The-Time-Value-of-Money課件(PPT 69頁(yè))

1、The Time Value of Money9第1頁(yè)，共69頁(yè)。Chapter OutlineTime value associated with moneyDetermining future value based on number of periods over which funds are to be compounded at given interest ratePresent value based on current value of funds to be receivedTables for future and present values, their need

2、 in computations, determination of yieldCompounding or discounting occurring on a less than annual basis2第2頁(yè)，共69頁(yè)。Relationship to the Capital Outlay DecisionDetermine whether future benefits are sufficiently large to justify current outlaysMathematical tools help in making capital allocation decisio

3、ns3第3頁(yè)，共69頁(yè)。Future Value Single AmountMeasuring value of an amount that is allowed to grow at a given interest over a period of time is necessaryAssuming that the worth of $1,000 needs to be calculated after 4 years at a 10% interest per year, we have:1st year$1,000 X 1.10 = $1,1002nd year.$1,100 X

4、1.10 = $1,2103rd year$1,210 X 1.10 = $1,3314th year$1,331 X 1.10 = $1,4644第4頁(yè)，共69頁(yè)。Future Value Single Amount (contd)A generalized formula is: Where FV = Future valuePV = Present valuei = Interest raten = Number of periods;In the previous case, PV = $1,000, i = 10%, n = 4, hence;5第5頁(yè)，共69頁(yè)。Future Val

5、ue of $1(FVIF)6第6頁(yè)，共69頁(yè)。Future Value Single Amount (contd)In determining future value, the following can be used:Where = The interest factorIf $10,000 were invested for 10 years at 8%, the future value would be:7第7頁(yè)，共69頁(yè)。Present Value Single AmountA sum payable in the future is worth less today than

6、 the stated amountThe formula for the present value is derived from the original formula for future value:The present value can be determined by solving for a mathematical solution to the formula above, thus restating the formula as:Assuming8第8頁(yè)，共69頁(yè)。Present Value of $1(PVIF)9第9頁(yè)，共69頁(yè)。Relationship o

7、f Present and Future Value10第10頁(yè)，共69頁(yè)。Future Value AnnuityA series of consecutive payments or receipts of equal amountThe future value of each payment can be totaled to find the future value of an annuityAssuming, A = $1,000, n = 4, and i = 10%11第11頁(yè)，共69頁(yè)。Future Value of an Annuity of $1(FVIFA)12第12

8、頁(yè)，共69頁(yè)。Compounding Process for Annuity13第13頁(yè)，共69頁(yè)。Present Value Annuity Calculated by discounting each individual payment back to the present and then all of these payments are added upAssuming that A = $1,000, n = 4, i = 10%, we have: 14第14頁(yè)，共69頁(yè)。Presentation of Time Value RelationshipRequires vari

9、ous comparison which include:The relationship between present value and future valueThe relationship between the present value of a single amount and the present value of an annuityFuture value related to future value of annuity15第15頁(yè)，共69頁(yè)。Annuity Equaling a Future ValueAssuming that at a 10% intere

10、st rate, after 4 years, an $4,641 needs to accumulated:For n = 4, and i = 10%, is 4.641. This A equals $1,00016第16頁(yè)，共69頁(yè)。Annuity Equaling a Present ValueDetermining what size annuity can be equated to a given amount:Assuming n = 4, i = 6%:17第17頁(yè)，共69頁(yè)。Relationship of Present Value to Annuity18第18頁(yè)，共6

11、9頁(yè)。Annuity Equaling a Present Value (contd)Determining the necessary repayments on a loan:Assuming n 20, i = 8%, Total payments ($4,074 for 20 years).$81,480Repayment of principal. 40,000 Payments applied to interest.$41,48019第19頁(yè)，共69頁(yè)。Payoff Table for Loan (amortization table)20第20頁(yè)，共69頁(yè)。Review21第2

