1. FINANCE 3. Present Value Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2004

2. MBA 2004 |2|2 Present Value Objectives for this session : 1. Introduce present value calculation in a simple 1-period setting 2. Extend present value calculation to several periods 3.Analyse the impact of the compounding periods 4.Introduce shortcut formulas for PV calculations

3. MBA 2004 |3|3 Present Value: teaching strategy 1-period: FV, PV, 1-year discount factor DF 1 NPV, IRR Several periods: start from Strips –Zero-coupons & disc. Factors General PV formula –From prices to interest rates: spot rates –Shortcut formulas: perpetuities and annuities –Compounding interval

4. MBA 2004 |4|4 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 = 0.9523 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 = 1.0526 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

5. MBA 2004 |5|5 Using Present Value Consider simple investment project: Interest rate r = 5%, DF 1 = 0.9523 125 -100 0 1

6. MBA 2004 |6|6 Net Future Value NFV = +125 - 100  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 -100 +100 +125 -105 01

7. MBA 2004 |7|7 Net Present Value NPV = - 100 + 125/1.05 = + 19 = - I + C 1 /(1+r) = - I + C 1  DF 1 = - 100+125  0.9524 = +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>0 -100 +125 -125 +119

8. MBA 2004 |8|8 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 = (125 - 100)/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

9. MBA 2004 |9|9 IRR: a general definition The Internal Rate of Return is the discount rate such that the NPV is equal to zero. -I + C 1 /(1+IRR)  0 In our example: -100 + 125/(1+IRR)=0  IRR=25%

10. MBA 2004 | 10 Present Value Calculation with Uncertainty Consider the following project: C 0 = -I = -100Cash flow year 1: The expected future cash flow is C 1 = 0.5 * 50 + 0.5 * 200 = 125 The discount rate to use is the expected return of a stock with similar risk r = Risk-free rate + Risk premium = 5% + 6% (this is an example) NPV = -100 + 125 / (1.11) = 12.6 +50 with probability ½ +200 with probability ½

11. MBA 2004 | 11 A simple investment problem Consider the following project: Cash flow t = 0C 0 = - I = - 100 Cash flow t = 5C 5 = + 150 (risk-free) How to calculate the economic profit? Compare initial investment with the market value of the future cash flow. Market value of C 5 = Present value of C 5 = C 5 * Present value of $1 in year 5 = C 5 * 5-year discount factor = C 5 * DF 5 Profit = Net Present Value = - I + C 5 * DF 5

12. MBA 2004 | 12 Using prices of U.S. Treasury STRIPS Separate Trading of Registered Interest and Principal of Securities Prices of zero-coupons Example: Suppose you observe the following prices MaturityPrice for $100 face value 198.03 294.65 390.44 486.48 580.00 The market price of $1 in 5 years is DF 5 = 0.80 NPV = - 100 + 150 * 0.80 = - 100 + 120 = +20

13. MBA 2004 | 13 Present Value: general formula Cash flows: C 1, C 2, C 3, …,C t, … C T Discount factors: DF 1, DF 2, …,DF t, …, DF T Present value:PV = C 1 × DF 1 + C 2 × DF 2 + … + C T × DF T An example: Year 0 1 2 3 Cash flow-100406030 Discount factor 1.0000.98030.94650.9044 Present value-10039.2156.7927.13 NPV = - 100 + 123.13 = 23.13

14. MBA 2004 | 14 Several periods: future value and compounding Invests for €1,000 two years (r = 8%) with annual compounding After one year FV 1 = C 0 × (1+r) = 1,080 After two years FV 2 = FV 1 × (1+r) = C 0 × (1+r) × (1+r) = C 0 × (1+r)² = 1,166.40 Decomposition of FV2 C 0 Principal amount 1,000 C 0 × 2 × r Simple interest 160 C 0 × r² Interest on interest 6.40 Investing for t yearsFV t = C 0 (1+r) t Example: Invest €1,000 for 10 years with annual compounding FV 10 = 1,000 (1.08) 10 = 2,158.82 Principal amount1,000 Simple interest 800 Interest on interest 358.82

