JEM034 Corporate Finance Winter Semester 2017/2018 Lecture #1 Olga Bychkova
Topics Covered Today Review of key finance concepts Present value (chapter 2 in BMA) Valuation of bonds (chapter 3 in BMA)
Present Value: Topics Covered Future Values and Present Values Looking for Shortcuts Perpetuities and Annuities More Shortcuts Growing Perpetuities and Annuities How Interest Is Paid and Quoted
Present and Future Value What the future value is? Amount to which an investment will grow after earning interest. What the present value is? The value of the future cash flows (inflows/outflows) determined as of the date of valuation. Why do we need the present value notion? To be able to compare projects/investment decisions that bring costs and benefits in different points in time in the future.
Future Value FV = C(1 + r) t, where C is the cash flow at time period 0, r is a discount rate. Example What is the future value of $100 if interest is compounded annually at a rate of 7% for two years? FV = $100(1 + 0.07) 2 = $114.49
Future Values with Compounding
Present Value PV = C t (1 + r) t, where C t is the cash flow at time period t, r is a discount rate. Discount factor = 1 = PV of $1 received in year t. (1 + r) t Example What is the present value of $114.49 received in year 2 if interest is compounded annually at a rate of 7%? PV = $114.49 (1 + 0.07) 2 = $100
Valuing an Office Building Step 1: Forecast cash flows Cost of building = C 0 = $370, 000 Sale price in year 1 = C 1 = $420, 000 Step 2: Estimate opportunity cost of capital If equally risky investments in the capital market offer a return of 5%, then Cost of capital = r = 5% Step 3: Discount future cash flows PV = C 1 1 + r = $420, 000 1 + 0.05 = $400, 000 Step 4: Go ahead if PV of payoff exceeds investment NPV = PV C 0 = $400, 000 $370, 000 = $30, 000
Net Present Value NPV = PV required investment NPV = C 0 + C 1 1 + r Net Present Value Rule: Accept investments that have positive net present values.
Risk and Present Value Higher risk projects require a higher rate of return. Higher required rates of return cause lower PVs. PV of $420, 000 at 5% = vs. PV of $420, 000 at 12% = $420, 000 1 + 0.05 $420, 000 1 + 0.12 = $400, 000 = $375, 000
Rate of Return Rule Rule: Accept investments that offer rates of return in excess of their opportunity cost of capital. Example In the project listed above, foregone investment opportunity is 12%. Should we do the project? Return = profit investment = $420, 000 $370, 000 $370, 000 = 0.135 or 13.5%
Multiple Cash Flows For multiple periods, we have the Discounted Cash Flow (DCF) formula: PV = C 1 1 + r + C 2 (1 + r) 2 + C 3 (1 + r) 3 + + C T T (1 + r) T = C t (1 + r) t t=1 To find the net present value, we add the (usually negative) initial cash flow: NPV = C 0 + PV = C 0 + T t=1 C T t (1 + r) t = C t (1 + r) t t=0
Perpetuity Perpetuity is a financial concept in which a cash flow is theoretically received forever. PV (perpetuity) = C 1 + r + C (1 + r) 2 + C C + + (1 + r) 3 (1 + r) t + PV (perpetuity) = C r
Perpetuity: Example What is the present value of $1 billion every year, for all eternity, if you estimate the perpetual discount rate to be 10%? PV = $1 billion 0.1 = $10 billion What if the investment does not start making money for 3 years? PV = $1 billion 0.1 1 = $7.51 billion (1 + 0.1) 3
Annuity Annuity is an asset that pays a fixed sum each year for a specified number of years. PV (annuity lasting t years) = C 1 + r + C (1 + r) 2 + C C + + (1 + r) 3 (1 + r) t PV (annuity lasting t years) = C r C r 1 (1 + r) t
Annuity as a Difference of Perpetuities
Annuity: Example #1 Tiburon Autos offers you easy payments of $5,000 per year, at the end of each year for 5 years. If interest rates are 7% per year, what is the cost of the car? PV = ( ) $5, 000 0.07 1 1 (1 + 0.07) 5 = $20, 501
Annuity: Example #2 You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease? Lease cost = $300 ( ) 0.005 1 1 (1 + 0.005) 4 12 = $12, 774.1
Future Value of Annuity FV of annuity = PV of annuity (1 + r) t = = C ( ) ( 1 (1 + r) 1 r (1 + r) t (1 + r) t t ) 1 = C r
Constant Growth Perpetuity = C 1 + r PV (growing perpetuity) = C(1 + g) C(1 + g)2 C(1 + g)t 1 + + (1 + r) 2 (1 + r) 3 + + (1 + r) t + PV (perpetuity) = C r g, where g is the annual growth rate of the cash flow.
Growing Perpetuity: Example What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and a constant growth rate of 4%? PV = $1 billion = $16.667 billion 0.1 0.04 What if the investment does not start making money for 3 years? PV = $1 billion (1 + 0.04)3 0.1 0.04 1 = $14.085 billion (1 + 0.1) 3
Constant Growth Annuity = C 1 + r PV (growing annuity lasting t years) = C(1 + g) C(1 + g)2 C(1 + g)t 1 + + (1 + r) 2 (1 + r) 3 + + (1 + r) t PV (growing annuity lasting t years) = C r g C r g (1 + g)t (1 + r) t
Growing Annuity as a Difference of Growing Perpetuities A three-year stream of cash flows that grow at the rate g is equal to the difference between two growing perpetuities.
