

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  1. 8.5 Annuities (regular withdrawal and finding time)

  2. 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

  3. 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.

  4. b) How much interest will he have paid over the termof 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.

  5. 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.

  6. Book P.522 (18)—Finding time Kyla must repay student loans that total $ 17,000. She can afford to make $325 monthly payments. The bank is charging an interest rate of 7.2% a compounded monthly. How long will it take Kyla to repay her loans? Solution: i = 0.072/12= 0.006 R=$ 325 per month PV= $ 17,000 PV=R × (1－(1+ i)^-n ) /i 17000= 325 × (1-(1+0.006)^ -n ) /0.006 ( 1-( 1.006)^-n )/0.006 ≈ 52.308 1- (1.006)^-n= 0.313848 (1.006)^-n=0.686152 -n㏒1.006= ㏒0.686152 -n ≈ -62.964 n ≈ 62.964 ∵ n=years × 12 ∴years= n/12= 62.964/12 = 5.247 0.247×12 =2.964 ≈3months ∴ It take Kyla to repay her loans about 5 years and 3 months .

  7. Thank you for listening! Take a look at your handout for solution!

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