3 - Time Value of Money - Part 2 (Annuities, DCF Valuation)

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Chapter 3 The Time Value of Money 货币的时间价值财务管理(双语版) 教学课件

年金现值：PVAn（Present Value of an Ordinary Annuity）
n 1
11i n
PV nR A t 11it R i
R PVi,nI
PVIFAi,n 年金现值系数（the Present Value Interest Factor of an Annuity at i for n periods）
P0V P 0FnV 1inFnV PVi,nIF
7. PVIFi,n : （the Present Value Interest Factor at i for n
periods）
8.
复利现值系数，利率为i,期数为n
6. Annuities 年金
系列、等额的收付
Ordinary Annuity 普通年金收付款发生于每期期末 Annuity Due 预(即)付年金收付款发生于每期期初
Perpetuity 永续年金
无限期支付的普通年金
Ordinary Annuity(普通年金)
0
R
R
R
R
年金终值：FVAn（Future Value of an Annuity）
FV nR A t n11intR1iin1RFVi,nIFA
FVIFAi,n 年金终值系数（the Future Value Interest Factor of an Annuity at i for n periods）
1
每年计息一次时产生的利息=名义利率每年计息m次时产生的利息
FV n A R F DV i,n 1 I 1 FA
现值（Present Value）
比普通年金少折现一期
PV n A R P D V i,nI 1 F i A

货币的时间价值

+50000=281855 元
14
普通年金终值计算：
更一般的，假设每年收付的金额为A，利率为i, 期数为n, 则计算复利终值的年金终值FVAn的公式如下：
从这个公式可以看出来，这是一个等比数列求和公式，简化后可以得到：
记：
15
普通年金终值计算：
例6：某父母为孩子在10年后上大学准备5万元钱，假设银行的存款年利率为6%，复利计息，那么在这10年中，每年年末要存入多少元？
Lecture 2：货币的时间价值 (Time Value of Money)
潘贵芳宁波诺丁汉大学
金融与投资
1
教学内容：
货币的时间价值单利/复利的现值与终值年金（普通年金、预付年金、递延年金、永
续年金）货币时间价值应用中的几个问题（利率或
折现率、期限、名义利率和实Fra bibliotek利率）2
货币的时间价值概述：
10
复利计息现值与终值：
例4 （改例3， p6）：王先生计划于5年后买车，需购车款13万元，王先生打算现在存笔钱到银行， 5年后正好用于购车，银行目前的存款利率为6%, 且复利计息，一年一计，则王先生需要存入的金额为：
PV=13*（P/F，6%，5）=13*0.7473=9.71 （万元）按单利计息时，王先生需要存入10万元，而当
这一问题，第二年计算利息的本金是上一年的年末终值，也就是1080，又再按8%的利息计算，也就是说 FV2=FV1*（1+i）=PV*（1+i）*（1+i）=PV*（1+i）2 =1000*（1+8%）2 =1166.4元第三年末，该账户金额为： FV3=FV2*（1+i）=PV*（1+i）2（1+i）=PV（1+i）3=1000*（1+8%） 3=1259.7元所以，更一般的，在第n年末，该账户的终值FVn应该为： FVn=PV (1+i)n=PV (F/P, i, n) (p.25)

time value of money

• Now assume that interest is paid every six months. That is interest is compounded semi-annually.
• How much money do you have at the end of the year?
• The number of compounding period (n) may not be the same as the number of years.
14
Periodic vs. Annual
• The annual rate must be converted to periodic rate if compounding is taking place more often than once a year.
• The number of Compounding periods (n).
6
SINGLE AMOUNT
• Consider only two cash flows, one at the beginning and one at the end of the period.
• Application - place some money in your saving account today and withdraw the proceeds in the future.
26
Present Value of Perpetuity
• Present Value of Perpetuity
PV = PMT/r
27
Annuity
• An annuity is a series of payments of equal amount.

财务管理基础课件：The Time Value of Money

A generalized formula for Future Value:
Where FV = Future value PV = Present value i = Interest rate n = Number of periods;
In the previous case, PV = $1,000, i = 10%, n = 4, hence;
1st year……$1,000 X 1.10 = $1,100 2nd year…...$1,100 X 1.10 = $1,210 3rd year……$1,210 X 1.10 = $1,331 4th year……$1,331 X 1.10 = $1,464
9-4
Future Value – Single Amount (Cont’d)
• The time value of money is used to determine whether future benefits are sufficiently large to justify current outlays
• Mathematical tools of the time value of money are used in making capital allocation decisions
annuity
9-25
Yield – Present Value of a Single Amount
• To calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:
Table 9–5

Time value of money课后习题答案

2. a Present value of $5,000 today = $5,000 b. Present value of $15,000 received in 5 years at 9%: PV0 = $15,000(PVIF.09,5) = $15,000 (0.650) = $9,750 (calculator solution = $9,748.97) c. Present value of a 15 year, $1,000 annuity at 9%: PVAN0 = $1,000 (PVIFA.09,15) = $1,000(8.061) = $8,061 (calculator solution = $8,060.69) Therefore, you prefer $15,000 in five years because it has the highest present value.
CHAPTER 4 THE TIME VALUE The investment paying five percent compound interest is more attractive because you will receive interest not only on the principal amount each year, but interest will be earned on the previous year's interest as well. 2. The future value interest factor for 10 percent and two years is 1.210, whereas the present value interest factor for 10 percent and two years is 0.826. 3. As the interest rate increases, any annuity amount is being discounted by a higher value, thereby reducing the present value of the annuity. This can be seen in Table IV by looking across any row of successively higher interest rates. In contrast, the future value of an annuity increases as the interest (compounding) rate increases. (See Table III.) 4. Daily compounding is preferred because you will earn interest on the interest earned in the account each day. Table 4-6 illustrates this. 5. Annuity due computations are common for lease contracts and insurance policies, where payments are generally made at the beginning of each period. 6. As can be seen in Table 4-7, the more frequent the compounding period, the lower the present values. 7. a A marketing manager might use present value concepts to evaluate the success of an advertising or other promotional campaign, the benefits of which are likely to extend beyond one year in time. Also, a firm selling capital goods must be familiar with the type of present value economic analysis that customers will use to evaluate purchases.