Effective Interest Rates Effective Annual Interest Rate interest rate that is annualized using compound interest. Annual Percentage Rate interest rate that is annualized using simple interest. Example Given a monthly rate of 1%, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)? EAR = (1 + 0.01) 12 1 = 0.1268 or 12.68% APR = 0.01 12 = 0.12 or 12%
Present Value: Problem 38, Chapter 2 of BMA Textbook You own an oil pipeline that will generate a $2 million cash return over the coming year. The pipeline s operating costs are negligible, and it is expected to last for a very long time. Unfortunately, the volume of oil shipped is declining, and cash flows are expected to decline by 4% per year. The discount rate is 10%. (a) What is the PV of the pipeline s cash flows if its cash flows are assumed to last forever? This calls for the growing perpetuity formula with a negative growth rate (g = 0.04): PV = $2 million = $14.29 million. 0.1 ( 0.04) (b) What is the PV of the cash flows if the pipeline is scrapped after 20 years? This calls for the growing annuity lasting 20 years: PV = $2 million 0.1 ( 0.04) ( 1 (1 0.04)20 (1 + 0.1) 20 ) = $13.35 million.
Valuing Bonds: Topics Covered Using the Present Value Formula to Value Bonds How Bond Prices Vary with Interest Rates The Term Structure of Interest Rates Explaining the Term Structure Corporate Bonds and the Risk of Default
Valuing a Bond If you own a bond, you are entitled to a fixed set of cash payoffs. Every year until the bond matures, you collect regular interest payments. At maturity, when you get the final interest payment, you also get back the face value of the bond, which is called the bond s principal. PV (bond) = PV (annuity of coupon payments) + + PV (final payment of principal)
Valuing a Bond: Example If today is April 1, 2015, what is the value of the following bond? An IBM Bond pays $115 every March 31 for 5 years. In March 2020 it pays an additional $1,000 and retires the bond. The bond is rated AAA (WSJ YTM 7.5%). Cash Flows March 2016 2017 2018 2019 2020 $115 $115 $115 $115 $1,115 PV = $115 ( 1 0.075 ) 1 (1 + 0.075) 5 + $1, 000 = $1, 161.84 (1 + 0.075) 5
Maturity and Prices The price of long term bonds is more sensitive to changes in the interest rate than is the price of short term bonds.
Duration Duration is the average time for which an investor must wait to receive the cash flows from the bond = the weighted average of the times when the bond s cash payments are received. The times are the future years 1, 2, 3, etc., extending to the final maturity date, which we call T. The weight for each year is the present value of the cash flow received at that time divided by the total present value of the bond. Duration = 1 PV (C 1) + 2 PV (C 2) + 3 PV (C 3) + + T PV (C T ) PV PV PV PV Duration measures how bond prices change when interest rates change: D = P y where P = bond price, y = YTM. (1 + y), P
Volatility Modified duration = volatility (%) = duration 1 + yield Modified duration measures the percentage change in bond price for a 1 percentage point change in yield.
Term Structure of Interest Rates The relationship between short and long term interest rates is called the term structure of interest rates. Spot Rate the actual interest rate today (t = 0). Forward Rate the interest rate, fixed today, on a loan made in the future at a fixed time. Future Rate the spot rate that is expected in the future. Yield To Maturity (YTM) the IRR on an interest bearing instrument = the interest rate which ensures that the NPV of a bond investment is equal to zero.
Law of One Price The law of one price states that the same commodity must sell at the same price in a well functioning market. Therefore, all safe cash payments delivered on the same date must be discounted at the same spot rate.
What Determines the Shape of Term Structure? Suppose that you held a portfolio of one year U.S. Treasuries in February 2009. Here are three possible reasons why you might decide to hold on to them, despite their low rate of return: 1. You believe that short term interest rates will be higher in the future (Expectations Theory). 2. You worry about the greater exposure of long term bonds to changes in interest rates (Preferred Maturity Theory). 3. You worry about the risk of higher future inflation (Inflation Theory).
Bond Ratings Key to bond ratings: The highest quality bonds are rated triple A. Bonds rated triple B or above are investment grade. Lower rated bonds are called high yield, or junk, bonds.
Bonds: Problem 26, Chapter 3 of BMA Textbook Assume spot rates shown below. Year 1 Year 2 Year 3 Year 4 Spot rates 4.6% 4.4% 4.2% 4% Suppose that someone told you that the 5 year spot interest rate was 2.5%. Why would you not believe him? How could you make money if he was right? What is the minimum sensible value for the 5 year spot rate? We will borrow $1,000 at a 5 year loan rate of 2.5% and buy a 4 year strip (zero-coupon bond) paying 4%. We may not know what interest rates we will earn on the last year (4 5), but our $1,000 will come due and we put it in under our mattress earning 0% if necessary to pay off the loan.
Bonds: Problem 26, Chapter 3 of BMA Textbook Let s turn to present value calculations: $1, 000 The cost of the strip = (1 + 0.04) 4 = $854.8. We will receive proceeds from the 2.5% loan $1, 000 = (1 + 0.025) 5 = $883.9. Pocket the difference of $29.1, smile, and repeat. The minimum sensible value would set the discount factors used in year 5 equal to that of year 4, which would assume a 0% interest rate from year 4 to 5. We can solve for the interest rate, where 1 (1 + r) 5 = 1, which is roughly 3.19%. (1 + 0.04) 4
Where Are We in the Corporate Finance Discussion?