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**Moving Cash Flows: Review**

Moeller-Finance Moving Cash Flows: Review Notes-Bond & Stock Valuation

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Moeller-Finance Formulas Notes-Bond & Stock Valuation

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**Growing Annuity Annuities are a constant cash flow over time**

Moeller-Finance Growing Annuity Annuities are a constant cash flow over time Growing annuities are a constant growth cash flow over time Notes-Bond & Stock Valuation

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**What are you worth today?**

Moeller-Finance What are you worth today? You will make $100,000 the first year. You expect to work for 40 years, get 9% raises every year and 20% per year on investments. Example: You receive a job offer with a $100,000 salary. You expect the salary to increase by 9% per year until your retirement in 40 years. Given an interest rate of 20%, what is the present value of your lifetime salary? Notes-Bond & Stock Valuation

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Moeller-Finance Cash Flow Timing When does the first cash flow occur relative to the present value of the _______ Perpetuity? Growing perpetuity? Annuity? Growing annuity? One period later! Implicitly, the first cash flow in these formulas is one period from today so when you are trying to estimate the appropriate cash flow remember that you are looking for next periods cash flow. For example, to the perpetuity formula show below, subscripts have been added to illustrate the timing issue where t represents the period you are discounting the cash flows to (typically time 0). Notes-Bond & Stock Valuation

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**Review: Bond Features Coupon Payments: Regular interest payments**

Moeller-Finance Review: Bond Features Coupon Payments: Regular interest payments Semi annual for most US corporate bonds Types of Coupon payments Fixed Rate: 8% per year Floating Rate: 6-mo. Treasury bill rate basis points. Face or Par Value: $1,000/bond Maturity: no. of years from issue date until principal is paid Coupon Rate Bond Features: Regular coupon or interest payments every period until the bond matures. i. fixed-rate bond ii. floating-rate bond Face or Par value: amount of money to be repaid at the end of the loan Maturity: number of years until the principal is paid back Coupon Rate: annual coupon payment divided by the face value of the bond. Example: If a bond has five years to maturity, an annual coupon of $100 and a face value of $1,000, its cash flows are as follows. In years 1-4, the bondholder receives $100 per year. In year 5, the bondholder receives the $100 coupon plus $1000 (par value). What is the coupon rate for this bond? 10% (0.10=100/1000) If the above bond had a semi-annual coupon of $50 each six months, the coupon rate for the bond would still be 10% (0.10=((50+50)/1000)) Notes-Bond & Stock Valuation

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**Bond Valuation Annuity Formula Moeller-Finance Bond Values and Yields**

Discount Valuation Method: The cash flows of a bond are typically the coupon payments and the face value. Finding the value of the bond requires discounting the coupons and the face value at the market rate. Yield-to-Maturity (YTM): The required rate of return or interest rate that makes the discounted cash flows from a bond equal to the bond's price. General Expression for the Value of a Bond Bond value = Present value of coupons + present value of face value Bond Value = PV(Annuity) + PV(Face Value) where C is the coupon payment, YTM is the yield-to-maturity, t is the number of periods until maturity, and par is the par or face value of the bond. Annuity Formula Notes-Bond & Stock Valuation

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Moeller-Finance What is the price of a $1000 bond maturing in ten years with a 12% coupon that is paid semiannually if the YTM is 10% Holden Book: Chapter 6.1 Notes-Bond & Stock Valuation

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Moeller-Finance Stock Valuation Notes-Bond & Stock Valuation

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**Common Stock Valuation is Difficult**

Moeller-Finance Common Stock Valuation is Difficult Uncertain cash flows Equity is the residual claim on the firm’s cash flows Life of the firm is forever Rate of return (the appropriate discount rate) is not easily observed Notes-Bond & Stock Valuation

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**Differential Growth Dividend Model**

