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This document provides an overview of key concepts related to cash flows and valuation. It discusses the accountant's approach to the statement of cash flows, which explains changes in cash balance rather than measuring firm value. The financial analyst's approach is more concerned with cash flows to equity and the firm. Present and future value concepts are introduced, along with discount rates and how they relate to risk. Annuities, including growing annuities, are defined and the formulas for present and future value are provided. Applications to valuing bonds and saving for college are discussed. The document aims to explain fundamental valuation concepts.

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Fundamentals of Valuation P.V. Viswanath Based on Damodaran’s Corporate Finance
P.V. Viswanath 2 Cash Flows: The Accountant’s Approach  The objective of the Statement of Cash Flows, prepared by accountants, is to explain changes in the cash balance rather than to measure the health or value of the firm
P.V. Viswanath 3 The Statement of Cash Flows Cash Flows From Operations + Cash Flows From Investing + Cash Flows from Financing Net cash flow from operations, after taxes and interest expenses Includes divestiture and acquisition of real assets (capital expenditures) and disposal and purchase of financial assets. Also includes acquisitions of other firms. Net cash flow from the issue and repurchase of equity, from the issue and repayment of debt and after dividend payments = Net Change in Cash Balance Figure 4.3: Statement of Cash Flows
P.V. Viswanath 4 Cash Flows: The Financial Analyst’s Approach  In financial analysis, we are much more concerned about  Cash flows to Equity: These are the cash flows generated by the asset after all expenses and taxes, and also after payments due on the debt. Cash flows to equity, which are after cash flows to debt but prior to cash flows to equity  Cash flow to Firm: This cash flow is before debt payments but after operating expenses and taxes. This looks at not just the equity investor in the asset, but at the total cash flows generated by the asset for both the equity investor and the lender.  These cash flow measures can be used to value assets, the firm’s equity and the entire firm itself.
P.V. Viswanath 5 Present and Future Value  Present Value – earlier money on a time line  Future Value – later money on a time line 0 2 3 5 4 6 1 100 100 100 100 100 100  If a project yields $100 a year for 6 years, we may want to know the value of those flows as of year 1; then the year 1 value would be a present value.  If we want to know the value of those flows as of year 6, that year 6 value would be a future value.  If we wanted to know the value of the year 4 payment of $100 as of year 2, then we are thinking of the year 4 money as future value, and the year 2 dollars as present value.
P.V. Viswanath 6 Rates and Prices  A rate is a “price” used to convert earlier money into later money, and vice-versa.  If $1 of today’s money is equal in value to $1.05 of next period’s money, then the conversion rate is 0.05 or 5%.  Equivalently, the price of today’s dollar in terms of next period money is 1.05. The excess of next period’s monetary value over this period’s value (1.05 – 1.00 or 0.05) is often referred to, as interest.  The price of next period’s money in terms of today’s money would be 1/1.05 or 95.24 cents.  This price reflects two elements: (1) Preference for current consumption (Greater =>Higher Discount Rate) (2) the uncertainty in the future cash flows (Higher Risk =>Higher Discount Rate)
P.V. Viswanath 7 Rate Terminology  Interest rate – “exchange rate” between earlier money and later money (normally the later money is certain).  Discount Rate – rate used to convert future value to present value.  Compounding rate – rate used to convert present value to future value.  Cost of capital – the rate at which the firm obtains funds for investment.  Opportunity cost of capital – the rate that the firm has to pay investors in order to obtain an additional $ of funds.  Required rate of return – the rate of return that investors demand for providing the firm with funds for investment.
P.V. Viswanath 8 Relation between rates  If capital markets are in equilibrium, the rate that the firm has to pay to obtain additional funds will be equal to the rate that investors will demand for providing those funds. This will be “the” market rate.  Hence this is the rate that should be used to convert future values to present values and vice-versa.  Hence this should be the discount rate used to convert future project (or security) cashflows into present values.
P.V. Viswanath 9 Two essential concepts 1. Cash flows at different points in time cannot be compared and aggregated. All cash flows have to be brought to the same point in time, before comparisons and aggregations are made. 2. The concept of a Time Line:
P.V. Viswanath 10 Discount Rates and Risk  In reality there is no single discount rate that can be used to evaluate all future cashflows.  The reason is that future cashflows differ not only in terms of when they occur, but also in terms of riskiness.  Hence, one needs to either convert future risky cashflows into certainty-equivalent cashflows, or, as is more commonly done, add a risk premium to the “certain-future-cashflows” discount rate to get the discount rate appropriate for risky-future- cashflows.
