第4章的预备 金钱的时间价值.pptx

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1、The Time Value of MoneyDong HuiSHUFE2013 FallA mini caseToyota Motor Credit Corporation要在2008年3月28日对市场公开发售一种投资产品，合约如下：投资者在2008年3月28日以每份$24,099的价格从TMCC购买这种产品；TMCC承诺在2038年3月28日以每份$100,000的价格从投资者手中买回这种产品。这个投资的好处是，你每投入1美金就可以得到大约4美金。不好的地方是，你要等30年才能得到这笔回报。你应该怎样看待这笔交易？Time value of moneyTime value of money

2、指的是，和将来每个时间才能拿到的1美金相比，今天拿在手中的1美金的价值更高。Example:Would you rather receive$1000 today,or next year?Example:You could deposit the money today and it would be worth than$1000 next year.The most basic principle of Finance is that a dollar today is worth more than a dollar in the future.Time value of money今

3、天的钱和将来某个时刻的钱的权衡（trade-off），取决于今天你可以把钱以什么样的利率（rate）进行投资。我们的目标就是理解和掌握这种价值的转换。Basic Definitions现值（Present Value）：earlier money on a time lineExample:“明年能得到的1000美金在今天值多少钱？”终值（Future Value）：Later money on a time line给定利率（回报率），你的投资在将来的某个时刻的价值是什么？Example:“如果今天你有1000美金，那么这笔钱在明年值多少？”Basic Definitions利率：“exch

4、ange rate”between earlier money and later moneyDiscount rate折现率Cost of capital资本成本Opportunity cost of capital资本的机会成本Required return要求回报率An Example:Moneys Time UnitePresent dollars and future dollars are different kinds of money.Think of money as having a“time unit”,denoting when it is received(or pa

5、id)Just like currencyWe can only compare money in the same time units:It doesnt make sense to add$50 US to 50;andIt doesnt make sense to add$50 received today with$50 received next year.终值 Future values单期投资（Single Period）的例子：假如你今天投资了1000美金，期限1年，年利率5%。一年以后的终值是多少？利息=1,000(.05)=50一年以后的价值=本金+利息=1,000+50

6、=1,050Future Value(FV)=1,000(1+.05)=1,050我们说，1050美金是今天将1000美金以5%的利率投资一年以后的终值如果你今天投资$X，利率为r，一个时期以后你会得到$X+$X(r)=$X(1+r)终值多期投资接上例，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=1,102.50It indicates that$1,102.5 is the future v

7、alue of$1,000 invested for two years at 5%interest rate.终值一般来说，t个时期以后，终值（FV）=$X(1+r)tExample:Suppose you invest$100 at 10%for 10 years,how much will you receive after 10 years?FV=?Future Values:General FormulaFV=PV(1+r)t终值FV=future value现值PV=present value利率r=period interest rate期限t=number of periods

8、复利的作用Effects of Compounding单利Simple interest:利息没有被再投资复利Compound interest:利息将被进行再投资，so that you earn interest on interest.Consider the previous exampleFV with simple interest=1,000+50+50=1,100FV with compound interest=1,102.50The extra 2.50 comes from the interest of.05(50)=2.50 earned on the first i

9、nterest payment5F-12An exampleTo illustrate the growth of the compound interest in our example.The simple interest is constant each year,but the amount of compound interest gets bigger every year.Future Values Example 2Suppose you invest the$1,000 with 5%of interest rate for 5 years.How much would y

10、ou have?FV=?How much if with simple interest?FV=?The effect of compounding is small for a small number of periods,but increases as the number of periods increases,which we can see from the difference5F-14Future Values Example 3Suppose you had a relative deposit$10 at 5.5%interest 200 years ago.How m

11、uch would the investment be worth today?FV=?What is the effect of compounding?Simple interest=?Compounding added$to the value of the investment5F-15Quick Quiz Part ISuppose you have$500 to invest and you believe that you can earn 8%per year over the next 15 years.How much would you have at the end o

12、f 15 years using compound interest?How much would you have using simple interest?Would you rather receive$1,000 today or$1,240 in two years if you can earn 10%per year on the$1,000?Timeline and Future Value=?5F-16现值（Present Values）Example:明年能收到的1000美金，在今天来看值多少钱？今天我应该投资多少钱才能满足将来的需要？FV=PV(1+r)tRearran

13、ge to solve for PV=FV/(1+r)t现值当提到“贴现”（discounting）时，我们指的是找到未来财富的现在价值当提到“价值”，通常指资产的“现值”。Present Value One Period Example假设1年以后你需要1万美金作为买车的首付，如果你每年的投资回报率为7%，那么你今天应该投资多少钱？请计算一年之后的110美元以及两年以后的121美元的现值，假定利率为10%。Present Values Example 2You want to begin saving for your daughters college education and you