Moeller-Finance Differential Growth Dividend Model Forecasted Dividends grow at a constant rate, g1 for a certain number of years and then grow at a second growth rate, g2. Example: The dividend of a company was $1 yesterday. During the next 18 years the dividend will grow at 14% per year. After that the dividend will grow at 10% per year. What is the price of the stock if the required return is 15%? First recognize that you are asked to calculate the PV of the stream of future dividends (= cash flows) for this company. Also recognize that the first half of the problem is basically a growing annuity followed by a growing perpetuity. Finally, try to set up a timeline, this helps keeping track of the different dividend payments as they occur over time. Holden Book: Chapter 8 Notes-Bond & Stock Valuation

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**The first dividend regime is a growing annuity**

Moeller-Finance The first dividend regime is a growing annuity The second dividend regime is a growing perpetuity Notes-Bond & Stock Valuation

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**Now, we need to sum the two dividend regime values.**

Moeller-Finance Now, we need to sum the two dividend regime values. Notes-Bond & Stock Valuation

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**EPS and Dividends Dividends (share repurchase) are a function of…**

Moeller-Finance EPS and Dividends Dividends (share repurchase) are a function of… Ability to pay: Cash flow uncertainty Decision to pay: Managerial uncertainty Why does a manager retain earnings? Has better investment opportunities than the shareholder Makes a sub-optimal decision for the shareholder What is a “better investment opportunity”? Investment has a NPV>0 Stock Price, EPS and the Present Value of Growth Opportunities Remember dividends are a function of two factors, a firm’s ability to pay and the amount a firm chooses to pay. If a firm pays out 100% of it’s earnings in dividends and earnings are expected to be constant, then the value of the firm equals the earnings per share divided by the appropriate r (P=EPS/r). However, what if a manager decides to keep a portion of the earnings? Before we discuss valuing this type of firm, why would a manager choose to keep some of the earnings? 1.) The manager believes that have better investment opportunities than the stockholders. 2.) The manager behaves non-optimally and keeps the earnings even though they do not believe the firm has better investment opportunities. Aside: How does the manager decide if the firm has “better” investment opportunities? Notes-Bond & Stock Valuation

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**Value a firm that retains earnings?**

Moeller-Finance Value a firm that retains earnings? Fundamental valuation equation: Sum of the discounted cash flows First component: PV(no-growth earnings stream) Remember EPS=Net income/Shareholders equity Second component: PV of growth opportunities Look for pricing shortcuts: perpetuity, annuity, etc. Rule: As long as PV(GO) > 0, price increases If the firm retains some earnings how do we value the stock? As always, the value of the firm is the sum of the discounted cash flows of the firm but how do we express it in the terms of earnings and investment opportunities? (Note: these calculations can be done on a per share or per firm basis. If you do it per firm, then instead of EPS use earnings and calculate PV(GO) per firm rather than per share.) where PV(GO) is the present value of growth opportunities. First component: (EPS/r) is just the present value of the no-growth earnings stream. (If a firm does not alter its investment policy, what the firm is worth.) Remember: EPS (earnings per share) = Net Income/Shareholders equity. Second component: The present value of growth opportunities. (Just calculate what the NPV of the additional investment of the firm will yield. Remember, always look for shortcuts to price these. Does the new investment opportunity behave like a perpetuity, growing perpetuity, annuity?) RULE: As long as the PV(GO) >0 then the price of the firm increases. Notes-Bond & Stock Valuation

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**One Time Investment Opportunity**

Moeller-Finance One Time Investment Opportunity Firm expects $1 million in earnings in perpetuity without new investments. Firm has 100,000 shares outstanding. Firm has investment opportunity at t=1 to invest $1 million in a project expected to increase future earnings by $210,000 per year. The firm’s discount rate is 10%. What is the share price with and without the project? What is the value of the firm if it doesn’t invest? Since the firm expects $1 million in earnings forever, the cash flows are a perpetuity. Therefore we can either first calculate EPS to get the share price or the value of the firm then divide it by the shares outstanding: What is the value of the firm if it does the investment? Notes-Bond & Stock Valuation

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**Constant Growth, Constant Investing**

