Future value of an annuity

Transcription

1 Announcements The secon hour-exam wll be hel on Fray, July 12. The use of cell phones an other wreless evces s not permtte on the exam. You wll nee to brng a separate calculator for the exam. Sharng of calculators s not permtte urng the exam.

2 Future value of an annuty A = accumulate (future) value = unform epost per compounng pero r = nomnal annual nterest rate t = number of years = r/n = nterest rate per compounng pero nt = number of compounng peros A 1 nt 1

3 Annuty Example An nvual saves $100 per month, eposte rectly nto her cret unon account on payay, the last ay of the month. The account earns 6% per year, compoune monthly. How much wll she have at the en of 5 years, assumng that the cret unon contnuous to pay the same nterest rate? A Answer: (1 A ) nt $ = $100 = 0.06/12 = nt = 12 5 = $

4 Example Suppose that a couple begns savng a regular amount of per month after ther chl s born an want to have $100,000 avalable for college when the chl turns 18. How much o they have to save each month, f ther account earns 6.5% nterest per year, compoune monthly? Soluton: The monthly rate s 0.065/ A = $100,000, , an nt = = 216. $100,000 1 nt ( ) $100,000 ( ) $100, $244.89

5 Conventonal Installment Loan Interest s compoune. P = prncpal (amount borrowe) r = nomnal annual rate of nterest t = number of years n = number of nstallments per year = r/n = nterest rate per compounng pero nt = number of nstallments What payment shoul be mae at the en of each compounng pero n orer to pay off the loan an any accumulate nterest n t years?

6 Conventonal Loan Payments How much prncpal an nterest woul be ue f all were pa at the en of t years? A = P(1 + ) nt Usng the savngs plan, how much of a epost,, shoul one save n tmes per year to accumulate A at the en of t years? A 1 nt 1

7 Amortzaton Formula P = prncpal (amount borrowe) = r/n = nterest rate per compounng pero nt = number of nstallments = payment mae at en of each pero P 1 nt nt 1 1 P 1 nt 1

8 Conventonal Loan Example Suppose that $9,000 s borrowe for 5 years at 6% APR. What s the monthly payment for a conventonal loan? nt nt (1 ) 1 P(1 ) 9, , ,

9 Stuent Loan Example Suppose a stuent borrows $10,000 at an nterest rate of 6.9% compoune monthly. The loan accrues nterest for two years before repayment begns. If the borrower has ten years to repay the accrue balance, what are the monthly payments? After two years, the accrue balance s A 10, , , nt nt (1 ) 1 P(1 ) $11, $ 22, $ 22, / $132.65

10 Example: Home Mortgage The prce of a home s $195,000. The bank requres a 10% own payment an two ponts at the tme of closng. The cost of the home s fnance wth a 30-year fxe rate mortgage at 7.5%. The requre own payment s 10% of $195,000: 0.10 $195,000 = $19,500. The amount of the mortgage s the fference between the prce of the home an the own payment: $195,000 $19,500 = $175,500

11 Example contnue: ponts Each pont s a one-tme charge that equals 1% of the loan amount. The cost of two ponts on a mortgage of $175,500 s 0.02 $175,500 = $3510. The own payment ($19,500) s pa to the seller an the cost of two ponts ($3510) s pa to the lenng nsttuton.

12 Example contnue: monthly payment We nee to fn the monthly mortgage payment for $175,500 at 7.5% for 30 years. We use the loan payment formula for nstallment loans. $175, $ The monthly mortgage payment for prncpal an nterest s approxmately $ In 30 years ths totals $441, n payments. Of ths, $441, ,500 = $266,263 s nterest.

13 30-Year Mortgage Suppose that a famly buys a $120,000 house wth $0 own payment an a 30-year fxe-rate mortgage wth 6% APR an monthly payments. The monthly payment s $ What proporton of each payment s nterest? In the frst month, the nterest s 0.5% of $120,000 = $600 or about 83% of the frst payment. In the last month, the nterest s a lttle less than 0.5% of the last payment or $3.58.

14 Amortzaton scheule (monthly payment = $719.46) Month Interest Prncpal Balance 120, , , , , , startng balance ,000 = = after 15 years pa off half Last payment s a lttle more ue to rounng.

15 Payng off the loan The loan s pa off slowly at frst snce nterest s a large porton of the ntal payments. After half of the term of a mortgage has passe, the borrower has pa off less than half of the loan (n our example, after 15 years, the prncpal s only reuce by $34, or about 30% of $120,000). It mght take 2/3 of the term of the mortgage to pay off 50% of the loan (n our example, t takes about 21 of 30 years to pay off $60,000).

16 30-Year vs. 15-Year Mortgage Suppose that a famly buys a $120,000 house wth $0 own payment an a 30-year fxe-rate mortgage wth 6% APR an monthly payments. What s the monthly payment? $ What woul the monthly payment be at the same 6% APR for a 15-year fxe-rate mortgage? $ In both cases, $600 of the frst payment s nterest.