15. MBA 2004 | 15 Present value and discounting How much would an investor pay today to receive €C t in t years given market interest rate r t ? We know that 1 € 0 => (1+r t ) t € t Hence PV  (1+r t ) t = C t => PV = C t /(1+r t ) t = C t  DF t The process of calculating the present value of future cash flows is called discounting. The present value of a future cash flow is obtained by multiplying this cash flow by a discount factor (or present value factor) DF t The general formula for the t-year discount factor is:

16. MBA 2004 | 16 Discount factors Interest rate per year # years 1%2%3%4%5%6%7%8%9%10% 10.99010.98040.97090.96150.95240.94340.93460.92590.91740.9091 20.98030.96120.94260.92460.90700.89000.87340.85730.84170.8264 30.97060.94230.91510.88900.86380.83960.81630.79380.77220.7513 40.96100.92380.88850.85480.82270.79210.76290.73500.70840.6830 50.95150.90570.86260.82190.78350.74730.71300.68060.64990.6209 60.94200.88800.83750.79030.74620.70500.66630.63020.59630.5645 70.93270.87060.81310.75990.71070.66510.62270.58350.54700.5132 80.92350.85350.78940.73070.67680.62740.58200.54030.50190.4665 90.91430.83680.76640.70260.64460.59190.54390.50020.46040.4241 100.90530.82030.74410.67560.61390.55840.50830.46320.42240.3855

17. MBA 2004 | 17 Spot interest rates Back to STRIPS. Suppose that the price of a 5-year zero-coupon with face value equal to 100 is 75. What is the underlying interest rate? The yield-to-maturity on a zero-coupon is the discount rate such that the market value is equal to the present value of future cash flows. We know that 75 = 100 * DF 5 and DF 5 = 1/(1+r 5 ) 5 The YTM r 5 is the solution of: The solution is: This is the 5-year spot interest rate

18. MBA 2004 | 18 Term structure of interest rate Relationship between spot interest rate and maturity. Example: MaturityPrice for €100 face valueYTM (Spot rate) 198.03r 1 = 2.00% 294.65r 2 = 2.79% 390.44r 3 = 3.41% 486.48r 4 = 3.70% 580.00r 5 = 4.56% Term structure is: Upward sloping if r t > r t-1 for all t Flat if r t = r t-1 for all t Downward sloping (or inverted) if r t < r t-1 for all t

19. MBA 2004 | 19 Using one single discount rate When analyzing risk-free cash flows, it is important to capture the current term structure of interest rates: discount rates should vary with maturity. When dealing with risky cash flows, the term structure is often ignored. Present value are calculated using a single discount rate r, the same for all maturities. Remember: this discount rate represents the expected return. = Risk-free interest rate + Risk premium This simplifying assumption leads to a few useful formulas for: Perpetuities(constant or growing at a constant rate) Annuities(constant or growing at a constant rate)

20. MBA 2004 | 20 Constant perpetuity C t =C for t =1, 2, 3,..... Examples: Preferred stock (Stock paying a fixed dividend) Suppose r =10% Yearly dividend =50 Market value P0? Note: expected price next year = Expected return = Proof: PV = C d + C d² + C d3 + … PV(1+r) = C + C d + C d² + … PV(1+r)– PV = C PV = C/r

21. MBA 2004 | 21 Growing perpetuity C t =C 1 (1+g) t-1 for t=1, 2, 3,..... r>g Example: Stock valuation based on: Next dividend div1, long term growth of dividend g If r = 10%, div 1 = 50, g = 5% Note: expected price next year = Expected return =

22. MBA 2004 | 22 Constant annuity A level stream of cash flows for a fixed numbers of periods C 1 = C 2 = … = C T = C Examples: Equal-payment house mortgage Installment credit agreements PV = C * DF 1 + C * DF 2 + … + C * DF T + = C * [DF 1 + DF 2 + … + DF T ] = C * Annuity Factor Annuity Factor = present value of €1 paid at the end of each T periods.

