ALLOCATING-RESOURCES-OVE課件

1、CHAPTER 4: ALLOCATING RESOURCES OVER TIME1ObjectiveExplain the concept of compounding and discounting and to provide examples of real lifeapplicationsNOTE:Students getting time value of money for the first time need to double or triple exposure to this basic skillExercise are extremely important2CHA

2、PTER 4: ALLOCATING RESOURCES OVER TIME4.1 compounding4.2 the frequency of compounding4.3 present value and discounting4.4 Alternative discounted cash flow decision rules4.5 multiple cash flows4.6 annuities4.8 loan amortization4.10 inflation and discounted cash flow analysis 3INTRODUCTION: TIME VALUE

3、 OF MONEY (TVM)$20 today is worth more than the expectation of $20 tomorrow because: a bank would pay interest on the $20inflation makes tomorrows $20 less valuable than todaysuncertainty of receiving tomorrows $2044.1 COMPOUNDINGWhat is compounding? PV and FVAssume that the interest rate is 10% p.a

4、.What this means is that if you invest $1 for one year, you have been promised $1*(1+10/100) or $1.10 next yearInvesting $1.10 for yet another year promises to produce 1.10 *(1+10/100) or $1.21 in 2-years5GENERALIZING THE METHODGeneralizing the method requires some definitions. Let:i be the interest

5、 rate n be the life of the lump sum investmentPV be the present value FV be the future value6EXAMPLE: FUTURE VALUE OF A LUMP SUMYour bank offers a deposit with an interest rate of 4% for a 3 year investment. You wish to invest 1,500 Yuan for 3 years, how much will your investment be worth:1. if Comp

6、ounding?2. if 4% is Simple Interest?7VALUE OF INVESTING $18Continuing in this manner you will find that the following amounts will be earned:FUTURE VALUE AND COMPOUND INTEREST9FUTURE VALUE OF A LUMP SUM10Table 4.2: future value factor（終值因子）Calculation in Excel: FV(i,n,0,-PV)Figure 4.2: FV under diff

7、erent i and n RULE OF 72This rule says that the number of years it takes for a sum of money to double in value (“the doubling time”) is approximately equal to the number 72 divided by the interest rate expressed in percent per yearDoubling Time = 72Interest Rate114.1.2 saving for old agePut 100 now,

8、 whats the value when you are 65? Under 8%, and 9%4.1.3 reinvesting at a different rateWhen you face multiple investment periods12LUMP SUMS FORMULAEWe have solved a present value and a future value of a lump sum. There remains two other variables that may be solved forinterest, inumber of periods, n

9、134.2 THE FREQUENCY OF COMPOUNDINGYou have a credit card that carries a rate of interest of 18% per year compounded monthly. What is the interest rate compounded annually?That is, if you borrowed $1 with the card, what would you owe at the end of a year?14THE FREQUENCY OF COMPOUNDINGWhen a rate is e

10、xpressed in terms of year while compounded with a different microperiod(e.g. monthly), then it is a nominal or annual percentage rate (APR)If the frequency of compounding is year based, then the rate is referred to as a the real or effective annual rate15THE FREQUENCY OF COMPOUNDINGAssume compounded

11、 m times in a year and a nominal rate k per year(APR), that means the effective rate is k/m per microperiodQ: Invest $1 for one year, obtain how much?A:$1*(1+k/m)mQ:What is the effective rate over a year? A:($1*(1+k/m)m - $1)/$1 = (1+k/m)m - 116CREDIT CARD If the credit card pays an APR of 18% per y

12、ear compounded monthly. The (real) monthly rate is 18%/12 = 1.5% so the real annual rate is (1+0.015)12 - 1 = 19.56%The two equal APR with different frequency of compounding have different effective annual rates:17THE FREQUENCY OF COMPOUNDINGA bank determines that it needs an effective rate of 12% o

13、n car loans to medium risk borrowersWhat annual percentage rates may it offer?18THE FREQUENCY OF COMPOUNDING19Q: What if m goes to infinity? THE FREQUENCY OF COMPOUNDING20THE FREQUENCY OF COMPOUNDINGMany lenders and borrowers do not have a clear understanding of APRs, but institutional lenders and b

14、orrowers doInstitutions are therefore able to extract a few basis points from consumers, but why bother?21THE FREQUENCY OF COMPOUNDINGFinancial intermediaries profit from differences in the lending and borrowing rates. Overheads, bad loans and competition results in a narrow margin. Small rate gains

15、 therefore result in a large increases in institutional profitsIn the long term, ill-informed consumers lose because of compounding224.3 PRESENT VALUE AND DISCOUNTING Q: Sometimes we want to know how much we have to invest or deposit to have a certain amount value of FVDiscount23PRESENT VALUE OF A L

16、UMP SUM24Discounted cash flow analysisPresent value factorEXAMPLE: PRESENT VALUE OF A LUMP SUMYou have been offered $40,000 for your printing business, payable in 2 years. Given the risk, you require a return of 8%. What is the present value of the offer?254.3.1 when a $100 gift is not really $1004.