12、1頁(yè)，共69頁(yè)。Yield Present Value of a Single AmountTo calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:We see that for n = 4 and = 0.683, the interest rate or yield is 10%22第22頁(yè)，共69頁(yè)。Yield Present Value of a Single Amount (contd)Interpolation may also b

13、e used to find a more precise answerDifference between the value at the lowest interest rate and the designated valueThe exact value can be determined thus:23第23頁(yè)，共69頁(yè)。Yield Present Value of an AnnuityTo calculate the yield on an investment of $10,000, producing $1,490 per annum for 10 years:Hence:2

14、4第24頁(yè)，共69頁(yè)。Special Considerations in Time Value AnalysisCertain contractual agreements may require semiannual, quarterly, or monthly compounding periodsIn such cases, to determine n, multiply the number of years by the number of compounding periods during the yearThe factor for i is determined by di

15、viding the quoted annual interest rate by the number of compounding periods25第25頁(yè)，共69頁(yè)。CasesCase 1: Determine the future value of a $1,000 investment after 5 years at 8% annual interest compounded semiannuallyWhere, n = 5 X 2 = 10; i = 8% / 2 = 4%Case 2: Determine the present value of 20 quarterly p

16、ayments of $2,000 each to be received over the next 5 years, where i = 8% per annumWhere, n = 20; i = 20%26第26頁(yè)，共69頁(yè)。Patterns of PaymentTime value of money evolves around a number of different payment or receipt patternsAssume a contract involving payments of different amounts each year for a three-

17、year periodTo determine the present value, each payment is discounted to the present and then totaled(Assume 8% discount rate)27第27頁(yè)，共69頁(yè)。Deferred AnnuityAssume, a contract involving payments of different amounts each year for a three year periodAn annuity of $1,000 is paid at the end of each year f

18、rom the fourth through the eighth yearTo determine the present value of the cash flows at 8% discount rate To determine the annuity28第28頁(yè)，共69頁(yè)。Deferred Annuity (contd)To discount the $3,993 back to the present, which falls at the beginning of the fourth period, in effect, the equivalent of the end o

19、f the third period, it is discounted back three periods, at 8% interest rate29第29頁(yè)，共69頁(yè)。Deferred Annuity (contd)30第30頁(yè)，共69頁(yè)。Alternate Method to Compute Deferred AnnuityDetermine the present value factor of an annuity for the total time period, where n = 8, i = 8%, the PVIFA = 5.747Determine the pres

20、ent value factor of an annuity for the total time period (8) minus the deferred annuity period (5). Here, 8 5 = 3; n = 3; i = 8%. Thus the value is 2.577Subtracting the value in step 2 from the value of step 1, and multiplying by A;31第31頁(yè)，共69頁(yè)。Alternate Method to Compute Deferred Annuity (contd)$3,1

21、70 is the same answer for the present value of the annuity as that reached by the first methodThe present value of the five-year annuity is added up to the present value of the inflows over the first three years to arrive at: 32第32頁(yè)，共69頁(yè)。Formula AppendixFuture valuesingle amount . . (9-1) A Present

22、valuesingle amount . (9-3) BFuture valueannuity . . . . . . . (9-4a) CFuture valueannuity in advance . . . . . . . . . . . . . . . . . . . (9-4b) Present valueannuity . . . . . . . (9-5a) DReview of formulas (a)PPT 9-533第33頁(yè)，共69頁(yè)。Formula AppendixPresent valueannuity inadvance . . . . . . . . . . . .

23、 . . . . (9-5b) Annuity equalling a futurevalue . . . . . . . . . . . . . . . . . . (9-6a) CAnnuity in advance equalling a future value . . . . . . . . . . . . (9-6b) Annuity equalling a presentvalue . . . . . . . . . . . . . . . . . . (9-7a) DAnnuity in advance equalling a present value . . . . . .