P.V. Viswanath 11 Discounted Cashflow Valuation where,  n = life of the asset  CFt = cashflow in period t  r = discount rate reflecting the riskiness of the estimated cashflows Value = CFt (1+ r)t t =1 t = n 
P.V. Viswanath 12 Cash Flow Types and Discounting Mechanics  There are five types of cash flows -  simple cash flows,  annuities,  growing annuities  perpetuities and  growing perpetuities
P.V. Viswanath 13 I. Simple Cash Flows  A simple cash flow is a single cash flow in a specified future time period. Cash Flow: CFt ________________________________________|____ Time Period: t  The present value of this cash flow is- PV of Simple Cash Flow = CFt / (1+r)t  The future value of a cash flow is - FV of Simple Cash Flow = CF0 (1+ r)t
P.V. Viswanath 14 Application: The power of compounding - Stocks, Bonds and Bills  Between 1926 and 1998, Ibbotson Associates found that stocks on the average made about 11% a year, while government bonds on average made about 5% a year.  If your holding period is one year,the difference in end-of-period values is small:  Value of $ 100 invested in stocks in one year = $ 111  Value of $ 100 invested in bonds in one year = $ 105
P.V. Viswanath 15 Holding Period and Value
P.V. Viswanath 16 The Frequency of Compounding  The frequency of compounding affects the future and present values of cash flows. The stated interest rate can deviate significantly from the true interest rate –  For instance, a 10% annual interest rate, if there is semiannual compounding, works out to- Effective Interest Rate = 1.052 - 1 = .10125 or 10.25%  The general formula is Effective Annualized Rate = (1+r/m)m – 1 where m is the frequency of compounding (# times per year), and r is the stated interest rate (or annualized percentage rate (APR) per year
P.V. Viswanath 17 The Frequency of Compounding Frequency Rate t Formula Effective Annual Rate Annual 10% 1 r 10.00% Semi-Annual 10% 2 (1+r/2)2-1 10.25% Monthly 10% 12 (1+r/12)12-1 10.47% Daily 10% 365 (1+r/365)365-1 10.52% Continuous 10% er-1 10.52%
P.V. Viswanath 18 II. Annuities  An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. Defining A to be the annuity, A A A A | | | | 0 1 2 3 4
P.V. Viswanath 19 Present Value of an Annuity  The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]           n r r A n r A PV Annuity an of PV ) 1 ( 1 1 ) , , (
P.V. Viswanath 20 Example: PV of an Annuity  The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -  The notation that will be used in the rest of these lecture notes for the present value of an annuity will be PV(A,r,n). PVof $1000 each year for next 5 years = $1000 1 - 1 (1.10)5 .10          $3,791
P.V. Viswanath 21 Annuity, given Present Value  The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n). Annuity given Present Value = A(PV, r,n) = PV r 1 - 1 (1 + r)n        
P.V. Viswanath 22 Computing Monthly Payment on a Mortgage  Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%.  The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation:  Monthly interest rate on loan = APR/12 = 0.08/12 = 0.0067 Monthly Payment on Mortgage = $200,000 0.0067 1 - 1 (1.0067)360          $1473.11
P.V. Viswanath 23 Future Value of an Annuity  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      
P.V. Viswanath 24 An Example  Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -  The notation that will be used for the future value of an annuity will be FV(A,r,n). FVof $1, 000 each year for next 5 years = $1000 (1.10)5 - 1 .10       = $6,105
P.V. Viswanath 25 Annuity, given Future Value  If you are given the future value and you are looking for an annuity - A(FV,r,n) in terms of notation - Annuity given Future Value = A(FV, r,n) = FV r (1+ r)n - 1       Note, however, that the two formulas, Annuity, given Future Value and Present Value, given annuity can be derived from each other, quite easily. You may want to simply work with a single formula.
P.V. Viswanath 26 Application : Saving for College Tuition  Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $16000/year now and that these costs are expected to rise 5% a year 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) = $127,537  If you need to set aside a lump sum now, the amount you would need to set aside would be -  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
P.V. Viswanath 27 Valuing a Straight Bond  You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%. PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) + 1000/1.08515 = $ 1186.85  If interest rates rise to 10%, PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015 = $1,057.05 Percentage change in price = -10.94%  If interest rate fall to 7%, PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 = $1,341.55 Percentage change in price = +13.03%
P.V. Viswanath 28 III. Growing Annuity  A growing annuity is a cash flow growing at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity looks as follows –
P.V. Viswanath 29 Present Value of a Growing Annuity  The present value of a growing annuity can be estimated in all cases, but one - where the growth rate is equal to the discount rate, using the following model:  In that specific case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect. PVof an Annuity = PV(A,r,g,n) = A(1 +g) 1 - (1+g) n (1+r)n (r - g)              
P.V. Viswanath 30 The Value of a Gold Mine  Consider the example of a gold mine, where you have the rights to the mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The price per ounce is $300 currently, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold that will be extracted from this mine can be estimated as follows – PVof extracted gold = $300* 5000 * (1.03) 1 - (1.03)20 (1.10)20 .10 - .03            $16,145,980
P.V. Viswanath 31 IV. Perpetuity  A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is- PV of Perpetuity = A r
P.V. Viswanath 32 Valuing a Consol Bond  A consol bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows - Value of Consol Bond = $60 / .09 = $667
P.V. Viswanath 33 V. Growing Perpetuities  A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is - where  CF1 is the expected cash flow next year,  g is the constant growth rate and  r is the discount rate. PV of Growing Perpetuity = CF1 (r - g)
P.V. Viswanath 34 Valuing a Stock with Growing Dividends  Southwestern Bell paid dividends per share of $2.73 in 1992. Its earnings and dividends have grown at 6% a year between 1988 and 1992, and are expected to grow at the same rate in the long 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 / (.1223 -.06) = $46.45
P.V. Viswanath 35 Two Measures of Discount Rates  Cost of Equity: This is the rate of return required by equity investors on an investment. It will incorporate a premium for equity risk -the greater the risk, the greater the premium. This is used to value equity.  Cost of capital: This is a composite cost of all of the capital invested in an asset or business. It will be a weighted average of the cost of equity and the after-tax cost of borrowing. This is used to value the entire firm.
