1 Intuition behind the Rule There are three reasons why a dollar tomorrow is worth less than a dollar today Individuals prefer present consumption to future consumption. To induce people to consumption you have to offer them more in the future. When there is monetary inflation, the value of currency decreases over time. The gr greater the difference in value between a dollar today and a dollar tomorrow. If there is any uncertainty (risk) associated with the cash flow in the future, the less valued. Other things remaining equal, the value of cash flows in future time periods will decr - the preference for current consumption increases. - expected inflation increases. - the uncertainty in the cash flow increases.
2 Discounting and Compounding: The Mechanism For Factoring In These E The mechanism for factoring in these elements is the discount rate. Discount Rate: The discount rate is a rate at which present and future cash flows are incorporates - (1) the preference for current consumption (Greater preference... Higher Disc (2) expected inflation (Higher inflation... Higher Discount Rate) (3) the uncertainty in the future cash flows (Higher Risk... Higher Discount A higher discount rate will lead to a lower value for cash flows in the future. The discount rate is also an opportunity cost, since it captures the returns that an ind made on the next best opportunity. Discounting future cash flows converts them into cash flows in present value dollars discounting converts future cash flows into present cash flows, Compounding converts present cash flows into future cash flows.
3 Principle 1 Cash flows at different points in time cannot be compared and aggregated. All cash brought to the same point in time, before comparisons and aggregations are made.
4 Cashflow Types and the Mechanics of Discounting There are four types of cash flows - simple cash flows, annuities, growing annuities perpetuities and growing perpetuities.
5 A. Simple Cash Flows: A simple cash flow is a single cash flow in a specified future ti Cash Flow: CFt Thus, the future value, in five years, of $10,000 today, assuming a 10% discount rate Time Period: t The present value of this cash flow is- PV of Simple Cash Flow = CFt / (1+r)t Thus, the present value of $5,000 in five years, assuming a 10% discount rate, is - PV of $5,000 in five years = $5,000 / = $3,105 The future value of a cash flow is - FV of Simple Cash Flow = CF0 (1+ r)t FV of $10,000 = $10,000 * = $16,105
6 Ibbotson and Sinquefield, in a study of returns on stocks and bonds between Application 1: The power of compounding - Stocks, Bonds and Bills on the average made 12.4%, treasury bonds made 5.2% and treasury bills made table provides the future values of $ 100 invested in each category at the end of periods - 1, 5, 10, 20, 30 and 40 years. Holding Period Stocks T. Bonds T.Bills 1 $ $ $ $ $ $ $ $ $ $1, $ $ $3, $ $ $10, $ $411.52
7 Concept Check: Most pension plans allow individuals to decide where their pensions fu invested - stocks, bonds or money market accounts. Where would you choose to invest funds? Will your allocation change as you get older? Why?
8 The frequency of compounding affects the future and present values of cash flows For instance, a 10% annual interest rate, if there is semiannual compounding, works ou The following table provides the effective rates as a function of the compounding freque Application 2: The Frequency of Compounding interest rate can deviate significantly from the true interest rate Effective Interest Rate = = or 10.25% Frequency Rate t Formula Effective Annual Rate Annual 10% 1 r 10% Semi-Annual 10% 2 (1+r/2) % Monthly 10% 12 (1+r/12) % Daily 10% 365 (1+r/365) % Continuous 10% expr %
9 B. Annuities: An annuity is a constant cash flow that occurs at regular intervals for a fi time. Defining A to be the annuity, A A A A
10 The present value of an annuity can be calculated by taking each cash flow and discoun the present, and adding up the present values. Alternatively, there is a short cut that can calculation - PV of an Annuity = PV(A, r,n) = A 1-1 (1+ r) n r where, A = Annuity r = Discount Rate n = Number of years
11 Thus, the present value of an annuity of $1,000 for the next five years, assuming a disco is - PV of $1000 each year for next 5 years = $ (1.10) 5.10 The notation that will be used in the rest of these lecture notes for the present value of a PV(A,r,n).