第三章货币的时间价值 chapter3- Time value of money

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 (general case):
3-2
To find the answer, you’ll have to know
CH3 The Time Value of Money
2019/12/18
Ch3 Time Value of Money
3-3
3.1 Compounding and Discounting Single Sums
3-9
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.
Today
Future
?
2019/12/18
Ch3 Time Value of Money
3-6
Time lines
0
1
2
3
i%
CF0
CF1
CF2
CF3
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 of the second period.

finance Time Value of Money(2)

Chapter 3 Time Value of Money: An IntroductionProblem 4Suppose Bank One offers a risk-free interest rate of 5.5% on both savings and loans, and Bank Enn offers a risk-free interest rate of 6% on both savings and loans.a.What arbitrage opportunity is available?b.Which bank would experience a surge in the demand for loans? Which bankwould receive a surge in deposits?c.What would you expect to happen to the interest rates the two banks areoffering?a.Take a loan from Bank One at 5.5% and save the money in Bank Enn at 6%.b.Bank One would experience a surge in the demand for loans, while Bank Ennwould receive a surge in deposits.c.Bank One would increase the interest rate, and/or Bank Enn would decrease itsrate.Problem 7Bubba is a shrimp farmer. In an ironic twist, Bubba is allergic to shellfish, so he cannot eat any shrimp. Each day he has one-ton supply of shrimp. The market price of shrimp is $10,000 per ton.a.What is the value of a ton of shrimp to him?b.Would this value change if he were not allergic to shrimp? Why or why not?a.The value of one ton of shrimp to Bubba is $10,000 because that is the marketprice.b.No. As long as he can buy or sell shrimp at $10,000 per ton, his personalpreference or use for shrimp is irrelevant to the value of the shrimp.Problem 11A friend asks to borrow $55 from you and in return will pay you $58 in one year. If your bank is offering a 6% interest rate on deposits and loans:a. How much would you have in one year if you deposited the $55 instead?b. How much money could you borrow today is you pay the bank $58 in one year?c. Should you loan the money to your friend or deposit it in the bank?a. I f you deposit the money in the bank today you will have:$1.06 in one year FV $55 today $58.30 in one year $ today ⎛⎫=⨯= ⎪⎝⎭b.If you lend the money to your friend for one year and borrow against the promised $58 repayment, then you could borrow:$1.06 in one year PV $58 in one year $54.72 today $ today ⎛⎫=÷= ⎪⎝⎭c. F rom a financial perspective, you should deposit the money in the bank, as it will result in more money for you at the end of the year.Problem 16Calculate the future value of $2000 ina. Five years at an interest rate of 5% per year.b. Ten years at an interest rate of 5% per year.c. Five years at an interest rate of 10% per year.d. Why is the amount of interest earned in part (a) less than half the amount of interest earned in part (b)?a. Timeline:0 1 2 555FV 2,000 1.052,552.56=⨯=b. Timeline:0 1 2 101010FV 2,000 1.053,257.79=⨯=c. Timeline:0 1 2 555FV 2,000 1.13,221.02=⨯=d. Because in the last 5 years you get interest on the interest earned in the first 5 years as well as interest on the original $2,000.Problem 22Your grandfather put some money in an account for you on the day you were born. You are now 18 years old and are allowed to withdraw the money for the first time. The account currently has $3996 in it and pays an 8% interest rate.a. How much money would be in the account if you left the money there until your 25th birthday?b. What if you left the money until your 65th birthday?c. How much money did your grandfather originally put in the account?a. Timeline:18 19 20 21 25 0 1 2 3 77FV 3,996(1.08)6,848.44==b. Timeline:18 19 20 21 65 0 1 2 3 4747FV 3,996(1.08)148,779==c. Timeline:0 1 2 3 4 18183,996PV 1,0001.08==Chapter 4 Time Value of Money: Valuing Cash Flow StreamsProblem 1You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs $5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would the timeline differ if you created it from the bank’s perspective?0 1 2 3 4 5From the bank’s perspective, the timeline is the same except all the signs are reversed.Problem 9The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year.a. What is the value of the bond immediately after a payment is made?b. What is the value of the bond immediately before a payment is made? Timeline:0 12 3a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula:100PV 2,500.0.04£==b. The value of the bond is equal to the present value of the cash flows. The cash flows are the perpetuity plus the payment that will be received immediately.PV =100+100=£2,600 Problem 30You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? Timeline:30 31 32 33 65 0 12 3 35FV = $2 millionThe PV of the cash flows must equal the PV of $2 million in 35 years. The cash flows consist of a 35-year annuity, plus the contribution today, so the PV is:()35C 1PV 1 C.0.05 1.05=-+⎛⎫ ⎪⎝⎭The PV of $2 million in 35 years is()352,000,000$362,580.57.1.05=Setting these equal gives:()()3535C 11C 362,580.570.05 1.05362,580.57C $20,868.91.11110.05 1.05-+=⇒==-+⎛⎫⎪⎝⎭⎛⎫⎪⎝⎭。

finance Time Value of Money(2)