23. MBA 2004 | 23 Constant Annuity C t = C for t = 1, 2, …,T Difference between two annuities: –Starting at t = 1 PV=C/r –Starting at t = T+1 PV = C/r ×[1/(1+r) T ] Example: 20-year mortgage Annual payment = €25,000 Borrowing rate = 10% PV =( 25,000/0.10)[1-1/(1.10) 20 ] = 25,000 * 10 *(1 – 0.1486) = 25,000 * 8.5136 = € 212,839

24. MBA 2004 | 24 Annuity Factors Interest rate per year # years 1%2%3%4%5%6%7%8%9%10% 10.99010.98040.97090.96150.95240.94340.93460.92590.91740.9091 21.97041.94161.91351.88611.85941.83341.80801.78331.75911.7355 32.94102.88392.82862.77512.72322.67302.62432.57712.53132.4869 43.90203.80773.71713.62993.54603.46513.38723.31213.23973.1699 54.85344.71354.57974.45184.32954.21244.10023.99273.88973.7908 65.79555.60145.41725.24215.07574.91734.76654.62294.48594.3553 76.72826.47206.23036.00215.78645.58245.38935.20645.03304.8684 87.65177.32557.01976.73276.46326.20985.97135.74665.53485.3349 98.56608.16227.78617.43537.10786.80176.51526.24695.99525.7590 109.47138.98268.53028.11097.72177.36017.02366.71016.41776.1446

25. MBA 2004 | 25 Growing annuity C t = C 1 (1+g) t-1 for t = 1, 2, …, Tr ≠ g This is again the difference between two growing annuities: –Starting at t = 1, first cash flow = C 1 –Starting at t = T+1 with first cash flow = C 1 (1+g) T Example: What is the NPV of the following project if r = 10%? Initial investment = 100, C 1 = 20, g = 8%, T = 10 NPV= – 100 + [20/(10% - 8%)]*[1 – (1.08/1.10) 10 ] = – 100 + 167.64 = + 67.64

26. MBA 2004 | 26 Review: general formula Cash flows: C 1, C 2, C 3, …,C t, … C T Discount factors: DF 1, DF 2, …,DF t, …, DF T Present value:PV = C 1 × DF 1 + C 2 × DF 2 + … + C T × DF T If r 1 = r 2 =...=r

27. MBA 2004 | 27 Review: Shortcut formulas Constant perpetuity: C t = C for all t Growing perpetuity: C t = C t-1 (1+g) r>g t = 1 to ∞ Constant annuity: C t =C t=1 to T Growing annuity: C t = C t-1 (1+g) t = 1 to T

28. MBA 2004 | 28 Compounding interval Up to now, interest paid annually If n payments per year, compounded value after 1 year : Example: Monthly payment : r = 12%, n = 12 Compounded value after 1 year : (1 + 0.12/12)12= 1.1268 Effective Annual Interest Rate: 12.68% Continuous compounding: [1+(r/n)] n →e r (e= 2.7183) Example : r = 12% e 12 = 1.1275 Effective Annual Interest Rate : 12.75%

29. MBA 2004 | 29 Juggling with compounding intervals The effective annual interest rate is 10% Consider a perpetuity with annual cash flow C = 12 –If this cash flow is paid once a year: PV = 12 / 0.10 = 120 Suppose know that the cash flow is paid once a month (the monthly cash flow is 12/12 = 1 each month). What is the present value? Solution 1: 1.Calculate the monthly interest rate (keeping EAR constant) (1+r monthly ) 12 = 1.10 → r monthly = 0.7974% 2.Use perpetuity formula: PV = 1 / 0.007974 = 125.40 Solution 2: 1.Calculate stated annual interest rate = 0.7974% * 12 = 9.568% 2.Use perpetuity formula: PV = 12 / 0.09568 = 125.40

30. MBA 2004 | 30 Interest rates and inflation: real interest rate Nominal interest rate = 10% Date 0 Date 1 Individual invests $ 1,000 Individual receives $ 1,100 Hamburger sells for $1 $1.06 Inflation rate = 6% Purchasing power (# hamburgers) H1,000 H1,038 Real interest rate = 3.8% (1+Nominal interest rate)=(1+Real interest rate)×(1+Inflation rate) Approximation: Real interest rate ≈ Nominal interest rate - Inflation rate