12 The reverse of this problem, is when the present value is known and the annuity is to b A(PV,r,n). r Annuity given = A(PV,r, n) = PV 1 - (1
13 The future value of an end-of-the-period annuity can also be calculated as follows- FV of an Annuity = FV(A, r, n) = A (1 +r) n - 1 r Thus, the future value of $1,000 each year for the next five years, at the end of the fifth a 10% discount rate) - FV of $1,000 each year for next 5 years = $1000 (1.10) The notation that will be used for the future value of an annuity will be FV(A,r,n). Alter are given the future value and you are looking for an annuity - A(FV,r,n) in terms of no r Annuity given Future Value = A(FV,r,n) = FV (1+ r)
14 Application 2: Saving for college tuition Assume that you want to send your newborn child to a private college (when he years old). The tuition costs are $ 16000/year now and that these costs are expected to for the next 18 years. Assume that you can invest, after taxes, at 8%. Expected tuition cost/year 18 years from now = 16000*(1.05)18 = $38,506 PV of four years of tuition costs at $38,506/year = $38,506 * PV(A,8%,4 years)= $127 [For simplicity, I have assumed that tuition costs are frozen at these levels for four years can be allowed to keep growing at 5% a year and the present value can be computed.] If you need to set aside a lump sum now, the amount you would need to set aside woul Amount one needs to set apart now = $127,357/(1.08)18 = $31,916 If set aside as an annuity each year, starting one year from now - If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405
15 Application 3: How much is your full-time MBA worth? Assume that you were earning $40,000/year before entering program and that tuition c Expected salary is $ 54,000/year after graduation. You can invest money at 8%. For simplicity, assume that the first payment of $16,000 has to be made at the start of second payment one year later. PV Of Cost Of MBA = $16,000+16,000/ * PV(A,8%,2 years) = $102,14 Assume that you will work 30 years after graduation, and that the salary differential ( $40000) will continue through this period. PV of Benefits Before Taxes = $14,000 * PV(A,8%,30 years) = $157,609 This has to be discounted back two years - $157,609/1.082 = $135,124 The present value of getting an MBA is = $135,124 - $102,145 = $32,979
16 Some Interesting Questions: 1. How much would your salary increment have to be for you to break even on your M 2. Keeping the increment constant, how many years would you have to work to break e
17 Application 4: Refinancing your mortgage - Is it worth it? Assume that you have a thirty-year mortgage for $200,000 that carries an interest The mortgage was taken three years ago. Since then, assume that interest rates have com 7.50%, and that you are thinking of refinancing. The cost of refinancing is expected to b loan. (This cost includes the points on the loan.) Assume also that you can invest your f Monthly payment based upon 9% mortgage rate (0.75% monthly rate) = $200,000 * A(PV,0.75%,360 months) = $1,609 Monthly payment based upon 7.50% mortgage rate (0.625% monthly rate) = $200,000 * A(PV,0.625%,360 months) = $1,398 Monthly Savings from refinancing = $1,609 - $1,398 = $211
18 If you plan to remain in this house indefinitely, of Savings (at 6% annually; 0.5% a month) = $211 * PV(A,0.5%,324 months) = $33,815 The savings will last for 27 years - the remaining life of the existing mortgage. You will need to make payments for three additional years as a consequence of the refi of Additional Mortgage payments - years 28,29 and 30 = $1,398 * PV(A,0.5%,36 months)/ = $9,532 Refinancing Cost = 2.5% of $200,000 = $5,000 Total Refinancing Cost = $9,532 + $5,000 = $14,532
19 Follow-up Questions: 1. How many years would you have to live in this house for you break even on this refi 2. We've ignored taxes in this analysis. How would it impact your decision?
20 Application 5: The Value of a Straight Bond You are trying to value a straight bond with a fifteen year maturity and a 10.75% current interest rate on bonds of this risk level is 8.5%. PV of cash flows on bond = * PV(A,8.5%,15 years) / = $ If interest rates rise to 10%, PV of cash flows on bond = * PV(A,10%,15 years)+ 1000/ = $1, Percentage change in price = % If interest rate fall to 7%, PV of cash flows on bond = * PV(A,7%,15 years)+ 1000/ = $1, Percentage change in price = % This asymmetric response to interest rate changes is called convexity.