货币的时间价值Time Value of Money

Time Value of Money ( 貨幣的時間價值 )0、今年的100元與明年的100元相比，何者的財富價值較高?答：今年的100元的財富價值較高，因為我如果將100元存放在台銀，一年後，以定存年利率1.5%計算，到期時我將擁有100*(1+1.5%)=101.5元>100元(明年)；所以明年的100元與今年的100元在作財富價值比較時，須將明年的100元以適當的折現率( Appropriate Discounting Rate )折現成現值( Present Value )，這就是折現的觀念。

反之如果明年我欲擁有100元，華銀的一年定存利率同為1.5%，則我只須存入100/(1+1.5%)=98.522元，此98.522元就是明年100元的折現值(在折現率1.5%條件下)。

1、Compound Interest ( 複利 ) and Future Value( 未年值或終值 ))1(r C + 2)1(r C +nr C )1(+n r C FV )1(+= Future Value ( 終值 ) C=現金流量，r =利率(or 報酬率2、Future Value ( 終值，未來值 ))(,n r n FVIF PV FV =FVIF =Future Value Interest Factor( 終期利率因子 )，r = 利率，n=期數範例 1、小明存入東和銀行30,000元，年利率=6%，每年複利一次，5年後會變成多少錢?)(,n r n FVIF PV FV =78.146,40338226.1000,30%)61(000,3055=⨯=+⨯=FV而338226.15%,6=FVIF 0 1 2 n3、Present Value ( 現值 )在某一時點之金錢價值折現( Discounting )成目前的金錢價值。

)1(1r C+ )1(2r C+ )1(r nC+)1(r nCPV +=PV=Present Value( 現值 )，C=現金流量，r =折現率，n =期數)()1(,n r n nnPVIF FV r FV PV =+=PVIF r,n =Present Value Interest Factor(現值利率因子 )範例：小明希望存一筆錢在東和銀行，3年後能有100,000元，定存年利率為1.7%，則小明此時應存入多少錢才能達成目標?)1(r nCPV +=61.068,959506861.0000,100000,100%)7.11(3=⨯==+PV而 9506861.03%,7.1=PVIF0 1 2n4、Annuity ( 年金 )是指在某固定時間點的等額金額支付。

chapter 2 time vaule of money

for r
FV = PV(1 + r)t r = (FV / PV)1/t – 1
Discount Rate 【例7 】
假设某项投资可以让你在6年内将投资额翻倍。你现在有10000元可以投资，这项投资的回报率是多少？
FV = 20 000 PV = 10 000 t = 6 years r=?
偿债基金是指为了在约定的未来一定时点清偿某笔债务或积聚一定数额的资金而必须分次等额提取的存款准备金
偿债基金与年金终值互为逆运算，其计算公式为
r A FV (1 r)t 1
先付年金-终值【例13 】
假设你正在存钱准备购房，你打算每年初投入10000元到某项年收益率为8%的项目上，第一笔钱今天投入，3年后你将会有多少钱？
2. 3一次性收付款项（lump sum）的复利现值
复利现值是复利终值的逆运算，它是指今后某一规定时间收到或付出的一笔款项，按贴现率i所计算的货币的现在价值。
0
1
2
3
10%
PV = ?
100
现值的计算【例4】
假设你今年21岁，有一种投资可以让你获得稳定的每年10%的收益，那么你现在要投资多少钱，才可以使你在65岁的时候能够拿到 100万元？
0
1
2
r%
100
3 year $100 ordinary annuity
0
1
2
3
r%
100
100
100
时间轴Time line2
Uneven cash flow stream
0
1
2
3
r%
-50
100
75
50
Interest rate is not constant

