The Time Value of Money.ppt

1、The Time Value of Money,9,1-2,Chapter Outline,Time value associated with money Determining future value based on number of periods over which funds are to be compounded at given interest rate Present value based on current value of funds to be received Tables for future and present values, their nee

2、d in computations, determination of yield Compounding or discounting occurring on a less than annual basis,1-3,Relationship to the Capital Outlay Decision,Determine whether future benefits are sufficiently large to justify current outlays Mathematical tools help in making capital allocation decision

3、s,1-4,Future Value Single Amount,Measuring value of an amount that is allowed to grow at a given interest over a period of time is necessary Assuming 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 1.1

4、0 = $1,2103rd year$1,210 X 1.10 = $1,3314th year$1,331 X 1.10 = $1,464,1-5,Future Value Single Amount (contd),A generalized formula is: 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;,1-6,Future Value of $1

5、(FVIF),1-7,Future Value Single Amount (contd),In determining future value, the following can be used: Where = The interest factor If $10,000 were invested for 10 years at 8%, the future value would be:,1-8,Present Value Single Amount,A sum payable in the future is worth less today than the stated am

6、ount The 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: Assuming,1-9,Present Value of $1(PVIF),1-10,Relationship of Present and Future

7、 Value,1-11,Future Value Annuity,A series of consecutive payments or receipts of equal amount The future value of each payment can be totaled to find the future value of an annuity Assuming, A = $1,000, n = 4, and i = 10%,1-12,Future Value of an Annuity of $1(FVIFA),1-13,Compounding Process for Annu

8、ity,1-14,Present Value Annuity,Calculated by discounting each individual payment back to the present and then all of these payments are added up Assuming that A = $1,000, n = 4, i = 10%, we have:,1-15,Presentation of Time Value Relationship,Requires various comparison which include: The relationship

9、 between present value and future value The relationship between the present value of a single amount and the present value of an annuity Future value related to future value of annuity,1-16,Annuity Equaling a Future Value,Assuming that at a 10% interest rate, after 4 years, an $4,641 needs to accum

10、ulated: For n = 4, and i = 10%, is 4.641. This A equals $1,000,1-17,Annuity Equaling a Present Value,Determining what size annuity can be equated to a given amount: Assuming n = 4, i = 6%:,1-18,Relationship of Present Value to Annuity,1-19,Annuity Equaling a Present Value (contd),Determining the nec

11、essary repayments on a loan: Assuming n 20, i = 8%, Total payments ($4,074 for 20 years).$81,480 Repayment of principal. 40,000 Payments applied to interest.$41,480,1-20,Payoff Table for Loan (amortization table),1-21,Review,1-22,Yield Present Value of a Single Amount,To calculate the yield on an in

12、vestment 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%,1-23,Yield Present Value of a Single Amount (contd),Interpolation may also be used to find a more precise answer Difference between the value at the lowest i

13、nterest rate and the designated value The exact value can be determined thus:,1-24,Yield Present Value of an Annuity,To calculate the yield on an investment of $10,000, producing $1,490 per annum for 10 years: Hence:,1-25,Special Considerations in Time Value Analysis,Certain contractual agreements m

14、ay require semiannual, quarterly, or monthly compounding periods In such cases, to determine n, multiply the number of years by the number of compounding periods during the year The factor for i is determined by dividing the quoted annual interest rate by the number of compounding periods,1-26,Cases

15、,Case 1: Determine the future value of a $1,000 investment after 5 years at 8% annual interest compounded semiannually Where, n = 5 X 2 = 10; i = 8% / 2 = 4% Case 2: Determine the present value of 20 quarterly payments of $2,000 each to be received over the next 5 years, where i = 8% per annum Where

16、, n = 20; i = 20%,1-27,Patterns of Payment,Time value of money evolves around a number of different payment or receipt patterns Assume a contract involving payments of different amounts each year for a three-year period To determine the present value, each payment is discounted to the present and th