14、estimate that she will need$150,000 in 17 years.If you feel confident that you can earn 8%per year,how much do you need to invest today?Present Values Example 3Your parents set up a trust fund for you 10 years ago that is now worth$19,671.51.If the fund earned 7%per year,how much did your parents in

15、vest?PV=?5F-21Present Value Important Relationship IFor a given interest rate the longer the time period,the lower the present valueWhat is the present value of$500 to be received in 5 years?10 years?The discount rate is 10%5 years:PV=?10 years:PV=?5F-22Present Value Important Relationship IIFor a g

16、iven time period the higher the interest rate,the smaller the present valueWhat is the present value of$500 received in 5 years if the interest rate is 10%?15%?Rate=10%:PV=500/(1.1)5=310.46Rate=15%;PV=500/(1.15)5=248.595F-23Quick Quiz Part IIWhat is the relationship between present value and future

17、value?Suppose you need$15,000 in 3 years.If you can earn 6%annually,how much do you need to invest today?If you could invest the money at 8%,would you have to invest more or less than at 6%?How much?5F-24The Basic PV Equation-RefresherPV=FV/(1+r)tThere are four parts to this equationPV,FV,r and tIf

18、we know any three,we can solve for the fourthgiven a 10%opportunity rate,$110 to be received in one year is equivalent to having$100 today(or$90.90 one year ago,or$82.64 two years ago,etc.).5F-25Compounding and DiscountingDiscounting and Compounding are the tools to manipulate moneys time units:Disc

19、ounting converts moneys time units back in timeCompounding converts moneys time units forward in timeExample:given a 10%opportunity rate,$110 to be received in one year is equivalent to having$100 today(or$90.90 one year ago,or$82.64 two years ago,etc.).At its most fundamental level,compounding and

20、discounting are nothing more than using a set of formulas to find equivalent values at any two points in time.Compounding and DiscountingIn economic terms,one might stress that equivalence just means that a rational person will be indifferent between$100 today and$110 in one year,given a 10%opportun

21、ity.(a)she could take the$100 today and invest it to have$110 in one year or(b)she could borrow$100 today and repay the loan with$110 in one year.Discount RateOften we will want to know what the implied interest rate is in an investmentRearrange the basic PV equation and solve for rFV=PV(1+r)tr=(FV/

22、PV)1/t 15F-28Discount Rate Example 1You are looking at an investment that will pay$1,200 in 5 years if you invest$1,000 today.What is the implied rate of interest?r=?5F-29Discount Rate Example 2Suppose you are offered an investment that will allow you to double your money in 6 years.You have$10,000

23、to invest.What is the implied rate of interest?r=?5F-30Discount Rate Example 3Suppose you have a 1-year old son and you want to provide$75,000 in 17 years towards his college education.You currently have$5,000 to invest.What interest rate must you earn to have the$75,000 when you need it?r=(75,000/5

24、,000)1/17 1=.172688=17.27%5F-31Quick Quiz Part IIIWhat are some situations in which you might want to know the implied interest rate?You are offered the following investments:You can invest$500 today and receive$600 in 5 years.The investment is considered low risk.You can invest the$500 in a bank ac

25、count paying 4%.What is the implied interest rate for the first choice and which investment should you choose?5F-32Quick Quiz Part IIIHow would the decision be different if you were looking at borrowing$500 today and either repaying at 4%or repaying$600?Finding the Number of PeriodsStart with the ba

26、sic equation and solve for t(remember your logs)FV=PV(1+r)tt=ln(FV/PV)/ln(1+r)The rule of 72:how long it will take to double your money#years to double=72/r where r is a percent.5F-34Rule of 72:ExampleYou are planning to travel to Paris for a week.The total travel fee will be about$4,000.Now you hav

27、e some savings of$2,000,and it can earn a 10%of the return every year.how long will it be before you have enough money to pay for your trip?t=72/r=7.2 years using rule of 72t=ln(4000/2000)/ln(1.1)=7.27 yearsNumber of Periods Example 1You want to purchase a new car,and you are willing to pay$20,000.I

28、f you can invest at 10%per year and you currently have$15,000,how long will it be before you have enough money to pay cash for the car?t=ln(20,000/15,000)/ln(1.1)=3.02 years5F-36Number of Periods Example 2Suppose you want to buy a new house.You currently have$15,000,and you figure you need to have a

29、 10%down payment plus an additional 5%of the loan amount for closing costs.Assume the type of house you want will cost about$150,000 and you can earn 7.5%per year,how long will it be before you have enough money for the down payment and closing costs?5F-37Number of Periods Example 2 ContinuedHow muc

30、h do you need to have in the future?Down payment=.1(150,000)=15,000Closing costs=.05(150,000 15,000)=6,750Total needed=15,000+6,750=21,750Compute the number of periodsUsing the formulat=ln(21,750/15,000)/ln(1.075)=5.14 years5F-38Quick Quiz Part IVWhen might you want to compute the number of periods?