THE TIME VALUE OF MONEY

O VERVIEWA dollar in the hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest that dollar and earn interest. Of all the techniques used in finance, none is more important than the concept of the time value of money,or discounted cash flow (DCF) analysis. The principles of time value analysis that are developed in this chapter have many applications, ranging from setting up schedules for paying off loans to decisions about whether to acquire new equipment.Future value and present value techniques can be applied to a single cash flow (lump sum), ordinary annuities, annuities due, and uneven cash flow streams. Future and present values can be calculated using a regular calculator or a calculator with financial functions. When compounding occurs more frequently than once a year, the effective rate of interest is greater than the quoted rate.The cash flow time line is one of the most important tools in time value of money analysis. Cash flow time lines help to visualize what is happening in a particular problem. Cash flows are placed directly below the tick marks, and interest rates are shown directly above the time line; unknown cash flows are indicated by question marks. Thus, to find the future value of $100 after 5 years at 5 percent interest, the following cash flow time line can be set up: Time: 0 1 2 3 4 5| | | | | | Cash flows: -100 FV5 = ?◆ A cash outflow is a payment, or disbursement, of cash for expenses, investments, and so on.◆ A cash inflow is a receipt of cash from an investment, an employer, or other sources.5%CHAPTER 3: THE TIME VALUE OF MONEY40Compounding is the process of determining the value of a cash flow or series of cash flows some time in the future when compound interest is applied. The future value is the amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate. The future value can be calculated asFV n = PV(1 + k)n,where PV = present value, or beginning amount; k = interest rate per period; and n = number of periods involved in the analysis. This equation can be solved in one of two ways: numerically or with a financial calculator. For calculations, assume the following data that were presented in the time line above: present value (PV) = $100, interest rate (k) = 5%, and number of years (n) = 5.◆Compounded interest is interest earned on interest.◆To solve numerically, use a regular calculator to find 1 + k = 1.05 raised to the fifth power,which equals 1.2763. Multiply this figure by PV = $100 to get the final answer of FV5 = $127.63.◆With a financial calculator, the future value can be found by using the time value of moneyinput keys, where N = number of periods, I = interest rate per period, PV = present value, PMT = annuity payment, and FV = future value. By entering N = 5, I = 5, PV = -100, and PMT = 0, and then pressing the FV key, the answer 127.63 is displayed.♦Some financial calculators require that all cash flows be designated as either inflows or outflows, thus an outflow must be entered as a negative number (for example, PV= -100 instead of PV = 100).♦Some calculators require you to press a “Compute” key before pressing the FV key.◆ A graph of the compounding process shows how any sum grows over time at variousinterest rates. The greater the rate of interest, the faster is the rate of growth.♦The interest rate is, in fact, a growth rate.♦The time value concepts can be applied to anything that is growing.Finding the present value of a cash flow or series of cash flows is called discounting, and it is simply the reverse of compounding. In general, the present value is the value today of a future cash flow or series of cash flows. By solving for PV in the future value equation, the present value, or discounting, equation can be developed and written in several forms:CHAPTER 3: THE TIME VALUE OF MONEY41rate, one can utilize either of the two solution methods:♦Numerical solution: Divide $127.63 by 1.05 five times to get PV = $100.♦Financial calculator solution: Enter N = 5, I = 5, PMT = 0, and FV = 127.63, and then press the PV key to get PV = -100.◆The opportunity cost rate is the rate of return on the best available alternative investment ofequal risk.◆ A graph of the discounting process shows how the present value of any sum to be receivedin the future diminishes and approaches zero as the payment date is extended farther into the future. At relatively high interest rates, funds due in the future are worth very little today, and even at a relatively low discount rate, the present value of a sum due in the very distant future is quite small.The compounding and discounting processes are reciprocals, or inverses, of one another. In addition, there are four variables in the time value of money equations: PV, FV, k, and n. If three of the four variables are known, you can find the value of the fourth.◆If we are given PV, FV, and n, we can determine k by substituting the known values intoeither the present value or future value equations, and then solving for k. Thus, if you can buy a security at a price of $78.35 which will pay you $100 after 5 years, what is the interest rate earned on the investment?♦Numerical solution: Use a trial and error process to reach the 5% value for k. This is a tedious and inefficient process. Alternatively, you could use algebra to solve the timevalue equation.♦Financial calculator solution: Enter N = 5, PV = -78.35, PMT = 0, and FV = 100, then press the I key, and I = 5 is displayed.◆Likewise, if we are given PV, FV, and k, we can determine n by substituting the knownvalues into either the present value or future value equations, and then solving for n. Thus, if you can buy a security with a 5 percent interest rate at a price of $78.35 today, how long will it take for your investment to return $100?♦Numerical solution: Use a trial and error process to reach the value of 5 for n. This is a tedious and inefficient process. The equation can also be solved algebraically.♦Financial calculator solution: Enter I = 5, PV = -78.35, PMT = 0, and FV = 100, then press the N key, and N = 5 is displayed.An annuity is a series of equal payments made at fixed intervals for a specified number ofCHAPTER 3: THE TIME VALUE OF MONEY 42periods. If the payments occur at the end of each period, as they typically do, the annuity is an ordinary, or deferred, annuity. If the payments occur at the beginning of each period, it is called an annuity due. ◆ The future value of an ordinary annuity, FVA n , is the total amount one would have at theend of the annuity period if each payment were invested at a given interest rate and held to the end of the annuity period.♦ Defining FVA n as the future value of an ordinary annuity of n years, and PMT as the periodic payment, we can writethe FV key, and 315.25 is displayed.♦ For an annuity due, each payment is compounded for one additional period, so the future value of the entire annuity is equal to the future value of an ordinary annuity compounded for one additional period. Thus:FVA (DUE)n = PMT ⎥⎦⎤⎢⎣⎡+⨯⎭⎬⎫⎩⎨⎧-+)k 1(k 1)k 1(n .♦ Most financial calculators have a switch, or key, marked “DUE” or “BEG” that permitsyou to switch from end-of-period payments (an ordinary annuity) to beginning-of-period payments (an annuity due). Switch your calculator to “BEG” mode, and calculate as you would for an ordinary annuity. Do not forget to switch your calculator back to “END” mode when you are finished. ◆The present value of an ordinary annuity, PVA n , is the single (lump sum) payment today that would be equivalent to the annuity payments spread over the annuity period. It is the amount today that would permit withdrawals of an equal amount (PMT) at the end (or beginning for an annuity due) of each period for n periods.♦ Defining PVA n as the present value of an ordinary annuity of n years and PMT as the periodic payment, we can writePVA n = PMT ⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛+∑=n 1t t )k 1(1 = PMT ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+-k )k 1(11n = PMT ⎥⎦⎤⎢⎣⎡+--k )k 1(1n .CHAPTER 3: THE TIME VALUE OF MONEY43♦ Using a financial calculator, enter N = 3, I = 5, PMT = -100, and FV = 0, and then press the PV key, for an answer of $272.32.♦ One especially important application of the annuity concept relates to loans with constant payments, such as mortgages and auto loans. With these amortized loans the amount borrowed is the present value of an ordinary annuity, and the payments constitute the annuity stream. ◆The present value for an annuity due isPVA (DUE)n = PMT ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⨯⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧+-)k 1(k )k 1(11n .♦ Using a financial calculator, switch to the “BEG” mode, and then enter N = 3, I = 5,PMT = -100, and FV = 0, and then press PV to get the answer, $285.94. Again, do not forget to switch your calculator back to “END” mode when you are finished.