Finance Concepts I: Present Discounted Value, Risk/Return Tradeoff

Transcription

1 Finance Concepts I: Present Discounted Value, Risk/Return Tradeoff Federal Reserve Bank of New York Central Banking Seminar Preparatory Workshop in Financial Markets, Instruments and Institutions Anthony Rodrigues October 17, 2005

2 Outline Comparison of Payments Received at Different Times Interest Rates, Present Discounted Value Comparison of Uncertain Payments Risk, Expected Return

3 Payments Received at Different Times A future payment has less value than the same payment received today. Is $10 today worth as much to you as $10 received next year? Cash received today can be used anytime between now and next year so it s more valuable than cash received a year from now.

4 Price of Future Payments Drops for More Distant Payment Dates Treasury Strips Prices on 8/26/ Price of $100 at a Specific Future Date /14/04 10/10/06 07/06/09 04/01/12 12/27/14 09/22/17 06/18/20 03/15/23 12/09/25 09/04/28 06/01/31 Payment Date Source: Bloomberg

5 Evaluating Future Payments How to compare payments at different times? The Present Discounted Value (PDV) of the payments converts future payments to current values that can be compared. When you can borrow or save at a particular interest rate, that rate can be used to discount future payments. Years from Now PDV of $1000 at 10% $1000 $909 $826 $751 PDV of $1000 at 15% $1000 $870 $756 $658 More distant payments typically have lower PDVs.

6 Evaluating Future Payments: Examples Suppose that my borrowing/saving rate is 10% (per year). The future values of $1000 today: Years from Now Future Value of $1000 at 10% $1000 $1100 $1210 $1331 Multiply $1000 by: (1+0.10) 0 (1+0.10) 1 (1+0.10) 2 (1+0.10) 3 Present values of $1000 at different times in the future: How much could I borrow today and repay with $1000 in the future? Years from Now PDV of $1000 at 10% $1000 $909 $826 $751 Divide $1000 by: (1+0.10) 0 (1+0.10) 1 (1+0.10) 2 (1+0.10) 3 At 15%? Years from Now PDV of $1000 at 15% $1000 $870 $756 $658 Divide $1000 by: (1+0.15) 0 (1+0.15) 1 (1+0.15) 2 (1+0.15) 3

7 Discount Rates When comparing known future payment streams, a natural discount rate is the opportunity cost of raising the funds. An alternative measure is the return if the funds were used in alternative investments. Individuals would use either the rate on the loan that finances a purchase or the return on funds used for the purchase. One approach for businesses is the user-cost-of-capital, the average interest rate for raising funds. See Capital Budgeting in Corporate Finance by Stephen Ross and Randolph Westerfield. Discount rates often vary with the time horizon.

8 Discount Rates from Treasury Strips Rates from Treasury Strips Ask Yields on 8/26/ Percent /14/04 10/10/06 7/6/09 4/1/12 12/27/14 9/22/17 6/18/20 3/15/23 12/9/25 9/4/28 6/1/31 Maturity Date Source: Bloomberg

9 Net Present Value How to evaluate a stream of payments (for example from an investment)? Compute the Net Present Value (NPV) of the payments. The NPV is the sum of present values of each cash flow. Example (Two $100 investments that pay out over 2-3 years): Time (years) Cash Flow I -$100 $50 $50 $50 Cash Flow II -$200 $120 $130 $0 Discount Rate 1% 5% 10% 15% NPV I $47 $36 $24 $14 NPV II $46 $32 $17 $5

10 Net Present Value More Illustrations: Example (three projects with $100 initial investment): Time (years) Cash Flow A -$100 $90 $30 Cash Flow B -$100 $30 $90 Cash Flow C -$100 $10 $110 Discount Rate 1% 5% 10% 15% NPV A $19 $13 $7 $1 NPV B $18 $10 $2 -$6 NPV C $18 $9 $0 -$8 These projects differ in the timing of their payoffs. The NPV of projects with slow payoffs is more sensitive to changes in the discount rate. Every project has a break-even rate where the NPV is zero. That break-even rate is usually higher for projects with quick payoffs.

11 Related Concepts Internal Rate of Return (IRR): Every project has a break-even discount rate that makes the NPV of an investment or project zero. IRR is that rate. The problem is nonlinear and is solved using a calculator, spreadsheet, or other approach to solving nonlinear equations.

12 Internal Rate of Return Examples What is the IRR of an investment that costs $100 today and pays $50 for the next three years? 23% What is the IRR of an investment that costs $200 today and pays $120 next year and $130 two years from now? 16% Projects can be compared on the basis of IRR. Which of the two sample projects is preferred? The one with the highest IRR. If the discount rate is 14%, which project should be undertaken? If the discount rate is 20%, which project should be undertaken?

13 Discount Rates Implicit in Market Rates Simple Loans Treasury Bills, Treasury Strips Treasury Notes, Bonds Other bonds (issued by corporations or other borrowers) not discussed today but relevant for investment decisions

14 Discount Rates Implicit in Market Rates Simple Loans Treasury Bills, Treasury Strips Cash Flows for a Loan Cash Flows for a Treasury Bill time time White Arrow = Receipt, Red Arrow = Payment

15 Simple Loan (borrow a sum today; pay back another amount later). Examples: Borrow $10 today, pay back $11 a year from now. Interest rate implicit in loan: 11/10 = 1+r; r =.10 (or 10%). Borrow $10 today, pay back $12 two years from now. Interest rate implicit in loan: 12/10 = (1+r) 2 ; r.095 (or about 9.5%). Borrow $10 today, pay back $14 three years from now. Interest rate implicit in loan: 14/10 = (1+r) 3 ; r.119 (or 11.9%).