17、3.2 discounting with compounding more frequently than annually264.4 ALTERNATIVE DISCOUNTED CASH FLOW DECISION RULESFV formula and net present value(NPV) rule“The NPV is the difference between the present value of all cash inflows minus cash outflows Reject a project if its NPV is negative.”Example:

18、P11827QUESTION 1:Is there any similar rules related to NPV?Ans: Future value rules and IRR rules28QUESTION 2:If the bank interest rate is 4% for 3 year deposit, at the same time a friend of you told you that he had a good project and want get some financing from you, he gives you an interest rate of

19、 8% for 3 yearQ:Does it necessarily mean you will be better off investing on your friend instead of bank?4.4.1 Investing in Land294.5 MULTIPLE CASH FLOWSTime LinesFuture Value of a Stream of Cash FlowPresent Value of a Stream of Cash FlowsInvesting with Multiple Cash Flows30TIME LINE31PRESENT VALUE

20、OF MULTIPLE CASH FLOWS324.6 ANNUITIESa sequence of equally spaced identical cash flows is a common occurrence, so automation pays offE.g.a mortgagea saving planan investment project 33CASH FLOW DIAGRAM OF ANNUITIES34ANNUITY FORMULA NOTATIONPV = the present value of the annuityFV = the future value o

21、f the annuityi = interest rate to be earned over the life of the annuityn = the number of paymentspmt = the periodic paymentNote: i should be set corresponding to n35FV ANNUITY FORMULA AND PAYMENT(REG.)36Page. 125PV ANNUITY FORMULA AND PAYMENT(REG.)37E.g. 4.6.4 Taking a Mortgage Loan p.1274.7 PERPET

22、UAL ANNUITIESRecall the annuity formula:38 Let n infinity with i 0:4.8 LOAN AMORTIZATIONE.g. MortgageLoan Amortization: The process of paying off a loans principal gradually over its termHow much is repayment of principal and how much is interest of outstanding balance?39OUTSTANDING BALANCE AS A FUN

23、CTION OF TIMEThe following graphs illustrate that in the early years, monthly payment are mostly interest. In latter years, the payments are mostly principle40414.9 EXCHANGE RATES AND TIME VALUE OF MONEYYou are considering the choice:Investing $10,000 in dollar-denominated bonds offering 10% / yearI

24、nvesting $10,000 in yen-denominated bonds offering 3% / yearAssume an exchange rate of 0.014243$10,000$11,000 1,000,0001,030,000Time10% $/$ (direct)100 /$3% / ? $/U.S.A.USDJapanJPY44$10,000$11,000 $11,000 1,000,0001,030,000Time10% $/$ (direct)100 /$3% / 93.636/$U.S.A.USDJapanJPYCONCLUSIONIf the yen

25、price actually rises by more than 6.8% during the coming year then the yen bond is a better investment454.9.1 COMPUTING NPV IN DIFFERENT CURRENCIESIn any time-value-of-money calculation, the cash flows and interest rates must be denominated in the same currencyUSA project U requires an investment of

26、 $10,000, as does a Japanese project J. U generates $6,000/year for 5 years, and project J generates 575,000/year for 5 yearsThe US interest is 6%, the Japanese interest is 4%, and the current exchange rate is 0.0146SOLUTIONUsing your financial calculatorDetermine the present value of U in $ by disc

27、ounting the 5 payments at 6%, and subtract the initial investment of $10,000Determine the present value of J in by discounting the 5 payments at 4%, and subtract the initial investment of 1,000,000Obtain $15,274 & 1,5599,798 respectively47SOLUTIONConvert the 1,5599,798 to $ using the current exchang

28、e rate to obtain $15,600 The Japanese NPV of of $15,600 is higher than the USA NPV or$15,274, so invest in the Japanese project484.10 INFLATION AND DISCOUNTED CASH FLOW ANALYSISWe will use the notationin the rate of interest in nominal termsir the rate of interest in real termsr the rate of inflatio

29、nThen we have the relationshipQ: Is there any simpler approximation of the formula?494.10.2 SAVING FOR COLLEGE:1Is the money your parents saved for your overseas study enough to cover your spend in 3 years?P.133504.10.4 WHY DEBTORS GAIN FROM UNANTICIPATED INFLATIONYou borrow $10,000 at 8% interest.

30、The todays spending power of the repayment is $10,000*1.08/ (1+inflation)If the actual inflation is the expected 6%, then the real cost of the loan in todays money is $10,188.68If the actual inflation is 10%, then the loans real cost (in todays values) is $9,818.18Unexpected inflation benefits borrower51INFLATION AND PRESENT VALUEA