◆You can solve for the interest rate (rate of return) earned on an annuity.♦ To solve numerically, you must use the trial-and-error process and plug in different values for k in the annuity equation to solve for the interest rate.♦ You can use the financial calculator by entering the appropriate values for N, PMT, and either FV or PV, and then pressing I to solve for the interest rate.◆You can solve for the number of periods (N) in an annuity.♦ To solve numerically, you must use the trial-and-error process and plug in different values for N in the annuity equation to solve for the number of periods.♦ You can use the financial calculator by entering the appropriate values for I, PMT, and either FV or PV, and then pressing N to solve for the number of periods.A perpetuity is a stream of equal payments expected to continue forever. ◆ The present value of a perpetuity is:PVP =kPMTrate Interest Payment =.♦ For example, if the interest rate were 12 percent, a perpetuity of $1,000 a year wouldhave a present value of $1,000/0.12 = $8,333.33. ◆ A consol is a perpetual bond issued by the British government to consolidate past debts; in general, any perpetual bond.CHAPTER 3: THE TIME VALUE OF MONEY44◆The value of a perpetuity changes dramatically when interest rates change.Many financial decisions require the analysis of uneven, or nonconstant, cash flows rather than a stream of fixed payments such as an annuity. An uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next.◆The term payment, PMT, designates constant cash flows, while the term CF designates cashflows in general, including uneven cash flows.◆The present value of an uneven cash flow stream is the sum of the PVs of the individualcash flows of the stream.♦The PV is found by applying the following general present value equation:PV = ∑=⎪⎪⎭⎫⎝⎛+n1 ttt)k1(1 CF.♦With a financial calculator, enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CF j, enter the appropriate interest rate, and then press theNPV key to obtain the PV of the cash flow stream.♦Be sure to clear the cash flow register before starting a new problem.◆Similarly, the future value of an uneven cash flow stream, or terminal value, is the sum ofthe FVs of the individual cash flows of the stream.♦The FV can be found by applying the following general future value equation:FV n = ∑=-+n 1 ttnt)k1(CF.♦Some calculators have a net future value (NFV) key which allows you to obtain the FV of an uneven cash flow stream.◆We generally are more interested in the present value of an asset’s cash flow stream than inthe future value because the prese nt value represents today’s value, which we can compare with the price of the asset.♦Once we know its present value, we can find the future value of an uneven cash flow stream by treating the present value as a lump sum amount and compounding it to thefuture period.◆If one knows the relevant cash flows, the effective interest rate can be calculated efficientlywith a financial calculator. Enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CF j, and then press the IRR key to obtain the interest rate of anCHAPTER 3: THE TIME VALUE OF MONEY45uneven cash flow stream.♦ IRR stands for internal rate of return , which is the return on an investment.Annual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. Semiannual, quarterly, and other compounding periods more frequent than on an annual basis are often used in financial transactions. Compounding on a nonannual basis requires an adjustment to both the compounding and discounting procedures discussed previously. Moreover, when comparing securities with different compounding periods, they need to be put on a common basis. This requires distinguishing between the simple, or quoted, interest rate and the effective annual rate. ◆ The simple , or quoted, interest rate is the contracted, or quoted, interest rate that is used tocalculate the interest paid per period. ◆ The periodic rate is the interest rate charged per period.Periodic rate = Stated annual interest rate/Number of periods per year.◆ The annual percentage rate, APR, is the periodic rate times the number of periods per year. ◆The effective annual rate, EAR, is the rate that would have produced the final compounded value under annual compounding. The effective annual rate is given by the following formula:Effective annual rate (EAR) = ,0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+where k SIMPLE is the simple, or quoted, interest rate (that is, the APR), and m is the numberof compounding periods (interest payments) per year. The EAR is useful in comparing securities with different compounding periods. ◆For example, to find the effective annual rate if the simple rate is 6 percent and semiannual compounding is used, we have:EAR = (1 + 0.06/2)2 – 1.0 = 6.09%.◆For annual compounding use the formula to find the future value of a single payment (lump sum):FV n = PV(1 + k)n .CHAPTER 3: THE TIME VALUE OF MONEY 46♦ When compounding occurs more frequently than once a year, use this formula:FV n = PV nm SIMPLE m k 1⨯⎪⎭⎫⎝⎛+.Here m is the number of times per year compounding occurs, and n is the number of years.◆The amount to which $1,000 will grow after 5 years if quarterly compounding is applied to a nominal 8 percent interest rate is found as follows:FV n = $1,000(1 + 0.08/4)(4)(5) = $1,000(1.02)20 = $1,485.95.♦ Financial calculator solution: Enter N = 20, I = 2, PV = -1000, and PMT = 0, and then press the FV key to find FV = $1,485.95.◆The present value of a 5-year future investment equal to $1,485.95, with an 8 percent nominal interest rate, compounded quarterly, is found as follows:.000,1$)02.1(95.485,1$PV /4)08.01(PV 95.485,1$20(4)(5)==+=♦ Financial calculator solution: Enter N = 20, I = 2, PMT = 0, and FV = 1485.95, and then press the PV key to find PV = -$1,000.00.◆In general, nonannual compounding can be handled one of two ways.♦ State everything on a periodic rather than on an annual basis. Thus, n = 6 periods rather than n = 3 years and k = 3% instead of k = 6% with semiannual compounding.♦ Find the effective annual rate (EAR) with the equation below and then use the EAR as the rate over the given number of years.EAR = .0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+An important application of compound interest involves amortized loans, which are paid offin equal installments over the life of the loan.◆The amount of each payment, PMT, is found using a financial calculator by entering N (number of years), I (interest rate), PV (amount borrowed), and FV = 0, and then pressing the PMT key to find the periodic payment.◆ Each payment consists partly of interest and partly of repayment of the amount borrowed (principal). This breakdown is often developed in a loan amortization schedule .CHAPTER 3: THE TIME VALUE OF MONEY47♦ The interest component is largest in the first period, and it declines over the life of the loan as the outstanding balance of the loan decreases.♦ The repayment of principal is smallest in the first period, and it increases thereafter.The text discussion has involved three different interest rates. It is important to understand their differences.◆The simple , or quoted, rate, k SIMPLE , is the interest rate quoted by borrowers and lenders. This quotation must include the number of compounding periods per year.♦ This rate is never shown on a time line, and it is never used as an input in a financial calculator unless compounding occurs only once a year. ♦ k SIMPLE = Periodic rate ⨯ m = Annual percentage rate = APR.◆The periodic rate, k PER , is the rate charged by a lender or paid by a borrower each interest period. Periodic rate = k PER = k SIMPLE /m.♦ The periodic rate is used for calculations in problems where two conditions hold: (1) payments occur on a regular basis more frequently than once a year, and (2) a payment is made on each compounding (or discounting) date.♦ The APR, or annual percentage rate, represents the periodic rate stated on an annual basis without considering interest compounding. The APR never is used in actual calculations; it is simply reported to borrowers. ◆The effective annual rate, EAR, is the rate with which, under annual compounding, we would obtain the same result as if we had used a given periodic rate with m compounding periods per year.