Mathematics - The Mathematics of Finance Readings: Text...

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Unformatted text preview: The Mathematics of Finance Readings: Text Chapter 9. OVERVIEW Present Value and Future Value of a Single Cash Flow. Present Value and Future Value of Multiple Cash Flows. Conventions for Reporting Interest Rates. Using Your Financial Calculator. 1 Why "Mathematics" Matters Characteristics of financial securities: $ today in exchange for future cash flow promises. Control rights in case cash flows fail to materialize. Examples: Corporations issuing stocks. Corporations and Government issuing bonds. Individuals receiving bank/finance company loan What are these "promises" worth? Payments occur in the future. Payments may be risky. 2 Future Value & Compounding As of 11/25/07, Wachovia offered the following fixed-rate CD's. Assume you have $100,000 to invest for one year. What will your money grow to? 3 Future Value & Compounding Wachovia's one-year jumbo CD promises a return of 3.60% annually. Future Value = $100,000 * (1 + k ) FV = $100,000 * (1.036 ) = $103,600 4 Future Value & Compounding (II) Wachovia's 5 year CD (not listed) promises 3.69% annually C 100,000 C(1+k) 103,690 C(1+k)2 107,516 C(1+k)3 111,484 C(1+k)4 115,597 C(1+k)5 119,863 0 FV = $100,000 * (1 + k ) 5 1 2 3 5 4 5 = $100,000 * (1.0369) = $119,862.79 5 Compound Interest 6 The Power of Compounding HOUSTON 6/20/2005 - A retired public school teacher who was so frugal that he bought expired meat and secondhand clothing left $2.1 million for his alma mater, Prairie View A&M -- the school's largest gift from a single donor. Whitlowe R. Green, 88, died of cancer in 2002. He retired in 1983 from the Houston Independent School District, where he was making $28,000 a year as an economics teacher. His donation shocked family members and friends alike. 7 Present Value Example: You need to set aside enough to make a tuition payment of $7,000 one year from now. How much must you set aside if the money is placed in a savings account at 3% per year? X * (1.03) = 7,000 X = 7,000 / 1.03 = 6,796.12 8 Present Value (II) Assume that your parents are making the same calculation for your sibling's (who is just starting high school) first tuition payment. X * (1.03)4 = 7,000 X = 7,000 / (1.03)4 = 6,219.41 9 Financial Calculators Example: Calculate the PV of $100,000 to be paid 10 years from now if the interest rate is 5.47% per year. N I/Y PV PMT FV 100,000 FV 10 N 5.47 I/Y 0 PMT CPT PV = $58,709.79 10 Financial Calculators Example: Calculate the FV of $10,000 invested at 6.33% per year for 10 years. 10,000 PV 10 N 6.33 I/Y 0 PMT CPT FV = $18,473.88 11 Solving For k Or n Example: Assume you have $1,200 to invest and want to know how long it will take for that sum to grow to $3,000 if invested in a CD at 7%: Algebraic Solution. 1,200 * (1.07)n = 3,000 (1.07)n = 30/12 n*ln(1.07) = ln(30/12) n = ln(30/12) / ln(1.07) = 13.54 12 Solving For k Or n (II) Example: Assume you have $1,200 to invest and want to know how long it will take for that sum to grow to $3,000 if invested in a CD at 7%: Calculator Solution -1,200 PV 7.0 I/Y 0 PMT 3,000 FV CPT N = 13.54 13 Solving For k Or n (III) Example: Assume that you have $1,200 and want to know what the interest rate would have to be in order for you to have $3,000 in 10 years. Algebraic Solution. 1,200 * (1+k)10 = 3,000 (1+k)10 = 30/12 (1+k) = (30/12)1/10 k = .09596 14 Solving For k Or n (IV) Example: Assume that you have $1,200 to invest and want to know what the interest rate would have to be in order for you to have $3,000 in 10 years. Calculator Solution -1,200 PV 10 N 0 PMT 3,000 FV CPT I/Y = .09596 15 PV & FV Of Multiple Cash Flows Value Additivity: The PV (FV) of a stream of cash flows is the sum of the PV (FV) of individual cash flow. Example: You have told your sister, a high-school senior, to budget $400, $500, $600, $400 for books and educational supplies for her four years at college. What is the PV of these future cash flows if the appropriate interest rate is 7%? 400 500 600 400 + + + 1 2 3 (1.07 ) (1.07 ) (1.07 ) (1.07 ) 4 373.83 + 436.72 + 489.72 + 305.16 = 1,605.49 16 Short Cuts: Perpetuity Assume that you will receive a perpetual annual payment (Pmt) and that the appropriate rate to discount each cash flow (irrespective of how far in the future it is) is k. PV perpetuity Pmt = k Where does this formula come from? 17 PV of a Perpetuity Derivation Option 1, brute force: 6.25 6.25 6.25 PV = + + + ... 