24、 . . . . . (9-7b) Review of formulas (b)PPT 9-534第34頁(yè)，共69頁(yè)。Valuation and Rates of Return10第35頁(yè)，共69頁(yè)。Chapter OutlineValuation of assets, based on the present value of future cash flowsThe required rate of return in valuing an asset is based on the risk involvedBond valuation and its determinationStoc

25、k valuation and its determinationPrice-earnings ratio36第36頁(yè)，共69頁(yè)。Valuation of Financial Assets37第37頁(yè)，共69頁(yè)。Valuation ConceptsValuation of a financial asset is based on determining the present value of future cash flowsRequired rate of return (the discount rate)Depends on the markets perceived level o

26、f risk associated with the individual securityIt is also competitively determined among companies seeking financial capitalImplying that investors are willing to accept low return for low risk and vice versaEfficient use of capital in the past results in a lower required rate of return for investors

27、38第38頁(yè)，共69頁(yè)。Valuation of BondsA bond provides an annuity stream of interest payments and a principal payment at maturityCash flows are discounted at Y (yield to maturity).Value of Y is determined in the bond market.The price of the bond is:Equal to the present value of regular interest paymentsDisco

28、unted by the yield to maturity added to the present value of the principal39第39頁(yè)，共69頁(yè)。Valuation of Bonds (contd)Assuming interest payments ( ) = $100; principal payments at maturity ( ) = $1,000; yield to maturity (Y) = 10% and total number of periods (n) = 20. Thus, the price of binds ( );Where: =

29、Price of the bond; = Interest payments; = Principal payment at maturity; t = Number corresponding to a period (running from 1 to n); n = Number of periods; Y = Yield to maturity (or required rate of return)40第40頁(yè)，共69頁(yè)。Present Value of Interest PaymentsTo determine the present value of a $100 annuity

30、 for 20 years, with a discount rate of 10%We have:41第41頁(yè)，共69頁(yè)。Present Value of Principal Payment (Par Value) at MaturityPrincipal payment at maturity is used interchangeably with par value or face value of the bondDiscounting $1,000 back to the present at 10%, we have:The current price of the bond,

31、based on the present value of interest payments and the present value of the principal payment at maturity:Here, the price of the bond is essentially the same as its par, or stated value to be received at maturity of $1,00042第42頁(yè)，共69頁(yè)。Relationship Between Bond Prices and YieldsBond prices are invers

32、ely related to bond yields (move in opposite directions)As interest rates in the economy change, the price or value of a bond changes:if the required rate of return increases, the price of the bond will decreaseif the required rate of return decreases, the price of the bond will increasePPT 10-843第4

33、3頁(yè)，共69頁(yè)。 Bond price and required rate of return(yield to maturity)If the market rate is higher than the coupon rate (the annual interest payment divided by the par value), the bond will sell at discount (below par value)If the market rate is equal to the coupon rate, the bond will sell at par valueI

34、f the market rate is lower than the coupon rate, the bond will sell at premium ( above par value)44第44頁(yè)，共69頁(yè)。Concept of Yield to MaturityThe yield to maturity or the discount rate is the required rate of return required by bondholdersThree factors influence the required rate of return:Required real

35、rate of returnInflation premiumRisk premium45第45頁(yè)，共69頁(yè)。Concept of Yield to MaturityReal rate of return:Demanded by the investor against current use of the funds on a non-adjusted basisInflation premium: Compensation towards the negative effect of inflation on the value of a dollarRisk free rate of r

36、eturn compensates for the use of funds and loss due to inflationRisk Premium: Towards special risks of an investment46第46頁(yè)，共69頁(yè)。Risk Premium (contd)Business Risk: inability of the firm to retain its competitive position and stability and growthFinancial risk: inability of the firm to meet its debt o

37、bligations as and when dueAssuming the risk premium is 3%, an overall required rate of return of 10% can be computed;47第47頁(yè)，共69頁(yè)。Increase in Inflation PremiumAssume this goes up from 4 to 6%, with everything else being constantPresent value of interest payments: $100 annuity for 20 years at a discou