♦ EAR is found as follows:EAR = .0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+In Appendix 3A we discuss using spreadsheets to solve time value of money problems.In Appendix 3B we discuss using interest tables to solve time value of money problems.In Appendix 3C we discuss how to generate a loan amortization schedule using a financial calculator.S ELF-TEST Q UESTIONSDefinitionalCHAPTER 3: THE TIME VALUE OF MONEY481.A(n) ______ _________ is a payment, or disbursement, of cash for expenses, investments,and so on.2.A(n) ______ ________ is a receipt of cash from an investment, an employer, or othersources.3._____________ is the process of determining the value of a cash flow or series of cashflows some time in the future.4.The ________ _______ is the amount to which a cash flow or series of cash flows willgrow over a given period of time when compounded at a given interest rate.5.The beginning value of an account or investment in a project is known as its ________________.ing a savings account as an example, the difference between the account’s present valueand its future value at the end of the period is due to __________ earned during the period.7.The expression PV(1 + k)n determines the ________ _______ of a sum at the end of ___periods.8.Finding the present value of a cash flow or series of cash flows is often referred to as_____________, and it is simply the reverse of the _____________ process.9.The _____________ ______ ______ is the rate of return on the best alternative investmentof equal risk.10. A series of equal payments at fixed intervals for a specified number of periods is a(n)_________. If the payments occur at the end of each period it is a(n) __________ annuity, while if the payments occur at the beginning of each period it is an annuity _____.11.A(n) ____________ is a stream of equal payments expected to continue forever.12.The term PMT designates __________ cash flows, while the term CF designates cash flowsin general, including ________ cash flows.13.The present value of an uneven cash flow stream is the _____ of the PVs of the individualcash flows of the stream.49 14.Since different types of investments use different compounding periods, it is important todistinguish between the quoted, or ________, interest rate and the ___________ annual interest rate, the rate that would have produced the final compound value under annual compounding.15.____________ time periods are used when payments occur within periods, instead of ateither the beginning or the end of periods.16.___________ loans are paid off in equal installments over their lifetime and are animportant application of compound interest.17.The __________ rate is equal to the simple interest rate divided by the number ofcompounding periods per year.Conceptual18.If a bank uses quarterly compounding for savings accounts, the simple interest rate will begreater than the effective annual rate (EAR).a. Trueb. False19.If money has time value (that is, k > 0), the future value of some amount of money willalways be more than the amount invested. The present value of some amount to be received in the future is always less than the amount to be received.a. Trueb. False20.You have determined the profitability of a planned project by finding the present value ofall the cash flows from that project. Which of the following would cause the project to look less appealing, that is, have a lower present value?a.The discount rate decreases.b.The cash flows are received in later years (further into the future).c.The discount rate increases.d.Statements b and c are correct.e.Statements a and b are correct.21.As the discount rate increases without limit, the present value of a future cash inflowa.Gets larger without limit.b.Stays unchanged.50c.Approaches zero.d.Gets smaller without limit; that is, approaches minus infinity.e.Goes to e k n.5122.Which of the following statements is correct?a.Except in situations where compounding occurs annually, the periodic interest rateexceeds the simple interest rate.b.The effective annual rate always exceeds the simple interest rate, no matter how few ormany compounding periods occur each year.c.If compounding occurs more frequently than once a year, and if payments are made attimes other than at the end of compounding periods, it is impossible to determinepresent or future values, even with a financial calculator. The reason is that under theseconditions, the basic assumptions of discounted cash flow analysis are not met.d.Assume that compounding occurs quarterly, that the simple interest rate is 8 percent,and that you need to find the present value of $1,000 due 10 months from today. Youcould get the correct answer by discounting the $1,000 at 8.2432 percent for 10/12ths ofa year.e.Statements a, b, c, and d are all false.S ELF-TEST P ROBLEMS(Note: In working these problems, you may get an answer which differs from ours by a few cents due to differences in rounding. This should not concern you; just choose the closest answer.)1.Assume that you purchase a 6-year, 8 percent savings certificate for $1,000. If interest iscompounded annually, what will be the value of the certificate when it matures?a.$630.17b. $1,469.33c. $1,677.10d. $1,586.87e. $1,766.332. A savings certificate similar to the one in the previous problem is available with theexception that interest is compounded semiannually. What is the difference between the ending value of the savings certificate compounded semiannually and the one compounded annually?a.The semiannual certificate is worth $14.16 more than the annual certificate.b.The semiannual certificate is worth $14.16 less than the annual certificate.c.The semiannual certificate is worth $21.54 more than the annual certificate.d.The semiannual certificate is worth $21.54 less than the annual certificate.e.The semiannual certificate is worth the same as the annual certificate.523. A friend promises to pay you $600 two years from now if you loan him $500 today. Whatannual interest rate is your friend offering?a. 7.55%b. 8.50%c. 9.54%d. 10.75%e. 11.25%4.At an inflation rate of 9 percent, the purchasing power of $1 would be cut in half in just over 8years (some calculators round to 9 years). How long, to the nearest year, would it take for the purchasing power of $1 to be cut in half if the inflation rate were only 4 percent?a. 12 yearsb. 15 yearsc. 18 yearsd. 20 yearse. 23 years5.You are offered an investment opportunity with the “guarantee” that your investment willdouble in 5 years. Assuming annual compounding, what annual rate of return would this investment provide?a. 40.00%b. 100.00%c. 14.87%d. 20.00%e. 18.74%6.You decide to begin saving toward the purchase of a new car in 5 years. If you put $1,000at the end of each of the next 5 years in a savings account paying 6 percent compounded annually, how much will you accumulate after 5 years?a. $6,691.13b. $5,637.09c. $1,338.23d. $5,975.32e. $5,731.947.Refer to Self-Test Problem 6. What would be the ending amount if the payments weremade at the beginning of each year?a. $6,691.13b. $5,637.09c.$1,338.23d. $5,975.32e. $5,731.948.Refer to Self-Test Problem 6. What would be the ending amount if $500 payments weremade at the end of each 6-month period for 5 years and the account paid 6 percent compounded semiannually?a. $6,691.13b.$5,637.09c. $1,338.23d. $5,975.32e. $5,731.949.Calculate the present value of $1,000 to be received at the end of 8 years. Assume aninterest rate of 7 percent.a. $582.01b. $1,718.19c. $531.82d. $5,971.30e. $649.3753 10.Jane Smith has $20,000 in a brokerage account, and she plans to contribute an additional$7,500 to the account at the end of every year. The brokerage account has an expected annual return of 8 percent. If Jane’s goal is to a ccumulate $375,000 in the account, how many years will it take for Jane to reach her goal?a. 5.20b. 10.00c. 12.50d. 16.33e. 18.4011. How much would you be willing to pay today for an investment that would return $800each year at the end of each of the next 6 years? Assume a discount rate of 5 percent.a. $5,441.53b. $4,800.00c. $3,369.89d. $4,060.55e. $4,632.3712.You have applied for a mortgage of $60,000 to finance the purchase of a new home. Thebank will require you to make annual payments of $7,047.55 at the end of each of the next20 years. Determine the interest rate in effect on this mortgage.a. 8.0%b. 9.8%c. 10.0%d. 5.1%e. 11.2%13.If you would like to accumulate $7,500 over the next 5 years, how much must you depositeach six months, starting six months from now, given a 6 percent interest rate and semiannual compounding?a. $1,330.47b. $879.23c. $654.23d. $569.00e. $732.6714. A company is offering bonds that pay $100 per year indefinitely. If you require a 12 percentreturn on these bonds (that is, the discount rate is 12 percent), what is the value of each bond?a. $1,000.00b. $962.00c. $904.67d. $866.67e.$833.3315.What is the present value (t = 0) of the following cash flows if the discount rate is 12percent?0 1 2 3 4 512%| | | | | |0 2,000 2,000 2,000 3,000 -4,000a.$4,782.43b. $4,440.51c. $4,221.79d. $4,041.23e. $3,997.9816.What is the effective annual percentage rate (EAR) of 12 percent compounded monthly?。