21 Application 6: Contrasting short term versus long term bonds You are valuing four bonds - 1 year, 5 year, 15 year and 30 year- with a coupon rate o Price Changes as a function of Bond Maturities 20.00% 15.00% % Change in Price 10.00% 5.00% 0.00% -5.00% % Change if rate drop to 7% % Change if rate increases to 10% % % Bond Maturity
22 Bond Pricing Proposition 1: The longer the maturity of a bond, the more sensitive it i interest rates.
23 Application 7: Contrasting low coupon and high coupon bonds You are valuing four different bonds all with the same maturity - 15 years, but diff rates - 0%, 5%, 10.75% and 12%.
24 Bond Price Changes as a function of Coupon Rates % Price Change 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% -5.00% % % % 0% 5% 10.75% 12% Coupon Rate % Change if ra drops to 7% % Change if ra increases to 10
25 Bond Pricing Proposition 2: The lower the coupon rate on the bond, the more sensiti changes in interest rates.
26 C. Growing Annuity: A growing annuity is a cash flow growing at a constant rat period of time. If A is the current cash flow, and g is the expected growth rate, the time growing annuity looks as follows Figure 3.8: A Growing Annuity A(1+g) A(1+g) 2 A(1+g) 3 A(1+g) n n Note that to qualify as a growing annuity, the growth rate in each period has to be the s growth rate in the prior period.
27 PRESENT VALUE OF GROWING ANNUITY 1 - PV of a Growing Annuity = A(1 + g) (1+ g)n (1 + r) n r - g The present value of a growing annuity can be estimated in all cases, but one - wher is equal to the discount rate. In that specific case, the present value is equal to the nominal sums of the annuities o without the growth effect. PV of a Growing Annuity for n years (when r=g) = n A Note also that this formulation works even when the growth rate is greater than the disc 1 Both the denominator and the numerator in the formula will be negative, yielding a positive present value.
28 Application 8: The value of a gold mine Consider the example of a gold mine, where you have the rights to the mine for th over which period you plan to extract 5,000 ounces of gold every year. The price per o currently, but it is expected to increase 3% a year. The appropriate discount rate is 10% value of the gold that will be extracted from this mine can be estimated as follows PV of extracted gold = $300*5000 * (1.03) 1 - (1.03)20 (1.10) =
29 of Extracted Gold as a function of Growth Rate $50,000,000 $45,000,000 $40,000,000 of Extracted Gold $35,000,000 $30,000,000 $25,000,000 $20,000,000 $15,000,000 $10,000,000 $5,000,000 $- 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% Growth Rate in Gold Prices
30 Concept Check: If both the growth rate and the discount rate go up by 1%, will the pre gold to be extracted from this mine increase or decrease?
31 D. Perpetuities: A perpetuity is a constant cash flow at regular intervals forever. The p perpetuity is- PV of Perpetuity = A r where A is the perpetuity. The future value of a perpetuity is infinite.
32 Application 9: Valuing a Console Bond A console bond is a bond that has no maturity and pays a fixed coupon. Assume 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows Value of Console Bond = $60 /.09 = $667
33 E. Growing Perpetuities: A growing perpetuity is a cash flow that is expected to grow rate forever. The present value of a growing perpetuity is - PV of Growing Perpetuity = CF 1 (r - g) where CF1 is the expected cash flow next year, g is the constant growth rate and r is the discount rate.
34 Application 10: Valuing a Stock with Stable Growth in Dividends Southwestern Bell paid dividends per share of $2.73 in Its earnings and div grown at 6% a year between 1988 and 1992, and are expected to grow at the same rate term. The rate of return required by investors on stocks of equivalent risk is 12.23%. Current Dividends per share = $2.73 Expected Growth Rate in Earnings and Dividends = 6% Discount Rate = 12.23% Value of Stock = $2.73 *1.06 / ( ) = $46.45
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