31、Suppose you want to buy some new furniture for your family room.You currently have$500,and the furniture you want costs$600.If you can earn 6%,how long will you have to wait if you dont add any additional money?5F-39Multiple cash flowsLast class,we deal with cases of single cash flows.In reality,mos

32、t investments have multiple cash flows.To have some practical skills,like:To calculate your car payments or student loan paymentsTo determine how long it will take to pay off a credit card if you make the minimum payment each monthMultiple Cash Flows Timeline and future valueSuppose you deposit$100

33、today in an account paying 8%.In one year,you will deposit another$100.how much will you have in two years?Multiple Cash Flows Future Value Example 6.1You think you will be able to deposit$4,000 at the end of each of the next three years in a bank account paying 8%interest.You currently have$7,000 i

34、n the account.How much will you have in three years?In four years?Draw a timeline firstMultiple Cash Flows Future Value Example 6.1Find the value at year 3 of each cash flow and add them togetherToday(year 0):FV=7000(1.08)3=8,817.98Year 1:FV=4,000(1.08)2=4,665.60Year 2:FV=4,000(1.08)=4,320Year 3:val

35、ue=4,000Total value in 3 years=8,817.98+4,665.60+4,320+4,000=21,803.58Value at year 4=21,803.58(1.08)=23,547.876F-43Multiple Cash Flows Future ValueTwo different ways to calculate future values for multiple cash flows:1.Compound the accumulated balance forward one year at a time2.Calculate the futur

36、e value of each cash flow first and then add them upBoth give the same answerMultiple Cash Flows FV Example 2Suppose you invest$500 in a mutual fund today and$600 in one year.If the fund pays 9%annually,how much will you have in two years?FV=?6F-45Multiple Cash Flows Example 2 ContinuedHow much will

37、 you have in 5 years if you make no further deposits?First way:FV=500(1.09)5+600(1.09)4=1,616.26Second way use value at year 2:FV=?6F-46Multiple Cash Flows FV Example 3Suppose you plan to deposit$100 into an account in one year and$300 into the account in three years.How much will be in the account

38、in five years if the interest rate is 8%?FV=?6F-47Multiple Cash Flows Future Value ExerciseIf you deposit$100 in one year,$200 in two years,and$300 in three years,how much will you have in three years?How much is the interest?How much will you have in five years if you dont add additional amounts?As

39、sume a 7%interest rate.Be careful about the lengths of timeMultiple Cash Flows Present Value Example 6.3Find the PV of each cash flows and add themYear 1 CF:200/(1.12)1=178.57Year 2 CF:400/(1.12)2=318.88Year 3 CF:600/(1.12)3=427.07Year 4 CF:800/(1.12)4=508.41Total PV=178.57+318.88+427.07+508.41=1,43

40、2.936F-49Example 6.3 Timeline01234200400600800178.57318.88427.07508.411,432.936F-50Multiple Cash Flows PV Another ExampleYou are considering an investment that will pay you$1,000 in one year,$2,000 in two years and$3000 in three years.If you want to earn 10%on your money,how much would you be willin

41、g to pay?PV=1000/(1.1)1=909.09PV=2000/(1.1)2=1,652.89PV=3000/(1.1)3=2,253.94PV=909.09+1,652.89+2,253.94=4,815.926F-51Decisions,DecisionsYour broker calls you and tells you that he has this great investment opportunity.If you invest$100 today,you will receive$40 in one year and$75 in two years.If you

42、 require a 15%return on investments of this risk,should you take the investment?Use the CF keys to compute the value of the investmentCF;CF0=0;C01=40;F01=1;C02=75;F02=1NPV;I=15;CPT NPV=91.49*No the broker is charging more than you would be willing to pay.6F-52Saving For RetirementYou are offered the

43、 opportunity to put some money away for retirement.You will receive five annual payments of$25,000 each beginning in 40 years.How much would you be willing to invest today if you desire an interest rate of 12%?Use cash flow keys:CF;CF0=0;C01=0;F01=39;C02=25,000;F02=5;NPV;I=12;CPT NPV=1,084.716F-53Sa

44、ving For Retirement Timeline0 1 2 39 40 41 42 43 44 0 0 0 0 25K 25K 25K 25K 25KNotice that the year 0 cash flow=0(CF0=0)The cash flows in years 1 39 are 0(C01=0;F01=39)The cash flows in years 40 44 are 25,000(C02=25,000;F02=5)6F-54Quick Quiz Part ISuppose you are looking at the following possible cash flows:Year 1 CF=$100;Years 2 and 3 CFs=$200;Years 4 and 5 CFs=$300.The required discount rate is 7%.What is the value of the cash flows at year 5?What is the value of the cash flows today?What is the value of the cash flows at year 3?6F-55