1. Théorie Financière 2004-2005 2. Valeur actuelle Professeur André Farber

2. August 23, 2004 Tfin 2004 02 Present Value |2 Present Value Objectives for this session : 1. Review 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 5. Apply present value for pricing bonds

3. August 23, 2004 Tfin 2004 02 Present Value |3 Interest rates and present value: 1-period Review: 1-period 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

4. August 23, 2004 Tfin 2004 02 Present Value |4 Using Present Value Consider simple investment project: C 0 = -I = -100C 1 = +125 Interest rate r 1 = 5%, DF 1 = 0.9523

5. August 23, 2004 Tfin 2004 02 Present Value |5 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

6. August 23, 2004 Tfin 2004 02 Present Value |6 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%

7. August 23, 2004 Tfin 2004 02 Present Value |7 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

8. August 23, 2004 Tfin 2004 02 Present Value |8 Several periods: 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

9. August 23, 2004 Tfin 2004 02 Present Value |9 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

10. August 23, 2004 Tfin 2004 02 Present Value |10 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

11. August 23, 2004 Tfin 2004 02 Present Value |11 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

12. August 23, 2004 Tfin 2004 02 Present Value |12 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:

13. August 23, 2004 Tfin 2004 02 Present Value |13 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

14. August 23, 2004 Tfin 2004 02 Present Value |14 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

15. August 23, 2004 Tfin 2004 02 Present Value |15 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

16. August 23, 2004 Tfin 2004 02 Present Value |16 The Euro yield curve

17. August 23, 2004 Tfin 2004 02 Present Value |17 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)

18. August 23, 2004 Tfin 2004 02 Present Value |18 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

19. August 23, 2004 Tfin 2004 02 Present Value |19 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 =

20. August 23, 2004 Tfin 2004 02 Present Value |20 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.

21. August 23, 2004 Tfin 2004 02 Present Value |21 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

22. August 23, 2004 Tfin 2004 02 Present Value |22 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

23. August 23, 2004 Tfin 2004 02 Present Value |23 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

24. August 23, 2004 Tfin 2004 02 Present Value |24 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 If r 1 = r 2 =...=r

25. August 23, 2004 Tfin 2004 02 Present Value |25 Useful formulas: summary 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

26. August 23, 2004 Tfin 2004 02 Present Value |26 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%

27. August 23, 2004 Tfin 2004 02 Present Value |27 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

28. August 23, 2004 Tfin 2004 02 Present Value |28 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

29. August 23, 2004 Tfin 2004 02 Present Value |29 Bond Valuation Objectives for this session : –1.Introduce the main categories of bonds –2.Understand bond valuation –3.Analyse the link between interest rates and bond prices –4.Introduce the term structure of interest rates –5.Examine why interest rates might vary according to maturity

30. August 23, 2004 Tfin 2004 02 Present Value |30 Zero-coupon bond Pure discount bond - Bullet bond The bondholder has a right to receive: one future payment (the face value) F at a future date (the maturity) T Example : a 10-year zero-coupon bond with face value $1,000 Value of a zero-coupon bond: Example : If the 1-year interest rate is 5% and is assumed to remain constant the zero of the previous example would sell for

31. August 23, 2004 Tfin 2004 02 Present Value |31 Level-coupon bond Periodic interest payments (coupons) Europe : most often once a year US : every 6 months Coupon usually expressed as % of principal At maturity, repayment of principal Example : Government bond issued on March 31,2000 Coupon 6.50% Face value 100 Final maturity 2005 2000 2001 2002 2003 2004 2005 6.50 6.50 6.50 6.50 106.50

32. August 23, 2004 Tfin 2004 02 Present Value |32 Valuing a level coupon bond Example: If r = 5% Note: If P 0 > F: the bond is sold at a premium If P 0 <F: the bond is sold at a discount Expected price one year later P 1 = 105.32 Expected return: [6.50 + (105.32 – 106.49)]/106.49 = 5%

33. August 23, 2004 Tfin 2004 02 Present Value |33 When does a bond sell at a premium? Notations: C = coupon, F = face value, P = price Suppose C / F > r 1-year to maturity: 2-years to maturity: As: P 1 > F with

34. August 23, 2004 Tfin 2004 02 Present Value |34 A level coupon bond as a portfolio of zero- coupons « Cut » level coupon bond into 5 zero-coupon Face value Maturity Value Zero 1 6.50 1 6.19 Zero 2 6.50 2 5.89 Zero 3 6.50 3 5.61 Zero 4 6.50 4 5.35 Zero 5 106.50 5 83.44 Total 106.49

35. August 23, 2004 Tfin 2004 02 Present Value |35 Law of one price Suppose that you observe the following data: What are the underlying discount factors? Bootstrap method 100.97 = DF 1 104 105.72 = DF 1 7 + DF 2 107 101.56 = DF 1 5.5 + DF 2 5.5 + DF 3 105.5