P.V. Viswanath 36 Equity Valuation Assets Liabilities Assets in Place Debt Equity Discount rate reflects only the cost of raising equity financing Growth Assets Figure 5.5: Equity Valuation Cash flows considered are cashflows fromassets, after debt payments and after making reinvestments needed for future growth Present value is value of just the equity claims on the firm Free Cash Flow to Equity = Net Income – Net Reinvestment – Net Debt Paid (or + Net Debt Issued)
P.V. Viswanath 37 Valuing Equity in a Finite Life Asset  Assume that you are trying to value the Home Depot’s equity investment in a new store.  Assume that the cash flows from the store after debt payments and reinvestment needs are expected will be $850,000 a year, growing at 5% a year for the next 12 years.  In addition, assume that the salvage value of the store, after repaying remaining debt will be $ 1 million.  Finally, assume that the cost of equity is 9.78%. Value of Equity in Store = 850,000 (1.05) 1 - (1.05)12 (1.0978)12         (.0978 -.05) + 1,000,000 (1.0978)12 = $8,053,999
P.V. Viswanath 38 Firm Valuation Assets Liabilities Assets in Place Debt Equity Discount rate reflects the cost of raising both debt and equity financing, in proportion to their use Growth Assets Figure 5.6: Firm Valuation Cash flows considered are cashflows fromassets, prior to any debt payments but after firmhas reinvested to create growth assets Present value is value of the entire firm, and reflects the value of all claims on the firm. Free Cash Flow to the Firm = Earnings before Interest and Taxes (1-tax rate) – Net Reinvestment Net Reinvestment is defined as actual expenditures on short-term and long-term assets less depreciation. The tax benefits of debt are not included in FCFF because they are taken into account in the firm’s cost of capital.
P.V. Viswanath 39 Valuing a Finite-Life Asset  Consider the Home Depot's investment in a proposed store. The store is assumed to have a finite life of 12 years and is expected to have cash flows before debt payments and after reinvestment needs of $ 1 million, growing at 5% a year for the next 12 years.  The store is also expected to have a value of $ 2.5 million at the end of the 12th year (called the salvage value).  The Home Depot's cost of capital is 9.51%.
P.V. Viswanath 40 Expected Cash Flows and present value Year Expecte d Cash Flows Value at End PV at 9.51% 1 $ 1,050 ,000 $ 958 ,817 2 $ 1,102 ,500 $ 919 ,329 3 $ 1,157 ,625 $ 881 ,468 4 $ 1,215 ,506 $ 845 ,166 5 $ 1,276 ,282 $ 810 ,359 6 $ 1,340 ,096 $ 776 ,986 7 $ 1,407 ,100 $ 744 ,987 8 $ 1,477 ,455 $ 714 ,306 9 $ 1,551 ,328 $ 684 ,888 10 $ 1,628 ,895 $ 656 ,682 11 $ 1,710 ,339 $ 629 ,638 12 $ 1,795 ,856 $ 2,500 ,000 $ 1,444,124 Value of St ore = $ 10 ,066,749
P.V. Viswanath 41 Valuation with Infinite Life Cash flows Firm: Pre-debt cash flow Equity: After debt cash flows Expected Growth Firm: Growth in Operating Earnings Equity: Growth in Net Income/EPS CF1 CF2 CF3 CF4 CF5 Forever Firm is in stable growth: Grows at constant rate forever Terminal Value CFn ......... Discount Rate Firm:Cost of Capital Equity: Cost of Equity Value Firm: Value of Firm Equity: Value of Equity DISCOUNTED CASHFLOW VALUATION Length of Period of High Growth
P.V. Viswanath 42 Valuing the Home Depot’s Equity  Assume that we expect the free cash flows to equity at Home Depot to grow for the next 10 years at rates much higher than the growth rate for the economy. To estimate the free cash flows to equity for the next 10 years, we make the following assumptions:  The net income of $1,614 million will grow 15% a year each year for the next 10 years.  The firm will reinvest 75% of the net income back into new investments each year, and its net debt issued each year will be 10% of the reinvestment.  To estimate the terminal price, we assume that net income will grow 6% a year forever after year 10. Since lower growth will require less reinvestment, we will assume that the reinvestment rate after year 10 will be 40% of net income; net debt issued will remain 10% of reinvestment.
