FINANCE 2. Foundations Solvay Business School Université Libre de Bruxelles Fall 2007

June 10, 2015 MBA Foundations |2 Interest rates and present value: 1 period Suppose that the 1-year interest rate r 1 = 5% €1 at time 0 → €1.05 at time 1 €1/1.05 = at time 0 → €1 at time 1 1-year discount factor: DF 1 = 1 / (1+r 1 ) Suppose that the 1-year discount factor DF 1 = 0.95 €0.95 at time 0 → €1 at time 1 € 1 at time 0 → € 1/0.95 = at time 1 The 1-year interest rate r 1 = 5.26% Future value of C 0 : FV 1 (C 0 ) = C 0 ×(1+r 1 ) = C 0 / DF 1 Present value of C 1 : PV(C 1 ) = C 1 / (1+r 1 ) = C 1 × DF 1 Data: r 1 → DF 1 = 1/(1+r 1 ) or Data: DF 1 → r 1 = 1/DF 1 - 1

June 10, 2015 MBA Foundations |3 Using Present Value Consider simple investment project: Interest rate r = 5%, DF 1 =

June 10, 2015 MBA Foundations |4 Net Future Value NFV =  1.05 = 20 = + C 1 - I (1+r) Decision rule: invest if NFV>0 Justification: takes into cost of capital – cost of financing –opportunity cost

June 10, 2015 MBA Foundations |5 Net Present Value NPV = /1.05 = + 19 = - I + C 1 /(1+r) = - I + C 1  DF 1 =  = +19 DF 1 = 1-year discount factor a market price C 1  DF 1 =PV(C 1 ) Decision rule: invest if NPV>0 NPV>0  NFV>

June 10, 2015 MBA Foundations |6 Internal Rate of Return Alternative rule: compare the internal rate of return for the project to the opportunity cost of capital Definition of the Internal Rate of Return IRR : (1-period) IRR = Profit/Investment = (C 1 - I)/I In our example: IRR = ( )/100 = 25% The Rate of Return Rule: Invest if IRR > r In this simple setting, the NPV rule and the Rate of Return Rule lead to the same decision: NPV = -I+C 1 /(1+r) >0  C 1 >I(1+r)  (C 1 -I)/I>r  IRR>r

June 10, 2015 MBA Foundations |7 Economic foundations of net present value Euros now Euros next year Slope = - (1 + r) = - (1 + 5%) I. Fisher 1907, J. Hirshleifer 1958 Perfect capital markets Separate investment decisions from consumption decisions Y0Y0 Y1Y1

June 10, 2015 MBA Foundations |8 Net Present Value NPV = -I + DF 1  C 1 =  60 = 7.14 Consider the following investment project: Initial cost: I (50) Future cash flow: C 1 (60) Budget constraint with project:

June 10, 2015 MBA Foundations |9 Fisher Separation Theorem Euros now Euros next year NPV Slope = - (1 + r) = - (1 + 5%) -50 I. Fisher 1907, J. Hirshleifer 1958 Perfect capital markets Investment decision independent of: - initial allocation - preferences (utility functions)

June 10, 2015 MBA Foundations |10 Enterprise Valuation Suppose an all equity financed company is created for this project. Step 1: Creation Step 2: Equity offering + investment t = 0t = Assets50 Equity50 t = 0t = Cash flows Assets0 Equity0 Market Cap. NPV = I+NPV =

June 10, 2015 MBA Foundations |11 0 C1C1 -I-I Slope = -(1+r) NPV Market value of company

June 10, 2015 MBA Foundations |12 Entreprise Value Maximisation 0 Investment Euros today Euros next year NPV Investment opportunities Market value of company Numerical example

June 10, 2015 MBA Foundations |13 Arbitrage and the Law of One Price Risk-free interest rate : 5% If bond price = $940 If bond price = $960

June 10, 2015 MBA Foundations |14 No Arbitrage Price of a Security

June 10, 2015 MBA Foundations |15 Valuing a Portfolio: Value Additivity

June 10, 2015 MBA Foundations |16 No-Arbitrage Price of a Risk-free Security You observe the following data: What is the price of Bond C? Consider the following replicating portfolio: n A = 0.50 n B = 0.50 => Price of Bond C = 0.50 x x 98 = 99.50

June 10, 2015 MBA Foundations |17 Looking for discount factors What are the underlying discount factors? Bootstrap method = DF = DF DF No-Arbitrage & Law of One Price  there exist discount factors DF 1 and DF 2 such that for any security: Price(Bond)=DF 1 x CashFlow 1 + DF 2 x CashFlow 2 DF 1 = 101/106 = DF 2 = (98 – 4 x )/104 =0.9057

June 10, 2015 MBA Foundations |18

June 10, 2015 MBA Foundations |19 Risky securities Price of risk 50% Market index: Expected payoff = 50% Expected return = Risk-free bond: Risk-free return = risk-free interest rate = Market index’s risk premium = 10% - 4%= 6%

June 10, 2015 MBA Foundations |20 No-Arbitrage Price of a Risky Security Price calculation using replicating portfolio:

June 10, 2015 MBA Foundations |21 Risk is Relative to the Overall Market B 600 0

June 10, 2015 MBA Foundations |22 Risk and Risk Premiums for Different Securities