【正文】 ? Another TVM formula that can be found in a spreadsheet is the payment formula – PMT(rate,nper,pv,fv) – The same sign convention holds as for the PV and FV formulas ? Click on the Excel icon for an example 6F27 Finding the Number of Payments – Example ? Start with the equation, and remember your logs. – 1,000 = 20(1 – 1/) / .015 – .75 = 1 – 1 / – 1 / = .25 – 1 / .25 = – t = ln(1/.25) / ln() = months = years ? And this is only if you don’t charge anything more on the card! 6F28 Finding the Number of Payments – Another Example ? Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $. How long before you pay off the loan? – 2,000 = (1 – 1/) / .05 – .136161869 = 1 – 1/ – 1/ = .863838131 – = – t = ln() / ln() = 3 years 6F29 Finding the Rate ? Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate? – Sign convention matters!!! – 60 N – 10,000 PV – PMT – CPT I/Y = .75% 6F30 Annuity – Finding the Rate Without a Financial Calculator ? Trial and Error Process – Choose an interest rate and pute the PV of the payments based on this rate – Compare the puted PV with the actual loan amount – If the puted PV loan amount, then the interest rate is too low – If the puted PV loan amount, then the interest rate is too high – Adjust the rate and repeat the process until the puted PV and the loan amount are equal 6F31 Quick Quiz – Part III ? You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn % per month? ? What monthly rate would you need to earn if you only have $200,000 to deposit? ? Suppose you have $200,000 to deposit and can earn % per month. – How many months could you receive the $5,000 payment? – How much could you receive every month for 5 years? 6F32 Future Values for Annuities ? Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is %, how much will you have in 40 years? – FV = 2,000( – 1)/.075 = 454, 6F33 Annuity Due ? You are saving for a new house, and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? – FV = 10,000[( – 1) / .08]() = 35, 6F34 Annuity Due Timeline 0 1 2 3 10000 10000 10000 32,464 35, 6F35 Perpetuity – Example ? Perpetuity formula: PV = C / r ? Current required return: – 40 = 1 / r – r = .025 or % per quarter ? Dividend for new preferred: – 100 = C / .025 – C = per quarter 6F36 Quick Quiz – Part IV ? You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? ? What if the first payment is made today? ? You are considering preferred stock that pays a quarterly dividend of $. If your desired return is 3% per quarter, how much would you be willing to pay? 6F37 Work the Web Example ? Another online financial calculator can be found at MoneyChimp ? Click on the web surfer and work the following example – Choose calculator and then annuity – You just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years? – Money chimp assumes annuity due!!! – Payment = $313, 6F38 Table 6F39 Growing Annuity A growing stream of cash flows with a fixed maturity ttrgCrgCrCPV)1()1()1()1()1(12 ??????????????????????????????????trggrCPV)1()1(16F40 Growing Annuity: Example A definedbenefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by threepercent each year. What is the present value at retirement if the discount rate is 10 percent? ,265$1.000,20$40??????????????????PV6F41