P.V. Viswanath 43 Estimating cash flows to equity: The Home Depot Year Net Income Reinvestment Needs Net Debt Paid FCFE PV of FCFE 1 $ 1,856 $ 1,392 $ (139) $ 603 $ 549 2 $ 2,135 $ 1,601 $ (160) $ 694 $ 576 3 $ 2,455 $ 1,841 $ (184) $ 798 $ 603 4 $ 2,823 $ 2,117 $ (212) $ 917 $ 632 5 $ 3,246 $ 2,435 $ (243) $ 1,055 $ 662 6 $ 3,733 $ 2,800 $ (280) $ 1,213 $ 693 7 $ 4,293 $ 3,220 $ (322) $ 1,395 $ 726 8 $ 4,937 $ 3,703 $ (370) $ 1,605 $ 761 9 $ 5,678 $ 4,258 $ (426) $ 1,845 $ 797 10 $ 6,530 $ 4,897 $ (490) $ 2,122 $ 835 Sum of PV of FCFE = $6,833
P.V. Viswanath 44 Terminal Value and Value of Equity today  FCFE11 = Net Income11 – Reinvestment11 – Net Debt Paid (Issued)11 = $6,530 (1.06) – $6,530 (1.06) (0.40) – (-277) = $ 4,430 million  Terminal Price10 = FCFE11/(ke – g) = $ 4,430 / (.0978 - .06) = $117,186 million  The value per share today can be computed as the sum of the present values of the free cash flows to equity during the next 10 years and the present value of the terminal value at the end of the 10th year. Value of the Stock today = $ 6,833 million + $ 117,186/(1.0978)10 = $52,927 million
P.V. Viswanath 45 Valuing Boeing as a firm  Assume that you are valuing Boeing as a firm, and that Boeing has cash flows before debt payments but after reinvestment needs and taxes of $ 850 million in the current year.  Assume that these cash flows will grow at 15% a year for the next 5 years and at 5% thereafter.  Boeing has a cost of capital of 9.17%.
P.V. Viswanath 46 Expected Cash Flows and Firm Value  Terminal Value = $ 1710 (1.05)/(.0917-.05) = $ 43,049 million Year Cash Flow Terminal Value Present Value 1 $978 $895 2 $1,124 $943 3 $1,293 $994 4 $1,487 $1,047 5 $1,710 $43,049 $28,864 Value of Boeing as a firm = $32,743

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valuation.ppt

  • 1.
    Fundamentals of Valuation P.V.Viswanath Based on Damodaran’s Corporate Finance
  • 2.
    P.V. Viswanath 2 CashFlows: The Accountant’s Approach  The objective of the Statement of Cash Flows, prepared by accountants, is to explain changes in the cash balance rather than to measure the health or value of the firm
  • 3.
    P.V. Viswanath 3 TheStatement of Cash Flows Cash Flows From Operations + Cash Flows From Investing + Cash Flows from Financing Net cash flow from operations, after taxes and interest expenses Includes divestiture and acquisition of real assets (capital expenditures) and disposal and purchase of financial assets. Also includes acquisitions of other firms. Net cash flow from the issue and repurchase of equity, from the issue and repayment of debt and after dividend payments = Net Change in Cash Balance Figure 4.3: Statement of Cash Flows
  • 4.
    P.V. Viswanath 4 CashFlows: The Financial Analyst’s Approach  In financial analysis, we are much more concerned about  Cash flows to Equity: These are the cash flows generated by the asset after all expenses and taxes, and also after payments due on the debt. Cash flows to equity, which are after cash flows to debt but prior to cash flows to equity  Cash flow to Firm: This cash flow is before debt payments but after operating expenses and taxes. This looks at not just the equity investor in the asset, but at the total cash flows generated by the asset for both the equity investor and the lender.  These cash flow measures can be used to value assets, the firm’s equity and the entire firm itself.
  • 5.
    P.V. Viswanath 5 Presentand Future Value  Present Value – earlier money on a time line  Future Value – later money on a time line 0 2 3 5 4 6 1 100 100 100 100 100 100  If a project yields $100 a year for 6 years, we may want to know the value of those flows as of year 1; then the year 1 value would be a present value.  If we want to know the value of those flows as of year 6, that year 6 value would be a future value.  If we wanted to know the value of the year 4 payment of $100 as of year 2, then we are thinking of the year 4 money as future value, and the year 2 dollars as present value.
  • 6.
    P.V. Viswanath 6 Ratesand Prices  A rate is a “price” used to convert earlier money into later money, and vice-versa.  If $1 of today’s money is equal in value to $1.05 of next period’s money, then the conversion rate is 0.05 or 5%.  Equivalently, the price of today’s dollar in terms of next period money is 1.05. The excess of next period’s monetary value over this period’s value (1.05 – 1.00 or 0.05) is often referred to, as interest.  The price of next period’s money in terms of today’s money would be 1/1.05 or 95.24 cents.  This price reflects two elements: (1) Preference for current consumption (Greater =>Higher Discount Rate) (2) the uncertainty in the future cash flows (Higher Risk =>Higher Discount Rate)
  • 7.
    P.V. Viswanath 7 RateTerminology  Interest rate – “exchange rate” between earlier money and later money (normally the later money is certain).  Discount Rate – rate used to convert future value to present value.  Compounding rate – rate used to convert present value to future value.  Cost of capital – the rate at which the firm obtains funds for investment.  Opportunity cost of capital – the rate that the firm has to pay investors in order to obtain an additional $ of funds.  Required rate of return – the rate of return that investors demand for providing the firm with funds for investment.
  • 8.
    P.V. Viswanath 8 Relationbetween rates  If capital markets are in equilibrium, the rate that the firm has to pay to obtain additional funds will be equal to the rate that investors will demand for providing those funds. This will be “the” market rate.  Hence this is the rate that should be used to convert future values to present values and vice-versa.  Hence this should be the discount rate used to convert future project (or security) cashflows into present values.
  • 9.