17、en totaled (Assume 8% discount rate),1-28,Deferred Annuity,Assume, a contract involving payments of different amounts each year for a three year period An annuity of $1,000 is paid at the end of each year from the fourth through the eighth year To determine the present value of the cash flows at 8%

18、discount rate,To determine the annuity,1-29,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 of the third period, it is discounted back three periods, at 8% interest rate,1-30,Deferred Annuity

19、 (contd),1-31,Alternate Method to Compute Deferred Annuity,Determine the present value factor of an annuity for the total time period, where n = 8, i = 8%, the PVIFA = 5.747 Determine the present value factor of an annuity for the total time period (8) minus the deferred annuity period (5). Here, 8

20、5 = 3; n = 3; i = 8%. Thus the value is 2.577 Subtracting the value in step 2 from the value of step 1, and multiplying by A;,1-32,Alternate Method to Compute Deferred Annuity (contd),$3,170 is the same answer for the present value of the annuity as that reached by the first method The present value

21、 of the five-year annuity is added up to the present value of the inflows over the first three years to arrive at:,1-33,Formula Appendix Future valuesingle amount . . (9-1) A Present valuesingle amount . (9-3) B Future valueannuity . . . . . . . (9-4a) C Future valueannuity in advance . . . . . . .

22、. . . . . . . . . . . . (9-4b) Present valueannuity . . . . . . . (9-5a) D,Review of formulas (a),PPT 9-5,1-34,Formula Appendix Present valueannuity in advance . . . . . . . . . . . . . . . . (9-5b) Annuity equalling a future value . . . . . . . . . . . . . . . . . . (9-6a) C Annuity in advance equa

23、lling a future value . . . . . . . . . . . . (9-6b) Annuity equalling a present value . . . . . . . . . . . . . . . . . . (9-7a) D Annuity in advance equalling a present value . . . . . . . . . . . (9-7b) ,Review of formulas (b),PPT 9-5,Valuation and Rates of Return,10,1-36,Chapter Outline,Valuation

24、 of assets, based on the present value of future cash flows The required rate of return in valuing an asset is based on the risk involved Bond valuation and its determination Stock valuation and its determination Price-earnings ratio,1-37,Valuation of Financial Assets,1-38,Valuation Concepts,Valuati

25、on of a financial asset is based on determining the present value of future cash flows Required rate of return (the discount rate) Depends on the markets perceived level of risk associated with the individual security It is also competitively determined among companies seeking financial capital Impl

26、ying that investors are willing to accept low return for low risk and vice versa Efficient use of capital in the past results in a lower required rate of return for investors,1-39,Valuation of Bonds,A bond provides an annuity stream of interest payments and a principal payment at maturity Cash flows

27、 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 payments Discounted by the yield to maturity added to the present value of the principal,1-40,Valuation of Bonds (contd),Assuming interest p

28、ayments ( ) = $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: = Price of the bond; = Interest payments; = Principal payment at maturity; t = Number corresponding to a period (running from 1 to n

29、); n = Number of periods; Y = Yield to maturity (or required rate of return),1-41,Present Value of Interest Payments,To determine the present value of a $100 annuity for 20 years, with a discount rate of 10% We have:,1-42,Present Value of Principal Payment (Par Value) at Maturity,Principal payment a

30、t maturity is used interchangeably with par value or face value of the bond Discounting $1,000 back to the present at 10%, we have: The current price of the bond, based on the present value of interest payments and the present value of the principal payment at maturity: Here, the price of the bond i

31、s essentially the same as its par, or stated value to be received at maturity of $1,000,1-43,Relationship Between Bond Prices and Yields,Bond prices are inversely related to bond yields (move in opposite directions) As interest rates in the economy change, the price or value of a bond changes: if th

32、e required rate of return increases, the price of the bond will decrease if the required rate of return decreases, the price of the bond will increase,PPT 10-8,1-44,Bond price and required rate of return(yield to maturity),If the market rate is higher than the coupon rate (the annual interest paymen

