3-货币的时间价值10-3

nullChapter 3Chapter 3Time Value of MoneyLearning Objectives*Learning ObjectivesCompute the future value of an investment made today present value of cash to be received at some future date rate of return on an investment amount of time required for an investment to grow to a given value at a specified rate of interestTime Value of Money*Time Value of MoneyMotivation *Motivation Which one will you choose? Choice A Choice B1 HKD Paid Today1 HKD Paid one year laterIntroduction*Introduction$20 today is worth more than expectation of $20 next year because: a bank would pay interest on the $20 inflation makes next year’s $20 less valuable than today’s There may be uncertainty of receiving next year’s $20 Future Values: Example 1*Future Values: Example 1Suppose you invest $1000 for one year at 5% per year. How much will you receive in one year? Interest = $1,000(.05) = $50 Value in one year = principal + interest = $1,000 + $50 = $1050 Future Value (FV) = $1,000(1 + .05) = $1,050 Suppose you leave the money in for another year. How much will you have two years from now? FV = $1,000(1.05)(1.05) = $1,000(1.05)2 = $1,102.50Generalizing the method*Generalizing the methodLet 现值PV = present value 终值FV = future value r = period interest rate t = number of time periods of the lump sum investment; time periods can be measured in years, months or days Note that “r” should correspond to the period in question FV = PV(1 + r)t Future value interest factor FVIF(r,t) = (1 + r)t FV Basics*FV BasicsA dollar in hand today is worth more than a dollar promised at some future date Trade-off between money now and money later depends on Amount of PV versus FV Interest rate, r Length of time, t Time line specifies all four factorsFV: Compounding of Interests*FV: Compounding of InterestsSimple interest单利 interest is earned only on the original principal Compound interest复利 interest is earned on principal and on interest received Consider the previous example FV with simple interest = $1,000 + $50 + $50 = $1,100 FV with compound interest = $1,102.50 The extra $2.50 comes from the interest of .05($50) = $2.50 earned on the first interest amountFuture Values: Example 2*Future Values: Example 2Suppose you invest the $1000 from the previous example for 10 years. How much would you have? FV = $1,000(1.05)10 = $1,628.89 The effect of compounding is small for a small number of periods, but increases as the number of periods increases. Simple interest would only have a future value of $1,500, for a difference of $128.89.FV – Important Relationships*FV – Important RelationshipsFor a given interest rate, r, the longer the time period, the higher the future value For a given time period, t, the higher the interest rate, the larger the future valueExample: Fig 4.2*Example: Fig 4.2Effects of Compounding within a Period (Year)*Effects of Compounding within a Period (Year)Suppose a bank quotes you an annual deposit rate of 6%, what is the FV after 1 year if the bank compounds interests once a year? Twice a year? Four times a year? Or, m times a year in general. In the following we will show the effects of increasing m, and will revisit the subject in more detail later. Effects of Compounding (Cont.)*Effects of Compounding (Cont.)Note 1: Interest Rates*Note 1: Interest RatesUnless stated otherwise, interest rates given in problems is assumed to be an annual rate in practice, interest rates are typically stated or quoted on a per annum basis adjustments must be made for fractional years period interest rate must match with length of compounding period: for example, suppose the quoted annual rate is 5%, and there is monthly compounding, then the monthly rate is 5%/12 = 0.4167%Note 2: Avoid rounding off within a calculation*Note 2: Avoid rounding off within a calculationAvoid removing intermediate results from your calculator. Store them in a memory register. This avoids input and output copying errors Do not round off an intermediate computation e.g. an annual interest rate of 5.9% implies a periodic (monthly) rate of 5.9%/12 = 0.49167%. It will be necessary to input 0.49167% and not 0.49% to obtain a solution that is correct to two decimal places.Time Value of Money*Time Value of MoneyPV Basics*PV BasicsHow much do I have to invest today to have some amount in the future? FV = PV(1 + r)t Rearrange to solve for PV = FV / (1 + r)t = FV (1 + r)-t Present value interest factor PVIF(r, t) = 1 / (1 + r)t Discounting finding the present value of some future amount Discounted cash flow (DCF) valuation the “value” of something Generally referred to the present value unless specifically indicate that we want the future valuePresent Values: Example 1*Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% rate of return annually, how much do you need to invest today? PV = $10,000 / (1.07)1 = $9,346.79Present Values: Example 1Present Values: Example 2*Present Values: Example 2You want to begin saving for a flat and you estimate that it will cost $2 million in 10 years. If you feel confident that you can earn 8% per year on your savings, how much do you need to save today? PVIF(8%,10) = 1/(1.08)10 PV = $2,000,000 / (1.08)10 = $926,387.98PV – Important Relationships*PV – Important RelationshipsFor a given interest rate, r, the longer the time period, the lower the present value For a given time period, t, the higher the interest rate, the smaller the present valueExample: Fig 4.3*Example: Fig 4.3Time Value of Money*Time Value of MoneyDiscount Rate*Discount RateThe implied interest rate in an investment Rearrange the basic FV equation and solve for r FV = PV(1 + r)t r = (FV / PV)1/t – 1 If using a financial calculator, remember the sign convention ( - for cash outflow, and + for cash inflow) or you will receive an error when solving for r If using formulas, you will want to make use of both the yx and the 1/x keysDiscount Rate: Example 1*Discount Rate: Example 1If you invest $15,000 for ten years, you will receive $30,000 then. What is your annual return? 15,000*(1+r)10 = 30,000 r = (30,000 / 15,000)1/10 – 1 = 20.1 = 0.07177 = 7.18% Calculator – the sign convention matters!!! PV = -$15,000 FV = $30,000 N = 10 I/Y = ??Rule of 72: How long does it take to double your investment at interest rate r?*Rule of 72: How long does it take to double your investment at interest rate r?null*Rule of 72Time Value of Money*Time Value of MoneyUsing Logarithms to Find Number of Period (t)*Using Logarithms to Find Number of Period (t)Basic properties of logarithms useful in financeFinding the Number of Periods*Finding the Number of PeriodsStart with basic equation and solve for t Number of Periods: Example 1*Number of Periods: Example 1You want to purchase a new car and you are willing to pay $200,000. If you can earn 10% per year and you currently have $150,000, how long will it take for you to have enough money to pay cash for the car? t = ln(200,000 / 150,000) / ln(1.1) = 3.02 years Using financial calculator PV = -$150,000 FV = $200,000 I/Y = 10 CPT N = 3.02 years Number of Periods: Example 2*Number of Periods: Example 2Suppose you want to buy a new car, which costs about $150,000. Suppose that buying a car you need to put 10% down and pay 5% for options, etc. The rest you plan to finance with a car loan. You currently have $15,000 in the saving account, which earns 7.5% per year. How long will it be before you have enough money for the down payment and fees?Number of Periods: Example 2*Number of Periods: Example 2How much do you need to have in the future? Down payment = .1($150,000) = $15,000 Fees = .05($150,000 – $15,000) = $6,750 Total needed = $15,000 + $6,750 = $21,750 Using the formula t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years Using financial calculator PV = -$15,000 FV = $21,750 I/Y = 7.5 CPT N = 5.14 yearsSummary: Table 4.4*Summary: Table 4.4Learning Objectives*Learning Objectivescompute the future value of an investment made today present value of cash to be received at some future date rate of return on an investment amount of time required for an investment to grow to a given value at a specified rate of interest