    P.V. Viswanath 9 Twoessential concepts 1. Cash flows at different points in time cannot be compared and aggregated. All cash flows have to be brought to the same point in time, before comparisons and aggregations are made. 2. The concept of a Time Line:
  • 10.
    P.V. Viswanath 10 DiscountRates and Risk  In reality there is no single discount rate that can be used to evaluate all future cashflows.  The reason is that future cashflows differ not only in terms of when they occur, but also in terms of riskiness.  Hence, one needs to either convert future risky cashflows into certainty-equivalent cashflows, or, as is more commonly done, add a risk premium to the “certain-future-cashflows” discount rate to get the discount rate appropriate for risky-future- cashflows.
  • 11.
    P.V. Viswanath 11 DiscountedCashflow Valuation where,  n = life of the asset  CFt = cashflow in period t  r = discount rate reflecting the riskiness of the estimated cashflows Value = CFt (1+ r)t t =1 t = n 
  • 12.
    P.V. Viswanath 12 CashFlow Types and Discounting Mechanics  There are five types of cash flows -  simple cash flows,  annuities,  growing annuities  perpetuities and  growing perpetuities
  • 13.
    P.V. Viswanath 13 I.Simple Cash Flows  A simple cash flow is a single cash flow in a specified future time period. Cash Flow: CFt ________________________________________|____ Time Period: t  The present value of this cash flow is- PV of Simple Cash Flow = CFt / (1+r)t  The future value of a cash flow is - FV of Simple Cash Flow = CF0 (1+ r)t
  • 14.
    P.V. Viswanath 14 Application:The power of compounding - Stocks, Bonds and Bills  Between 1926 and 1998, Ibbotson Associates found that stocks on the average made about 11% a year, while government bonds on average made about 5% a year.  If your holding period is one year,the difference in end-of-period values is small:  Value of $ 100 invested in stocks in one year = $ 111  Value of $ 100 invested in bonds in one year = $ 105
  • 15.
    P.V. Viswanath 15 HoldingPeriod and Value
  • 16.
    P.V. Viswanath 16 TheFrequency of Compounding  The frequency of compounding affects the future and present values of cash flows. The stated interest rate can deviate significantly from the true interest rate –  For instance, a 10% annual interest rate, if there is semiannual compounding, works out to- Effective Interest Rate = 1.052 - 1 = .10125 or 10.25%  The general formula is Effective Annualized Rate = (1+r/m)m – 1 where m is the frequency of compounding (# times per year), and r is the stated interest rate (or annualized percentage rate (APR) per year
  • 17.
    P.V. Viswanath 17 TheFrequency of Compounding Frequency Rate t Formula Effective Annual Rate Annual 10% 1 r 10.00% Semi-Annual 10% 2 (1+r/2)2-1 10.25% Monthly 10% 12 (1+r/12)12-1 10.47% Daily 10% 365 (1+r/365)365-1 10.52% Continuous 10% er-1 10.52%
  • 18.
    P.V. Viswanath 18 II.Annuities  An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. Defining A to be the annuity, A A A A | | | | 0 1 2 3 4
  • 19.
    P.V. Viswanath 19 PresentValue of an Annuity  The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [A = Annuity; r = Discount Rate; n = Number of years]           n r r A n r A PV Annuity an of PV ) 1 ( 1 1 ) , , (
  • 20.
    P.V. Viswanath 20 Example:PV of an Annuity  The present value of an annuity of $1,000 at the end of each year for the next five years, assuming a discount rate of 10% is -  The notation that will be used in the rest of these lecture notes for the present value of an annuity will be PV(A,r,n). PVof $1000 each year for next 5 years = $1000 1 - 1 (1.10)5 .10          $3,791
  • 21.
    P.V. Viswanath 21 Annuity,given Present Value  The reverse of this problem, is when the present value is known and the annuity is to be estimated - A(PV,r,n). Annuity given Present Value = A(PV, r,n) = PV r 1 - 1 (1 + r)n        
  • 22.
    P.V. Viswanath 22 ComputingMonthly Payment on a Mortgage  Suppose you borrow $200,000 to buy a house on a 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%.  The monthly payments on this loan, with the payments occurring at the end of each month, can be calculated using this equation:  Monthly interest rate on loan = APR/12 = 0.08/12 = 0.0067 Monthly Payment on Mortgage = $200,000 0.0067 1 - 1 (1.0067)360          $1473.11
  • 23.
    P.V. Viswanath 23 FutureValue of an Annuity  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      
  • 24.
    P.V. Viswanath 24 AnExample  Thus, the future value of $1,000 at the end of each year for the next five years, at the end of the fifth year is (assuming a 10% discount rate) -  The notation that will be used for the future value of an annuity will be FV(A,r,n). FVof $1, 000 each year for next 5 years = $1000 (1.10)5 - 1 .10       = $6,105
  • 25.
    P.V. Viswanath 25 Annuity,given Future Value  If you are given the future value and you are looking for an annuity - A(FV,r,n) in terms of notation - Annuity given Future Value = A(FV, r,n) = FV r (1+ r)n - 1       Note, however, that the two formulas, Annuity, given Future Value and Present Value, given annuity can be derived from each other, quite easily. You may want to simply work with a single formula.
  • 26.
