Present Value for Finance

Present value (PV) of an investment: value is expressed in terms of dollars today. Future Value (FV) of an investment: value is expressed it in terms of dollars in the future. NPV=PV(Benefits)-PV(Cost) Rule 1 Only Values at the same point in time can compared or combined. Rule 2. To move a cash flow forward. FV=C*(1+r)^n Rule3 To move a cash flow backward. PV=C/(1+r)^n   Present Value (PV) of a Perpetuity simplifies to: PV=C/r If r >g, the Present Value (PV) of a Growing Perpetuity simplifies to: C/(r-g).  .    Converting an APR to an EAR  Growth in purchasing power = .           As we move from daily to hourly to compounding every second, we approach the limit of continuous compounding, in which we compound every instant. Used in asset pricing models. The EAR for a Continuously Compounded APR.  The Continuously Compounded APR for an EAR  APR=ln(1+EAR).  A bond is a security sold by governments and corporations to raise money from investors today in exchange for promised future payments. Bond Certificate: states the terms of the bond (amounts and dates of payment). Maturity Date: final repayment date. Term: the time remaining until the repayment date. Bondholders receive two types of payment: Coupon: Promised interest payments. Face Value: Notional amount used to compute the interest payments. Coupon Rate: Determines the amount of each coupon payment, expressed as an APR. Coupon Payment (𝐶𝑃𝑁) is then defined as: [pic 8] Zero-Coupon Bonds - Yield to Maturity (YTM).  YTM = the discount rate that sets the present value of the promised bond payments equal to the current market price of the bond.      .Example.  Problem. Consider a five-year bond with a $1000 face value and a 5% coupon rate. Interest payments are made semiannually.  Suppose you are told that its yield to maturity has increased to 6.30% (expressed as an APR with semiannual compounding). What price is the bond trading for now? Applying the bond valuation formula:   1.  N = 10   2. y (semi-annual rate) =   3.     4.    Coupon bonds.  Discount  A bond is selling at a discount if the price is less than the face value.  Par  A bond is selling at par if the price is equal to the face value.  Premium  A bond is selling at a premium if the price is greater than the face value.   Coupon bonds trading at a discount: Investor earns a return both from receiving the coupons and from receiving a face value that exceeds the price paid for the bond. YTM  > CPN Rate.  Coupon bonds trading at a premium: Investor earns a return from receiving the coupons but this return will be diminished by receiving a face value less than the price paid for the bond.   CPN Rate  > YTM          Consider three 30-year bonds with annual coupon payments. One bond has a 10% coupon rate, one has a 5% coupon rate, and one has a 3% coupon rate. If the yield to maturity of each bond is 5%, what is the price of each bond per $100 face value? Which bond trades at a premium, which trades at a discount, and which trades at par? Applying the bond valuation formula to solve for the price:      Holding all other things constant, a bond’s yield to maturity will not change over time Holding all other things constant, the price of discount or premium bond will move towards par value over time  If a bond’s yield to maturity has not changed, then the IRR of an investment in the bond equals its yield to maturity even if you sell the bond early  Example: 30-year zero-coupon bond, YTM = 5%, FV = $100        There is an inverse relationship between interest rates and bond prices  As interest rates and bond yields rise, bond prices fall   As interest rates and bond yields fall, bond prices rise.  Shorter-maturity zero-coupon bonds are less sensitive to changes in interest rates than longer-term zero coupon bonds. Bonds with higher coupon rate are less sensitive to interest rate changes than bonds with lower coupon bonds. Suppose you purchase a 10-year bond with 6% annual coupons. You hold the bond for four years, and sell it immediately after receiving the fourth coupon. If the bond’s yield to maturity was 5% when you purchased and sold the bond. What cash flows will you pay and receive from your investment in the bond per $100 face value? What is the rate of return of your investment?   First, we compute the initial price of the bond by discounting its 10 annual coupons of $6 and final face value of $100 at the 5% yield to maturityThus, the initial price of the bond = $107.72. (Note that the bond trades above par, as its coupon rate exceeds its yield.)Next, we compute the price at which the bond is sold, which is the present value of the bonds cash flows when only 6 years remain until maturity. Therefore, the bond was sold for a price of $105.08.  