2 3 (1 + k) (1 + k) (1 + k) Option 2, finesse: 6.25 6.25 (1 + k ) * PV = 6.25 + + + ... 2 (1 + k) (1 + k) 6.25 6.25 6.25 PV = + + + ... 2 3 (1 + k) (1 + k) (1 + k) - k * PV = 6.25 6.25 PV = k 18 Perpetuity Example Example: Your parents are considering buying a beach house as a rental property. Similar cottages rent for approximately $2,000 per week. Taxes, insurance, and upkeep costs are about $800 per week. If the appropriate discount rate is 8%, what is the cottage worth? PV = Pmt / k (1,200 * 52) / .08 = 780,000 19 Short Cuts: Annuity Annuity: A fixed cashflow for a finite number of periods. The present value of the annuity can be calculated "long hand" using the concept of value additivity. PVannuity Pmt Pmt Pmt = + + ..... + 1 2 (1 + k ) (1 + k ) (1 + k ) n 1 1 1 PVannuity = PMT (1 + k ) 1 + (1 + k ) 2 + ..... + (1 + k ) n 1 1- n 1 (1 + k ) n = PMT * = j PVIFAk ,n = j =1 (1 + k ) k 20 PV of an Annuity Example Example: Your parents must set aside enough money to cover your sister's annual college tuition payments of $10,000. She will enter school next year and the money will be invested at 5%. How much must they set aside? PVIFAk ,n 1 1 - (1.05) 4 = 35,459.51 = 10,000 * .05 21 PV of an Annuity Example (II) $59.5 Million Winner Comes Forward ! Christopher Ewen of Seymour, Connecticut, won the $59,500,000 Powerball jackpot on June 25, 2005... Ewen's payment options: 30 annual payments of $1,983,333 Lump sum = present value of payments @ 4.25% 1 1 - (1.0425) 30 1,983,333 * .0425 = 1,983,333 *16.7790 = 33,278,344.41 22 FV of an Annuity If we are investing an annuity payment, we often want to calculate what it will grow to. FVannuity = Pmt (1 + k ) n -1 j =0 n -1 + ..... + Pmt (1 + k ) j 0 = Pmt (1 + k ) FVIFAk ,n (1 + k ) = k n -1 23 FV of an Annuity Example Assume that you have decided to put $3,000 per year for the next 40 years in an IRA invested in a mutual fund that you expect will return 8% per year on average. How much will you have at the 40 year point? (1.08) 40 - 1 3,000 * .08 = 3,000 * 259.0565 = 777,169.50 24 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)? 3k 3k 3k 3k 3k 3k 5k 5k 5k 5k 0 1 2 3 14 15 16 28 29 30 25 More Complex Example (1.11) 15 - 1 3,000 * = 103,216.08 .11 103,216.08 * (1.11) = 493,846.56 15 (1.11) 15 - 1 5,000 * = 172,026.79 .11 493,846.56 + 172,026.79 = 665,873.35 26 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 usually advertise this as a "6% APR" compounded monthly Annual Percentage Rate (APR) of 6% = .5% * 12 27 Annual Percentage Rates If you invest $100 for 1 year, what will be the future value of your account? $100 * (1.005)12 = $100 * 1.0617 = $106.17 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 28 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? .12 APYA = 1 + - 1 = .1262 6 .1204 APYB = 1 + - 1 = .1259 4 29 6 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!) 30 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? 10.9% monthly rate = = .9083 12 1 ( .009083) 72 = 16,995 .009083 1- PMT 16,995 PMT = = 322.61 52.6796 31 Amortized Loan (III) Alternatively they offer 48 month 9.9% APR financing. What would be your monthly payment? 9.9% monthly rate = = .825% 12 1- PMT 1 ( .00825) 48 = 16,995 .00825 16,995 PMT = = 430.22 59.503 32 Amortized Loan (IV) How much of the first and second payment in the 48 month loan is principal versus interest? Interest due: 16,995 * .00825 = 140.21 Principal paid: 430.22 - 140.21 = 290.01 Next Month Interest due: (16,995 290.01) * .00825 = 137.82 Principal paid: 430.22 137.82 = 292.40 33 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? 34 Annuity FV Example $20,000 (1.06 ) = $114,869.82 30 $1,000,000 - 114,869.82 = 885,130.18 (1.10 ) 30 - 1 X = 885,130.18 .10 X 164.494 = 885,130.18 X = 5,380.93 35 Example (II) Assume that the $1,000,000 total is placed in the stock market mutual fund at 10% when you retire. How much can you withdraw every year for the next 25 years ending with $100,000 in your account when you make the final withdrawal? 1 1 - (1.10 ) 25 = 1,000,000 X .10 X 9.077 = 1,000,000 X = 110,168.07 36 Example (III) 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? I/Y = 10% PV = 0 PMT = -4,000 FV = 1,000,000 CPT N = 37 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 38 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? 39 ...
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This note was uploaded on 02/14/2011 for the course FINA 363 taught by Professor Masoudie during the Fall '10 term at South Carolina.

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