    P.V. Viswanath 26 Application: Saving for College Tuition  Assume that you want to send your newborn child to a private college (when he gets to be 18 years old). The tuition costs are $16000/year now and that these costs are expected to rise 5% a year 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) = $127,537  If you need to set aside a lump sum now, the amount you would need to set aside would be -  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
  • 27.
    P.V. Viswanath 27 Valuinga Straight Bond  You are trying to value a straight bond with a fifteen year maturity and a 10.75% coupon rate. The current interest rate on bonds of this risk level is 8.5%. PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) + 1000/1.08515 = $ 1186.85  If interest rates rise to 10%, PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015 = $1,057.05 Percentage change in price = -10.94%  If interest rate fall to 7%, PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 = $1,341.55 Percentage change in price = +13.03%
  • 28.
    P.V. Viswanath 28 III.Growing Annuity  A growing annuity is a cash flow growing at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity looks as follows –
  • 29.
    P.V. Viswanath 29 PresentValue of a Growing Annuity  The present value of a growing annuity can be estimated in all cases, but one - where the growth rate is equal to the discount rate, using the following model:  In that specific case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect. PVof an Annuity = PV(A,r,g,n) = A(1 +g) 1 - (1+g) n (1+r)n (r - g)              
  • 30.
    P.V. Viswanath 30 TheValue of a Gold Mine  Consider the example of a gold mine, where you have the rights to the mine for the next 20 years, over which period you plan to extract 5,000 ounces of gold every year. The price per ounce is $300 currently, but it is expected to increase 3% a year. The appropriate discount rate is 10%. The present value of the gold that will be extracted from this mine can be estimated as follows – PVof extracted gold = $300* 5000 * (1.03) 1 - (1.03)20 (1.10)20 .10 - .03            $16,145,980
  • 31.
    P.V. Viswanath 31 IV.Perpetuity  A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is- PV of Perpetuity = A r
  • 32.
    P.V. Viswanath 32 Valuinga Consol Bond  A consol bond is a bond that has no maturity and pays a fixed coupon. Assume that you have a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows - Value of Consol Bond = $60 / .09 = $667
  • 33.
    P.V. Viswanath 33 V.Growing Perpetuities  A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is - where  CF1 is the expected cash flow next year,  g is the constant growth rate and  r is the discount rate. PV of Growing Perpetuity = CF1 (r - g)
  • 34.
    P.V. Viswanath 34 Valuinga Stock with Growing Dividends  Southwestern Bell paid dividends per share of $2.73 in 1992. Its earnings and dividends have grown at 6% a year between 1988 and 1992, and are expected to grow at the same rate in the long 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 / (.1223 -.06) = $46.45
  • 35.
    P.V. Viswanath 35 TwoMeasures of Discount Rates  Cost of Equity: This is the rate of return required by equity investors on an investment. It will incorporate a premium for equity risk -the greater the risk, the greater the premium. This is used to value equity.  Cost of capital: This is a composite cost of all of the capital invested in an asset or business. It will be a weighted average of the cost of equity and the after-tax cost of borrowing. This is used to value the entire firm.
  • 36.
    P.V. Viswanath 36 EquityValuation Assets Liabilities Assets in Place Debt Equity Discount rate reflects only the cost of raising equity financing Growth Assets Figure 5.5: Equity Valuation Cash flows considered are cashflows fromassets, after debt payments and after making reinvestments needed for future growth Present value is value of just the equity claims on the firm Free Cash Flow to Equity = Net Income – Net Reinvestment – Net Debt Paid (or + Net Debt Issued)
  • 37.
    P.V. Viswanath 37 ValuingEquity in a Finite Life Asset  Assume that you are trying to value the Home Depot’s equity investment in a new store.  Assume that the cash flows from the store after debt payments and reinvestment needs are expected will be $850,000 a year, growing at 5% a year for the next 12 years.  In addition, assume that the salvage value of the store, after repaying remaining debt will be $ 1 million.  Finally, assume that the cost of equity is 9.78%. Value of Equity in Store = 850,000 (1.05) 1 - (1.05)12 (1.0978)12         (.0978 -.05) + 1,000,000 (1.0978)12 = $8,053,999
  • 38.
    P.V. Viswanath 38 FirmValuation Assets Liabilities Assets in Place Debt Equity Discount rate reflects the cost of raising both debt and equity financing, in proportion to their use Growth Assets Figure 5.6: Firm Valuation Cash flows considered are cashflows fromassets, prior to any debt payments but after firmhas reinvested to create growth assets Present value is value of the entire firm, and reflects the value of all claims on the firm. Free Cash Flow to the Firm = Earnings before Interest and Taxes (1-tax rate) – Net Reinvestment Net Reinvestment is defined as actual expenditures on short-term and long-term assets less depreciation. The tax benefits of debt are not included in FCFF because they are taken into account in the firm’s cost of capital.
  • 39.
    P.V. Viswanath 39 Valuinga Finite-Life Asset  Consider the Home Depot's investment in a proposed store. The store is assumed to have a finite life of 12 years and is expected to have cash flows before debt payments and after reinvestment needs of $ 1 million, growing at 5% a year for the next 12 years.  The store is also expected to have a value of $ 2.5 million at the end of the 12th year (called the salvage value).  The Home Depot's cost of capital is 9.51%.