NPV Investment Rule: When making an investment decision to accept or reject a stand-alone project,  accept a project if its NPV is positive and reject it if its NPV is negative. The value of accepting the project is given by its NPV, and the value of rejecting the project is zero. The IRR of a project provides information about the sensitivity of the project’s NPV to errors in the estimate of its cost of capital. In general: the difference between the cost of capital and the IRR is the maximum estimation error in the cost of capital that can exist without changing the original decision to invest or not. When the rules conflict, the NPV decision rule should be followed. Internal Rate of Return (IRR): average return earned by taking on the investment opportunity. IRR Investment Rule: Take any investment where the IRR exceeds the opportunity cost of capital. Turn down any investment whose IRR is less than the opportunity cost of capital. The IRR rule is valid only if the project has a positive NPV for every discount rate below the IRR. While the IRR rule has shortcomings for making investment decisions, the IRR itself remains useful. IRR measures the average return of the investment and the sensitivity of the NPV to any estimation error in the cost of capital. Payback Investment Rule: If the payback period is less than a pre-specified length of time, you accept the project. Otherwise, you reject the project. NPV Rule and Mutually Exclusive Projects: First: determine which projects have a positive NPV Second: rank the projects according to the NPV. NPV Rule: Select the project with the highest NPV. Equity Cost of Capital: expected return of other investments available in the market with equivalent risk to the shares of the firm.  If the current stock price were less than this amount, expect investors to rush in and buy it, driving up the stock’s price. If the stock price exceeded this amount, selling it would cause the stock price to quickly fall. Total return of the stock =  Dividend Yield + Capital Gain Rate     Suppose you expect Walgreen Company to pay dividends of $0.44 per share and trade for $33 per share at the end of the year. If investments with equivalent risk to Walgreen’s stock have an expected return of 8.5%, what is the most you would pay today for Walgreen’s stock? What dividend yield and capital gain rate would you expect at this price? Using the equation 𝑃_0=(〖𝐷𝑖𝑣〗_1+𝑃_1)/(1+𝑟_𝐸 )    where:〖𝐷𝑖𝑣〗_1=$0.44; 𝑟_𝐸=8.5%; 𝑃_1=$33  𝑃_0=(0.44+33)/(1+0.085)=$30.82  Dividend Yield = 0.44/30.82=1.43%  Capital Gain Rate = (𝑃_1−𝑃_0)/𝑃_0 =(33−30.82)/30.82=7.07%  Expected Total Return = 1.43%+7.07%=8.5%  Therefore: Expected Total Return = 𝑟_𝐸. Why?  Dividend-Discount Model: The price of any stock is equal to the present value of the expected future dividends it will pay. Consolidated Edison, Inc. (Con Edison) is a regulated utility company that services the NYC area. Suppose Con Edison plans to pay $2.36 per share in dividends in the coming year. If its equity cost of capital is 7.5% and dividends are expected to grow by 1.5% per year in the future, estimate the value of Con Edison’s stock. 〖𝐷𝑖𝑣〗_1=$2.36   𝑟_𝐸=7.5%  𝑔=1.5%   Crane Sporting Goods expects to have earnings per share of $6 in the coming year. Rather than reinvest these earnings and grow, the firm plans to pay out all of its earnings as a dividend.  With these expectations of no growth, Crane’s current share price is $60. Suppose Crane could cut its dividend payout rate to 75% for the foreseeable future and use the retained earnings to open new stores. The return on its investment in these stores is expected to be 12%. Assuming its equity cost of capital is unchanged, what effect would this new policy have on Crane’s stock price? Under the initial no-growth: determine equity cost of capital, using the current price and the “no growth” assumption:  Under the new policy, the dividend payout ratio will be 75%, and the retention rate will be 25%. Therefore:  The price under the new policy can be computed using the constant dividend growth model:  Suppose Crane decides to cut its dividend payout rate to 75% to invest in new stores, as in the previous example. But now suppose the return on these new investment is 8%, rather than 12%. Given its expected EPS this year of $6 and its equity cost of capital of 10%, what will happen to Crane’s current share price in this case? Solution Under the new policy, the dividend payout ratio will be 75%, and the retention rate will be 25%. Therefore:      The price under the new policy can be computed using the constant dividend growth model:    Changing Growth Rates We cannot use the constant dividend growth model to value a stock if the growth rate is not constant:  For example, young firms often have very high initial earnings growth rates. During this period of high growth, these firms often retain 100% of their earnings to exploit profitable investment opportunities. As they mature, their growth slows. At some point, their earnings exceed their investment needs and they begin to pay dividends. HOW TO MEASURE RISK AND RETURN? Expected (Mean) Return: Calculated as a weighted average of the possible returns, where the weights correspond to the probabilities     Variance: The expected squared deviation from the mean     Standard Deviation: the square root of the variance   Variance and Standard Deviation are measures of the risk of a probability distribution.[pic 1][pic 2][pic 3][pic 4][pic 5][pic 6][pic 7][pic 9][pic 10][pic 11][pic 12][pic 13][pic 14][pic 15][pic 16][pic 17][pic 18][pic 19][pic 20][pic 21][pic 22][pic 23][pic 24][pic 25][pic 26][pic 27][pic 28][pic 29][pic 30][pic 31][pic 32][pic 33][pic 34][pic 35]