33、t 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 value If the market rate is lower than the coupon rate, the bond will sell at premium ( above par value),1-45,Concept of Yield to Maturity,The yield

34、 to maturity or the discount rate is the required rate of return required by bondholders Three factors influence the required rate of return: Required real rate of return Inflation premium Risk premium,1-46,Concept of Yield to Maturity,Real rate of return: Demanded by the investor against current us

35、e of the funds on a non-adjusted basis Inflation premium: Compensation towards the negative effect of inflation on the value of a dollar Risk free rate of return compensates for the use of funds and loss due to inflation Risk Premium: Towards special risks of an investment,1-47,Risk Premium (contd),

36、Business Risk: inability of the firm to retain its competitive position and stability and growth Financial risk: inability of the firm to meet its debt obligations as and when due Assuming the risk premium is 3%, an overall required rate of return of 10% can be computed;,1-48,Increase in Inflation P

37、remium,Assume this goes up from 4 to 6%, with everything else being constant Present value of interest payments: $100 annuity for 20 years at a discount rate of 12%;,1-49,Increase in Inflation Premium (contd),Present value of principal payment at maturity: Present value of $1,000 after 20 years at a

38、 discount rate of 12%; Total present value: Assuming that increase inflation increases required rate of return and decreases the bond price by $150 approximately,1-50,Decrease in Inflation Premium,Assuming that the inflation premium declines: The required rate of return decrease to 8%, where the 20

39、year bond with a 10% interest rate would now sell for; Present value of interest payments,1-51,Decrease in Inflation Premium (contd),Present value of principal payment at maturity Total present value,1-52,Bond Price Table,1-53,Time to Maturity,Influences the impact of a change in yield to maturity o

40、n valuation Longer the maturity, the greater the impact of changes in yield,1-54,Impact of Time to Maturity on Bond Prices,1-55,Determining Yield to Maturity from the Bond Price,The yield to maturity (Y), that will equate the interest payments ( ) and the principal payments ( ) to the price of the b

41、ond ( ) Assuming that a 15 year bond pays $110 per year (11%) in interest and $1,000 after 15 years in principal repayment Choosing an initial percentage to try as a discount rate, we have:,1-56,1,300 1,200 1,100 1,000 900 800 700,Bond Price ($),30,25,15,Number of years to maturity,* The relationshi

42、p in the graph is not symmetrical in nature.,10% bond, $1,000 par value,Assumes 12% yield to maturity,5,0,Assumes 8% yield to maturity,PPT 10-10,Relationship between time to maturity and bond price*,1-57,Compute the yield to maturity,Trial and error process Interpolation method A less exact calculat

43、ion of the yield to maturity Principal -Price of the bond Approximate Yield = Annual interest payment + Number of years to maturity to Maturity . 6 (Price of the bond) +.4( Principal payment),1-58,Semiannual Interest and Bond Prices,A 10% interest rate may be paid as $50 twice a year in the case of

44、semiannual payments To make the conversion: Divide the annual interest rate by two Multiply the number of years by two Divide the annual yield to maturity by two Assuming 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

45、 $1,000 = %50 semiannual interest 20 X 2 = 40 periods to maturity 12%/2 = 6% yield to maturity, expressed on a semiannual basis,1-59,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 payments Present

46、value of principal payment at maturity Total present value,1-60,Valuation of Preferred Stock,Preferred stock: usually represents a perpetuity (something with no maturity date) has a fixed dividend payment is valued without any principal payment since it has no ending life is considered a hybrid secu

47、rity owners have a higher priority of claim than common shareholders price is based upon PV of future dividends,PPT 10-12,1-61,Valuation of Preferred Stock,1-62,Determining the Rate of Return (Yield) from the Market Price,Assuming the annual preferred dividend ( ) is $10 and the price of the preferr

48、ed stock ( ) is $100, the required rate of return (yield): A higher market price provides quite a decline in the yield:,1-63,Valuation of Common Stock,The value of common stock is the present value of a stream of future dividends Common stock dividends can vary, unlike preferred stock dividends There are 3 possible cases