  • 40.
    P.V. Viswanath 40 ExpectedCash Flows and present value Year Expecte d Cash Flows Value at End PV at 9.51% 1 $ 1,050 ,000 $ 958 ,817 2 $ 1,102 ,500 $ 919 ,329 3 $ 1,157 ,625 $ 881 ,468 4 $ 1,215 ,506 $ 845 ,166 5 $ 1,276 ,282 $ 810 ,359 6 $ 1,340 ,096 $ 776 ,986 7 $ 1,407 ,100 $ 744 ,987 8 $ 1,477 ,455 $ 714 ,306 9 $ 1,551 ,328 $ 684 ,888 10 $ 1,628 ,895 $ 656 ,682 11 $ 1,710 ,339 $ 629 ,638 12 $ 1,795 ,856 $ 2,500 ,000 $ 1,444,124 Value of St ore = $ 10 ,066,749
  • 41.
    P.V. Viswanath 41 Valuationwith Infinite Life Cash flows Firm: Pre-debt cash flow Equity: After debt cash flows Expected Growth Firm: Growth in Operating Earnings Equity: Growth in Net Income/EPS CF1 CF2 CF3 CF4 CF5 Forever Firm is in stable growth: Grows at constant rate forever Terminal Value CFn ......... Discount Rate Firm:Cost of Capital Equity: Cost of Equity Value Firm: Value of Firm Equity: Value of Equity DISCOUNTED CASHFLOW VALUATION Length of Period of High Growth
  • 42.
    P.V. Viswanath 42 Valuingthe Home Depot’s Equity  Assume that we expect the free cash flows to equity at Home Depot to grow for the next 10 years at rates much higher than the growth rate for the economy. To estimate the free cash flows to equity for the next 10 years, we make the following assumptions:  The net income of $1,614 million will grow 15% a year each year for the next 10 years.  The firm will reinvest 75% of the net income back into new investments each year, and its net debt issued each year will be 10% of the reinvestment.  To estimate the terminal price, we assume that net income will grow 6% a year forever after year 10. Since lower growth will require less reinvestment, we will assume that the reinvestment rate after year 10 will be 40% of net income; net debt issued will remain 10% of reinvestment.
  • 43.
    P.V. Viswanath 43 Estimatingcash flows to equity: The Home Depot Year Net Income Reinvestment Needs Net Debt Paid FCFE PV of FCFE 1 $ 1,856 $ 1,392 $ (139) $ 603 $ 549 2 $ 2,135 $ 1,601 $ (160) $ 694 $ 576 3 $ 2,455 $ 1,841 $ (184) $ 798 $ 603 4 $ 2,823 $ 2,117 $ (212) $ 917 $ 632 5 $ 3,246 $ 2,435 $ (243) $ 1,055 $ 662 6 $ 3,733 $ 2,800 $ (280) $ 1,213 $ 693 7 $ 4,293 $ 3,220 $ (322) $ 1,395 $ 726 8 $ 4,937 $ 3,703 $ (370) $ 1,605 $ 761 9 $ 5,678 $ 4,258 $ (426) $ 1,845 $ 797 10 $ 6,530 $ 4,897 $ (490) $ 2,122 $ 835 Sum of PV of FCFE = $6,833
  • 44.
    P.V. Viswanath 44 TerminalValue and Value of Equity today  FCFE11 = Net Income11 – Reinvestment11 – Net Debt Paid (Issued)11 = $6,530 (1.06) – $6,530 (1.06) (0.40) – (-277) = $ 4,430 million  Terminal Price10 = FCFE11/(ke – g) = $ 4,430 / (.0978 - .06) = $117,186 million  The value per share today can be computed as the sum of the present values of the free cash flows to equity during the next 10 years and the present value of the terminal value at the end of the 10th year. Value of the Stock today = $ 6,833 million + $ 117,186/(1.0978)10 = $52,927 million
  • 45.
    P.V. Viswanath 45 ValuingBoeing as a firm  Assume that you are valuing Boeing as a firm, and that Boeing has cash flows before debt payments but after reinvestment needs and taxes of $ 850 million in the current year.  Assume that these cash flows will grow at 15% a year for the next 5 years and at 5% thereafter.  Boeing has a cost of capital of 9.17%.
  • 46.
    P.V. Viswanath 46 ExpectedCash Flows and Firm Value  Terminal Value = $ 1710 (1.05)/(.0917-.05) = $ 43,049 million Year Cash Flow Terminal Value Present Value 1 $978 $895 2 $1,124 $943 3 $1,293 $994 4 $1,487 $1,047 5 $1,710 $43,049 $28,864 Value of Boeing as a firm = $32,743

Editor's Notes

  • #4 This show all cash flows. The objective is to explain changes in the cash balance rather than to measure the health or value of the firm.
  • #5 Cash flows to debt: interest payments, new debt issues, debt repayments Cash flows to equity: dividends, stock buybacks, new stock issues
  • #6 It’s important to point out that there are many different ways to refer to the interest rate that we use in time value of money calculations. Students often get confused with the terminology, especially since they tend to think of an “interest rate” only in terms of loans and savings accounts.