人教版高中英语必修三 (The Value of Money)Pronunciation 新课件

Learn
2）A：You willfinish the work，won'tyou？（↗） B：Yes，I will．（或No，I won't．）
A用升调提问，表示A心中没有把握，因此，得到的回答既可能是肯定的，也可能是否定的。
Learn
3．语调可以反应谈话人的互动性。如： A：Are you Mr．Blake？ B：Yes．（↙） A：Room twenty-six．
核心句式：强调句
教材原句 It was the ship that brought you to the England. 是这艘船把你带到了英国。句式讲解基本结构：It is/ was + 被强调部分(通常是主语、宾语或状语)+ that/ who(当强语且主语指人)+ 其他部分。 e.g. It was yesterday that he met Li.
1. Read the conversation with a partner and discuss what they intend to convey by using different intonation.
Owner： You know what? It’s a million-pound bank note. Waiter 1: Really? Waiter 2: Really! Waiter 3: Really?!
Jean用升调说“Sorry”，其意思是“I didn't hear you．Could you say that again， please？”
2）A：Jean，can you bring me the newspaper？ B：Sorry．（↙）

高等教育：个人理财 Section 1 (Time Value of Money)

Simple interest does not assume that interest is earned on interest.
If the previous example had assumed simple interest, the interest earned in years 1, 2, and 3 would only be $40 each year. Therefore, the future value would be $620 (not $629.86).
and why?
2-2
The Time Value of Money
A dollar today is worth more than a dollar in the future
Timeபைடு நூலகம்value terminology
Future value (FV) Present value (PV) Interest rate per compounding period (i) Number of compounding periods (n) Payment or annuity (PMT)
$3,000.
2-13
Present Value of a Single Amount
Interest Rate Period 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
1 0.990 0.981 0.971 0.962 0.952 0.943 0.935 0.926 0.917 2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 3 0.970 0.942 0.915 0.889 0.864 0.837 0.816 0.794 0.772