38、nt rate of 12%;48第48頁(yè)，共69頁(yè)。Increase in Inflation Premium (contd)Present value of principal payment at maturity: Present value of $1,000 after 20 years at a discount rate of 12%;Total present value: Assuming that increase inflation increases required rate of return and decreases the bond price by $15

39、0 approximately 49第49頁(yè)，共69頁(yè)。Decrease in Inflation PremiumAssuming that the inflation premium declines:The required rate of return decrease to 8%, where the 20 year bond with a 10% interest rate would now sell for;Present value of interest payments50第50頁(yè)，共69頁(yè)。Decrease in Inflation Premium (contd)Pres

40、ent value of principal payment at maturity Total present value51第51頁(yè)，共69頁(yè)。Bond Price Table52第52頁(yè)，共69頁(yè)。Time to MaturityInfluences the impact of a change in yield to maturity on valuationLonger the maturity, the greater the impact of changes in yield53第53頁(yè)，共69頁(yè)。Impact of Time to Maturity on Bond Price

41、s54第54頁(yè)，共69頁(yè)。Determining Yield to Maturity from the Bond PriceThe yield to maturity (Y), that will equate the interest payments ( ) and the principal payments ( ) to the price of the bond ( )Assuming that a 15 year bond pays $110 per year (11%) in interest and $1,000 after 15 years in principal repa

42、yment Choosing an initial percentage to try as a discount rate, we have:55第55頁(yè)，共69頁(yè)。1,3001,2001,1001,000900800 700Bond Price ($)302515Number of years to maturity* The relationship in the graph is not symmetrical in nature.10% bond, $1,000 par valueAssumes 12% yield to maturity50Assumes 8% yield to m

43、aturityPPT 10-10Relationship between time to maturity and bond price*56第56頁(yè)，共69頁(yè)。Compute the yield to maturityTrial and error processInterpolation methodA less exact calculation of the yield to maturity Principal -Price of the bond Approximate Yield = Annual interest payment + Number of years to mat

44、urity to Maturity . 6 (Price of the bond) +.4( Principal payment) 57第57頁(yè)，共69頁(yè)。Semiannual Interest and Bond PricesA 10% interest rate may be paid as $50 twice a year in the case of semiannual paymentsTo make the conversion:Divide the annual interest rate by twoMultiply the number of years by twoDivid

45、e the annual yield to maturity by twoAssuming a 10%, $1,000 par value bond has a maturity of 20 years, the annual yield at 12%:10%/2 = 5% semiannual interest rate; hence 5% X $1,000 = %50 semiannual interest20 X 2 = 40 periods to maturity12%/2 = 6% yield to maturity, expressed on a semiannual basis5

46、8第58頁(yè)，共69頁(yè)。Semiannual Interest and Bond Prices (contd)At a present value of a $50 annuity for the 40 periods, at discount rate of 6%:Present value of interest paymentsPresent value of principal payment at maturityTotal present value59第59頁(yè)，共69頁(yè)。Valuation of Preferred StockPreferred stock:usually repr

47、esents a perpetuity (something with no maturity date)has a fixed dividend paymentis valued without any principal payment since it has no ending lifeis considered a hybrid securityowners have a higher priority of claim than common shareholdersprice is based upon PV of future dividendsPPT 10-1260第60頁(yè)，

48、共69頁(yè)。Valuation of Preferred Stock61第61頁(yè)，共69頁(yè)。Determining the Rate of Return (Yield) from the Market PriceAssuming the annual preferred dividend ( ) is $10 and the price of the preferred stock ( ) is $100, the required rate of return (yield):A higher market price provides quite a decline in the yield: 62第62頁(yè)，共69頁(yè)。Valuation of Common StockThe value of common stock is the present value of a stream of future dividendsCommon stock dividends can vary, unlike preferred stock dividendsThere are 3 possible case