  • #10 All to often, much of what we read about cash flow violates this principle. As I write this note, I am reading that Alex Rodriguez has signed a 10-year contract for $ 252 million. While he will no doubt be a wealthy man, the contract’s value cannot be correctly estimated by adding cash flows over time…
  • #12 The value of any asset is the present value of the expected cash flows on the assets.
  • #13 For purely mechanical purposes, this is a way of categorizing cash flows for calculating present value.
  • #14 T can be a fraction of a year (say 0.25 years) or include fractions of years.. T can even be stated in terms of months or days, as long as the interest rate is specified accordingly (as a monthly or daily rate)
  • #16 As the holding period lengthens, the end-of-the-period value differences get larger.
  • #17 The loan sharks of yesteryear were on to something when they charged daily rates… A 1% daily rate translates into an annual rate of 3678%.
  • #19 For a cashflow to be an annuity, it has to be The same amount in each period The periods have to remain fixed (monthly, annual)
  • #20 Good place to link up to the financial calculator. The present value button does wonders…
  • #21 May help to separate the second term in the equation and state it as a present value factor. In the example above, the present value factor is 3.791. For those using present value tables, this makes the link to the present value equation. What if the cash flows had been at the beginning of each year? Present value = $ 1000 + $ 1000 (PV(A, 10%, 4 years) = $ 3,791 (1.1)
  • #22 Reverses the process. Here, we look for an annuity given a present value (Examples: Figuring out the monthly payment on a mortgage)
  • #23 Follow-up questions: If you pay at the beginning of each month, rather than the end, how much could you shorten the mortgage period? $1473 + 1473(PV(A,.67%,n)) = 200000; Trial and error yields n=350: Save 10 months of payments
  • #24 This is how much the value will be on the last day (when the last annuity comes due)
  • #25 Again, worth asking how much the future value would be if the cash flows were at the beginning of each year.
  • #26 Reverses the problem
  • #27 The assumptions of constant tuition each year for 4 years, and that the payments are made at the end of each year (no university will accept that payment plan) are purely for simplicity.
  • #28 A straight bond is a combination of an annuity and a simple cash flow. The interest rate used should reflect the risk of the bond. If it is a government bond, it will be a default-free interest rate. If it is a corporate bond, it should be a rate that reflects the default risk. We have used one interest rate to discount all of the cash flows. A more precise valuation may be obtained by discounting each cash flow at a rate that reflects when that cash flow comes in.. A 1-year rate for the 1-year cash flow, a 2-year rate for the 2-year cash flow…
  • #29 This is usually a cash flow that cannot be found on a financial calculator but is very common with equity investments…
  • #30 This formula will work for all cases, except when g=r. When g>r, both the numerator and denominator become negative allowing you to still compute the present value. When g=r, the present value= nA (the growth and discount rate effects cancel out)
  • #31 You could always compute the present value by taking each cash flow and discounting back to the present. You should get the same answer.
  • #32 The best way to think about a perpetuity is not as forever but as a very long time.. The present value of an annuity that last 50 or 60 years converges on this value…
  • #33 There are British and Canadian console bonds still in existence…
  • #34 This is a cash flow growing at a constant rate forever. Here, g<r. Since the growth rate is forever, it is constrained to be less than or equal to the growth rate of the economy. If we allow for that constraint, g will always be less than r.
  • #35 Note that we use dividends from next year in the numerator.
  • #36 The discount rate can be measured from the perspective of just equity investors or all suppliers of capital in the firm.
  • #37 The value of equity is the present value of cash flows to the equity investors discounted back at the rate of return that those equity investors need to make to break even (the cost of equity). In the strictest sense of the word, the only cash flow stockholders in a publicly traded firm get from their investment is dividends, and the dividend discount model is the simplest and most direct version of an equity valuation model. Reinvestment can be computed as change in non-cash working capital plus capital expenditures less depreciation
  • #38 Note that these cash flows are after debt payments and the discount rate used is the cost of equity.
  • #39 A firm includes not just the equity, but all claim holders. The cash flow to the firm is the collective cash flow that all claim holders make from the firm, and it is discounted at the weighted average of their different costs.
  • #40 This asset has a finite life. Since the cash flows are prior to debt payments, the cash flows are firm cash flows and the cost of capital is the appropriate discount rate.
  • #41 The value of this store is $ 10.066 million.
  • #42 With an asset with an infinite life, you need to stop estimating cash flows at some point and estimate a terminal value. In a discounted cash flow framework, this can be done when the growth rate in cash flows becomes perpetual (less than or equal to the growth rate of the economy) Sets up the basic inputs: 1. Discount rates 2. Cash flows 3. Expected Growth 4. Length of the period that they can sustain a growth rate higher than the growth rate of the economy.
  • #43 The growth rate and reinvestment rate assumptions are linked.
  • #44 The free cash flow to equity is the cash flow after net debt payment or issues and reinvestment. Note that the debt creates a cash inflow for equity investors because it reduces how much they need to reinvest back into the firm.
  • #45 The terminal value of equity is discounted back to today at the cost of equity. Note that the cost of equity of 9.78% is used to estimate the terminal value of equity.
  • #46 These are the basic inputs for the valuation.
  • #47 The cost of capital for Boeing of 9.17% is used as the discount rate for the cash flows. This is the value of Boeing as a firm and I would need to subtract out the outstanding debt to get to the value of equity.