Moeller-Finance Constant Growth, Constant Investing Firm Q has EPS of $10 at the end of the first year and a dividend pay-out ratio of 40%, rE = 16% and a return on investment of 20%. The firm takes advantage of its growth opportunities each year by investing retained earnings. PV(GO) model 1st investment = 0.6 × $10 = $6, which generates 0.2 × $6 = $1.20 Per share PVGO1 = -6 + (1.20/0.16) = $1.50 (at t=1) 2nd investment = 0.6 × $11.20 = $6.72, generating 0.2 × $6.72 = $1.344 Per share PVGO2 = (1.344/0.16) = $1.68 (at t=2) What is the present value of the first year’s investment (growth opportunity)? Since the dividend payout ratio is 40%, it means the company retains and invests 60% of its earnings. In the end of the first year the EPS is $10, so they invest $6 (10*(1-0.4)). That $6 investment yields 20% or $1.20 in the second year and the investment continues to earn $1.20 indefinitely. This type of cash flow, a constant payment forever, is a perpetuity, so we can value the first year’s investment as follows: What is the present value of the second year’s investment? At the end of the second year, our EPS is now $11.20 ( ) so we invest $6.72 (11.20*(1-0.4)). In the third year that $6.72 investment yields 20% or $1.344 indefinitely. Again, this is a perpetuity so the second year’s investment is valued as follows: Notes-Bond & Stock Valuation

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**Constant Growth, Constant Investing (cont)**

Moeller-Finance Constant Growth, Constant Investing (cont) Relationship between PV(GO)’s? 1.68 = (1+g) × g=0.12 Is there an easier way to estimate g for this case? G=ROI x Investment Rate=0.2 x (1-0.4)=0.12 PVGO0 = $1.50 / ( ) =$37.50 No-growth dividend value: $10/0.16 = $62.50 P = $ $37.50 = $100 Now we could continue to calculate the yearly investments present value but there should be a way to simplify this yearly series of investment cash flows. Since this is a stock, we assume that the investment opportunities will continue forever so the series of PV(GO)’s are perpetuities. Clearly they are not constant dollar perpetuities but it is possible that they are constant growth perpetuities. To prove they are constant growth perpetuities you could calculate one more and see if the growth rate between them is consistent or since we know that is a consistent relationship between the investment growth and return, it must grow at a constant rate. If so, what is that constant growth rate (g)? The growth rate for this firm can be expressed as: Let’s check and see if the 12% growth rate is correct. Since we calculated two of the present value of growth opportunities, did they grow by 12%? YES! Now that we know that the growth opportunities grow at a constant rate forever, we can price the cash flows from the investment as a growing perpetuity . Returning to the PV(GO) formula, we know the value of this firm is as follows: Notes-Bond & Stock Valuation

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**Constant Growth, Constant Investing (cont)**

Moeller-Finance Constant Growth, Constant Investing (cont) Can we price this firm a different way? Since the investment grows at a constant rate we can immediately estimate g Investment rate x ROI = 0.6 × 20% = 12% Then estimate PV(GO) as a growing perpetuity based on dividends rather than cash flow D1 / (rE - g) = $4 / ( ) = $100 So the entire firm is worth $100 Notes-Bond & Stock Valuation

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Moeller-Finance Another Example Firm X currently has expected earnings of $100,000 per year in perpetuity. Firm X is switching its policy and wants to invest 20% of its earnings in projects with a 10% return. The discount rate is 18%. No-growth price: P=$100,000/0.18 = $555,555 PV(GO) is a constant growth perpetuity What’s g? g=Investment rate x ROI = 0.2 × 10% = 2% What is the first year’s investment cash flow? Invest $20,000 and receive $2,000 forever -20,000+(2,000/0.18)= PV(GO) = (-8,888.89)/( ) = - 55,555 New Policy: P=$555, ,555 = $500,000 Because the information is give to us at a firm level, we will calculate all of these values at the firm level, not per share. The value of the firm before switching its investment policy: How to do we value the firm that is switching its policy? We add the value of the firm, before switching the policy to the present value of investing 20% of EPS at 10%. From the previous examples, we know that this new investment policy is a constant growth perpetuity. We calculate g and the value of the firm as follows: So this investment policy actually decreases the value of the firm!!! Notes-Bond & Stock Valuation

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