36. August 23, 2004 Tfin 2004 02 Present Value |36 Bond prices and interest rates Bond prices fall with a rise in interest rates and rise with a fall in interest rates

37. August 23, 2004 Tfin 2004 02 Present Value |37 Sensitivity of zero-coupons to interest rate

38. August 23, 2004 Tfin 2004 02 Present Value |38 Duration for Zero-coupons Consider a zero-coupon with t years to maturity: What happens if r changes? For given P, the change is proportional to the maturity. As a first approximation (for small change of r): Duration = Maturity

39. August 23, 2004 Tfin 2004 02 Present Value |39 Duration for coupon bonds Consider now a bond with cash flows: C 1,...,C T View as a portfolio of T zero-coupons. The value of the bond is: P = PV(C 1 ) + PV(C 2 ) +...+ PV(C T ) Fraction invested in zero-coupon t: w t = PV(C t ) / P Duration : weighted average maturity of zero-coupons D= w 1 × 1 + w 2 × 2 + w 3 × 3+…+w t × t +…+ w T ×T

40. August 23, 2004 Tfin 2004 02 Present Value |40 Duration - example Back to our 5-year 6.50% coupon bond. Face value Value w t Zero 1 6.50 6.19 5.81% Zero 2 6.50 5.89 5.53% Zero 3 6.50 5.61 5.27% Zero 4 6.50 5.35 5.02% Zero 5 106.50 83.44 78.35% Total 106.49 Duration D =.0581×1 + 0.0553×2 +.0527 ×3 +.0502 ×4 +.7835 ×5 = 4.44 For coupon bonds, duration < maturity

41. August 23, 2004 Tfin 2004 02 Present Value |41 Price change calculation based on duration General formula: In example: Duration = 4.44 (when r=5%) If Δr =+1% : Δ ×4.44 × 1% = - 4.23% Check: If r = 6%, P = 102.11 ΔP/P = (102.11 – 106.49)/106.49 = - 4.11% Difference due to convexity

42. August 23, 2004 Tfin 2004 02 Present Value |42 Duration -mathematics If the interest rate changes: Divide both terms by P to calculate a percentage change: As: we get:

43. August 23, 2004 Tfin 2004 02 Present Value |43 Yield to maturity Suppose that the bond price is known. Yield to maturity = implicit discount rate Solution of following equation:

44. August 23, 2004 Tfin 2004 02 Present Value |44 Spot rates Consider the following prices for zero-coupons (Face value = 100): Maturity Price 1-year 95.24 2-year 89.85 The one-year spot rate is obtained by solving: The two-year spot rate is calculated as follow: Buying a 2-year zero coupon means that you invest for two years at an average rate of 5.5%

45. August 23, 2004 Tfin 2004 02 Present Value |45 Forward rates You know that the 1-year rate is 5%. What rate do you lock in for the second year ? This rate is called the forward rate It is calculated as follow: 89.85 × (1.05) × (1+f 2 ) = 100 → f 2 = 6% In general: (1+r 1 )(1+f 2 ) = (1+r 2 )² Solving for f 2 : The general formula is:

46. August 23, 2004 Tfin 2004 02 Present Value |46 Forward rates :example Maturity Discount factor Spot rates Forward rates 1 0.9500 5.26 20.8968 5.60 5.93 30.8444 5.80 6.21 40.7951 5.90 6.20 50.7473 6.00 6.40 Details of calculation: 3-year spot rate : 1-year forward rate from 3 to 4

47. August 23, 2004 Tfin 2004 02 Present Value |47 Term structure of interest rates Why do spot rates for different maturities differ ? As r 1 r 1 = r 2 if f 2 = r 1 r 1 > r 2 if f 2 < r 1 The relationship of spot rates with different maturities is known as the term structure of interest rates Time to maturity Spot rate Upward sloping Flat Downward sloping

48. August 23, 2004 Tfin 2004 02 Present Value |48 Forward rates and expected future spot rates Assume risk neutrality 1-year spot rate r 1 = 5%, 2-year spot rate r 2 = 5.5% Suppose that the expected 1-year spot rate in 1 year E(r 1 ) = 6% STRATEGY 1 : ROLLOVER Expected future value of rollover strategy: ($100) invested for 2 years : 111.3 = 100 × 1.05 × 1.06 = 100 × (1+r 1 ) × (1+E(r 1 )) STRATEGY 2 : Buy 1.113 2-year zero coupon, face value = 100

49. August 23, 2004 Tfin 2004 02 Present Value |49 Equilibrium forward rate Both strategies lead to the same future expected cash flow → their costs should be identical In this simple setting, the foward rate is equal to the expected future spot rate f 2 =E(r 1 ) Forward rates contain information about the evolution of future spot rates