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Mathematics3 - More Complex Example Starting next year you...

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Unformatted text preview: More Complex Example Starting next year, you plan on investing $3,000 per year for 15 years. At year 16, you will begin investing $5,000 per year for an additional 15 years. If your investments return 11% per year, how much will you have on the date of your last payment (year 30)? 1 Non-Annual Compounding In practice, compounding (and discounting) is often done more frequently than annually. For comparability, however, rates are always reported on an annual basis. Example: Assume your bank savings account pays 0.5% compounded monthly. The bank will advertise this as a "6% APR" compounded monthly Annual Percentage Rate (APR) of 6% = .5% * 12 2 Annual Percentage Rates If you invest $100 for 1 year, what will be the future value of your account? APR's are not directly comparable if compounding periods vary. Often we convert to an Effective Interest Rate or Annual Percentage Yield. For an APR k, compounded m times per year: k APY = 1 + - 1 m 3 m APR & APY Example Example: You can invest in 1 of 2 accounts. Account A: APR of 12% compounded bi-monthly. Account B: APR of 12.04% compounded tri-monthly. Which account would you prefer? What is each account's APY? 4 APY & Amortized Loans Consumers loans use monthly payment periods, have interest rates reported as APR's and are examples of amortized loans (They enable you to buy things that you really can't afford!) 5 Amortized Loan (II) Example: Harley Haven's current list price on a V-Rod is $16,995. Financing is available: $0 down with 72 month 10.9% APR financing. What would be your monthly loan payment? 6 Amortized Loan Alternatively they offer 48 month 9.9% APR financing. What would be your monthly payment? 7 Amortized Loan How much of the first and second payment in the 48 month loan is principal versus interest? Interest due: Principal paid: Next Month Interest due: Principal paid: 8 Annuity FV Example You currently have $20,000 in the bank and have finally decided to get serious about saving for your retirement in 30 years. You have decided to invest the $20,000 in Treasury securities which you feel will generate a return on 6% per year. You also intend on investing additional money each year in a stock market mutual fund that you believe will return 10% per year. If you make 30 equal annual investments into the stock market mutual fund, how much will you need to invest each year in order to have a total of $1,000,000 in both accounts at the end of 30 years? 9 Rather than keep the 20,000 that you currently have invested in the Treasury Securities, you have decided to blow the $20,000 on a new car. For how many years would you now need to invest $4,000 every year at 10% in order to attain your goal of $1,000,000 when you retire? 10 Shortcut Summary Present Value Lump Sum Future Value 1 n (1 + k ) 1 1- (1 + k ) n k (1+ k ) (1 + k ) k n n Annuity -1 Perpetuity 1 k 11 PRACTICE PROBLEMS FOR START OF CLASS You are due $1,000 in 10 years. If the discount rate is 7%, what is the PV? You invest $1,000 today in an account returning 7% per year. What will the FV be in 15 years? Starting next year, you will owe $5,000 every year for the next 25 years. If the discount rate is 11%, what is the PV of these payments. What would the PV in the prior question be if payments started immediately? What would be the PV if, starting next year, you owed $5,000, every year, forever? (discount rate is 11%) Starting next year, a bond pays 30 annual coupon payments of $70 and a lump sum at maturity of $1,000. If the discount rate is 6.5%, what is the value of the bond? 12 ...
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