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1、Do you have these puzzles?,Buy a mobile by lump-sum payment or by installment ? Arrange savings for a future expenditure? What kind of loans to apply for? Buy or sell a bond you are holding? ,2019/9/13,Ch3 Time Value of Money,CHAPTER 3 Time Value of Money,Compounding and Discounting of Single Sums A
2、nnuities Types of Interest Rates,2019/9/13,Ch3 Time Value of Money,CH3 The Time Value of Money,To find the answer, youll have to know,2019/9/13,Ch3 Time Value of Money,3.1 Compounding and Discounting Single Sums,2019/9/13,Ch3 Time Value of Money,We know that receiving $1 today is worth more than $1
3、in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.,2019/9/13,Ch3 Time Value of Money,If we can measure this opportunity cost, we can:,Translate $1 today into its equivalent in the
4、future (compounding). Translate $1 in the future into its equivalent today (discounting).,2019/9/13,Ch3 Time Value of Money,Time lines,Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning
5、 of the second period.,CF0,CF1,CF3,CF2,0,1,2,3,i%,2019/9/13,Ch3 Time Value of Money,Drawing time lines: 1）$100 lump sum due in 2 years;,2019/9/13,Ch3 Time Value of Money,Drawing time lines: 2) Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50,2019/9/13,Ch3 Time Value of Money
6、,Compounding:,To find the Future Value (FV) of a cash flow, we suppose we earn interest on principal as well as on interest accumulated each term.,2019/9/13,Ch3 Time Value of Money,Example,What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? FV can be solved by using the ar
7、ithmetic, or Table A-3 (pp.A-6),2019/9/13,Ch3 Time Value of Money,Solving for FV: The arithmetic method,After 1 year: FV1 = PV ( 1 + i ) = $100 (1.10) = $110.00 After 2 years: FV2 = PV ( 1 + i )2 = $100 (1.10)2 =$121.00 After 3 years: FV3 = PV ( 1 + i )3 = $100 (1.10)3 =$133.10 After n years (genera
8、l case): FVn = PV ( 1 + i )n,2019/9/13,Ch3 Time Value of Money,Solving for FV: The table-checking method,FVn = PV *FVIF(i, n) FV3= $100* FVIF(10%,3) = $100*1.331 =$133.10,2019/9/13,Ch3 Time Value of Money,Discounting:,Finding the Present Value (PV) of a cash flow or series of cash flows when compoun
9、d interest is applied (the reverse of compounding).,2019/9/13,Ch3 Time Value of Money,PV = ?,100,Example,What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?,0,1,2,3,10%,2019/9/13,Ch3 Time Value of Money,Solving for PV: The arithmetic method,Solve the general FV equation for PV: PV
10、= FVn / ( 1 + i )n PV = FV3 / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.13,2019/9/13,Ch3 Time Value of Money,Solving for PV: The table-checking method,Solve the general FV equation for PV: PV = FVn * PVIF(i,n) Ref.: Table A-1 (PP.A-2) PV = FV3 * PVIF(10% ,3) = $100 *0.7513 = $75.13,2019/9/13,Ch3 Time Valu
11、e of Money,Solves the general FV or PV equation for N. Same as previous problems, but now solving for N. Eg. Pp. 104,Other application 1: Solving for N With I, PV, FV known to you,2019/9/13,Ch3 Time Value of Money,Solves the general FV or PV equation for i. Same as previous problems, but now solving
12、 for N. Eg. Pp. 103,Other application 2: Solving for I With n, PV, FV known to you,2019/9/13,Ch3 Time Value of Money,A magic 72 rule,When the compound interest rate is less than 20%, the years you need to double your todays wealth is 72/the interest rate number. Conversely ,if you want to double you
13、r wealth in X years, you have to ensure your investment earn 72/X% annually!,2019/9/13,Ch3 Time Value of Money,The Power of Compound Interest,compounding the 8th wonder! The power of compound interest exceeds the nuclear bomb!,2019/9/13,Ch3 Time Value of Money,Story of an ancient Athenian,This Athen
14、ian pocketed all the money except a single Drachma, which he invested in Athenian government bonds paying 3 percent compounded annually. He didnt live long enough to see the results, but after 2,000 years that Drachmae wound up being worth more than all the assets on the Earth!,2019/9/13,Ch3 Time Va
15、lue of Money,To make the “nuclear bomb” work, investing early, or, a long enough investment period is necessary!,2019/9/13,Ch3 Time Value of Money,Suppose the merchant invests only 3 years, he will have only 1*FVIF(3%,3)=1.0927 Drachmae,2019/9/13,Ch3 Time Value of Money,a sequence of equal cash flow
16、s, occurring at fixed intervals for a specified number of periods.,3.2 Annuities,2019/9/13,Ch3 Time Value of Money,a bonds semi-annual coupon interest payments over the life of the bond. Repaying bank loans: a stream of equal repayments.,Examples of Annuities:,2019/9/13,Ch3 Time Value of Money,Types of annuities:,Ordinary (deferred) annuity Annuity due Perpetuities,2019/9/13,Ch3 Time Value of Money,Difference between an ordinary annuity and an annuity due,2019