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2-3; CN19
You deposit $100 per year for three years, with the first deposit made one year from now (ordinary annuity) into an account earning 7% interest per year. How much will you have in the account after you make the last deposit?
$1000
$1000 (1.07)3
$1000
$934.58 873.44 816.30 762.90 712.99 $4,100.20
$1000 (1.07)4
$1000 (1.07)5
Time 0: No cash flow Time 1: First $1,000 payment; discount one period at 7%: $1,000/(1.07)1 $934.58 Time 2: Second $1,000 payment; discount two periods at 7%; $1,000/(1.07)2 873.44 Time 3: Third $1,000 payment; discount 2-11;three CN22 periods at 7%; $1,000/(1.07)3
Calculator solution: N I/Y
PV
PMT
FV
216 mos.
Annual PMT amount
.66 per mo.PMT =
0
CPT
$208.30
100,000
2,670.21 ≈ 222.52, which is > than 208.30 (the monthly payment) 12 mos.

Annuities DUE have the payments come due at the beginning of the period. Example: rent, commercial leases

2-2; CN19
Think of the future value of an annuity (FVA) as the SUM OF THE FUTURE VALUES OF EACH INDIVIDUAL CASH FLOW in the annuity stream. After all, each cash flow could be placed in a separate account, but when you go to withdraw your funds you will be concerned with the sum of all the future values together. We say that future values are ADDITIVE. FVIFA = FVIF – 1 = (1 + r)t – 1 r r Annuity future value = PMT x FVIFA
If
we call 1 – (1/(1 + r)t)
2-9; CN21
the present value interest
Example 2.11
(Present Value of an Ordinary Annuity and Annuity Due) 2.11a. is the end of each year for the next 5 years, discounted at (present value of an ordinary annuity) What
(Present Value of an Ordinary Annuity and Annuity Due) 2.11b (present value of an annuity due) If the payments were made at the beginning of each period, we bring the present value of each payment forward one period by multiplying (1 + r). This is called an annuity due. 4,100.20 x (1 + .07) = 4,387.21
PV1 = 100 = 100
(1.1)1 (1.1)2
+
PV2 = 100
+
PV3
(1.1)3
PVA
=
90.91 2-8; CN21 +
82.64
+
Example 2.10 (Cont.)
(Present Value of an Annuity)
Notice
how the present value of each payment decreases as the $100 is received farther and farther into the future (i.e., the PV of the first payment is greater than the second or third…). Thus, the present value of an annuity equals the sum of the present values of each payment in the annuity stream. That is what we mean by PVs being additive. We can show that PMT x [(1 – (1/(1 + r)t)] = Present value of r an Annuity (PVA)
PVADUE = PVAORD x (1 + r) = $4,100.20 x 1.07 = 4,387.21 • Likewise, the future value of an annuity due is calculated as follows: FVADUE = FVAORD x (1 + r)
2-13; CN23
Example 2.11 (Cont.)
2.11b Annuity Due
0 4 5
$1000 $0 0 (1.07)
1
2
3
$1000
$1000 (1.07)1
$1000
$1000 (1.07)2
$1000
$1000
$1000
$1,000.00 934.58 873.44 816.30 762.90 $4,387.21
Your display should NOT say “BGN”
N
I/Y
PV
PMT
FV
3 years
7 per yr.
0
-100
CPT
FVAORD =
$321.49
2-5; CN19
In order to accumulate $100,000 in the kid’s college fund by age 18, what is the annual payment that must be invested if your investment yields 8% per year? Start on the child’s first birthday (this is an ordinary annuity).
present value of $1,000 to be received at the
7%?
2-10; CN22
Example 2.11 (Cont.)
2.11a Ordinary Annuity
0 4
$0 $1000
1 5
$1000 $1000 1 (1.07)
2
3
$1000
$1000 (1.07)2
$1000 (1.07)3
$1000 (1.07)4
Time 0: First $1,000 payment; discount zero periods at 7%: $1,000/(1.07)0 $1,000.00 Time 1: Second $1,000 payment; discount one periods at 7%: $1,000/(1.07)1 = 934.58 Time 2: Third $1,000 payment; discount one periods at 7%; $1,000/(1.07)2 873.44 2-14; CN23
Annuities Annuity: A level stream of cash flows for a fixed period of time, a constant period apart.

ORDINARY annuities have the payments come at the end of the period. Example: car/house payment
0 $0
1
2
$100 x (1100 $100 x (1.07)1 $100.00 $100 x (1.07)2 107.00 114.49 2-4; CN19 FVA = $321.49
Use the financial calculator: FIRST: 2nd CLR TVM
=
=
Example 2.11 (Cont.)
• Rather than use the “BEG” feature on the financial calculator for an annuity due, just add one period of interest to the ordinary annuity result for each cash flow (to the overall total) as computed in 6.4a:

3 - Time Value of Money - Part 2 (Annuities, DCF Valuation)

Chapter 3 The Time Value of Money 货币的时间价值 财务管理(双语版) 教学课件

货币的时间价值

time value of money

财务管理基础课件：The Time Value of Money

Time value of money课后习题答案

第三章 货币的时间价值 chapter3- Time value of money

finance Time Value of Money(2)

finance Time Value of Money(2)

货币的时间价值Time Value of Money

chapter 2 time vaule of money

THE TIME VALUE OF MONEY

人教版高中英语必修三 (The Value of Money)Pronunciation 新课件

高等教育：个人理财 Section 1 (Time Value of Money)

Chapter 3 The Time Value of Money 货币的时间价值财务管理(双语版) 教学课件

第三章货币的时间价值 chapter3- Time value of money