16 Treasury Bills and Strips U.S Treasury Bills and Strips resemble simple loans. Investor buys the security and receives a single payment at maturity. This type of bond is also called a discount bond or a zero-coupon bond because it is sold at a discount from the face value and has no intermediate coupon payments before maturity. The yield-to-maturity (the interest rate received when the bond is held to maturity) can be computed from the current price. Example: On Tuesday, August 30, 2005, we consider a 3 month bill. The low price for a bill paying $100 in 177 days (the number of days until maturity on February 23, 2006) was $ For 177 days: 100 = 1+ r r or 1.81 % Annualized: 177 ( ) = 1+ r r or 3.77 % Note: This rate differs from the so-called bond equivalent yield on the Treasury bill.

17 U.S. Treasury Bonds and Notes U.S. Treasury bonds ( 30 year maturity) and notes (2, 3, 5, 10 year maturity) pay out a stream of fixed coupon payments every six months as well as a final payment of the face value of the security.

18 U.S. Treasury Bonds U.S. Treasury bonds ( 30 year maturity) and notes (2, 3, 5, 10 year maturity) pay out a stream of fixed coupon payments every six months as well as a final payment of the face value of the security. Cash flows for a Treasury note or bond (red arrow = purchase price, white arrow = payments from Treasury): Note & Bond Cash Flows time

19 U.S. Treasury Bonds U.S. Treasury bonds ( 30 year maturity) and notes (2, 3, 5, 10 year maturity) pay out a stream of fixed coupon payments every six months as well as a final payment of the face value of the security. The security is uniquely identified by the coupon rate (1/2 paid every six months) and the date of payments. Note & Bond Cash Flows time

20 U.S. Treasury Bonds U.S. Treasury bonds ( 30 year maturity) and notes (2, 3, 5, 10 year maturity) pay out a stream of fixed coupon payments every six months as well as a final payment of the face value of the security. The security is uniquely identified by the coupon rate (1/2 paid every six months) and the date of payments. Prices are reported daily in newspapers including the New York Times and the Wall Street Journal. The quotes are illustrated on the next page. Note & Bond Cash Flows time

21 From the Money and Investing section of the Wall Street Journal.

22 Bond Pricing Conventions Example (current 10-year note maturing on August 15, 2014 with 4 1/4 coupon): Bid Price Asked Price Ask Yield August 17, :25 99: August 29, :19 100: Note: The bid price is the price that a dealer would pay for a bond while the asked price is the price at which the bond would be purchased from a dealer. The fractional part of the price is reported in 32 nds (i.e. 99:25 means /32). The paper also reports the Yield-to-Maturity - Ask Yield - associated with the Asking Price. This is the interest rate that sets the PV of the bond payments equal to the current price. Note that price moves opposite to yield.

23 Bond Pricing The yield-to-maturity is a common way of describing the bond s value. The yield varies inversely with the price of the bond. The yield-to-maturity is the interest rate an investor would earn from holding the bond until maturity. Yield and price changes are approximately connected through the Duration of the bond. (Price)/Price - (Duration) Yield / (1+Yield) Duration is a weighted-average of the payment times for the bond, weighted by the discounted value of the payment. Negative sign because price and yield move in opposite directions.

24 Bond pricing II Example (current 10-year Treasury note): Bid Price Asked Price Ask Yield August 17, :25 99: August 29, :19 100: Duration on August 17 was about 8.34 years. Asked price changes from August 17 to August 29. The actual change in price was: ( )/ = or roughly %. The approximate percentage price change is (Price)/Price = - (Duration) Yield / (1+Yield) = ( )/( ) = or about %.

25 Bond Pricing III The return from owning a bond over short periods is largely determined by price (yield) changes during the holding period as well as coupon income and can be quite different from the yield-to-maturity. Price sensitivity to yield changes is largely but not totally captured by duration. The measure can be improved by accounting for convexity - the change in the slope of the price-yield curve as the yield changes, which is positive for standard bonds. More accurate measures of price sensitivity are derived by modeling the sensitivity of individual payments from a bond to yield changes and by adding to get the total effect.

26 Treasury Yield Curves Reflect Current Economic Conditions 7 7/24/ /26/2001 Yield (Percent) 4 3 9/13/2004 8/26/ /26/2002 7/25/ Source: Wall Street Journal Time to Maturity (Years)

27 Individual Treasury Yields Can Reflect Supply and Demand Conditions for Particular Securities Yield (Percent) Year 10-Year 30-Year Time to Maturity (Years) Source: FRB New York

28 Yield Curves Reflect the Credit Quality of the Issuer (8/26/2005) Swap AAA AA BBB BB Treasury Yield (Percent) Time to Maturity (Years) Sources: Wall Street Journal, Bloomberg and Board of Governors

29 Nominal and Real Interest Rates Why are unregulated interest rates usually higher in high inflation countries than in low inflation countries? Inflation is a reason why $1 received in the future is less valuable than $1 received today. The Fisher equation implies a relationship between interest rates and inflation: Nominal Interest Rate = Expected Inflation Rate + Real Interest Rate When expected future inflation rises, interest rates should rise to offset the loss in purchasing power induced by the inflation. The Real Interest Rate is a measure of return in purchasing power of goods available by saving.

30 Risk and Return Most financial decisions involve some uncertainty: Until a future event is resolved, the outcome is unknown. In many cases, decision-making requires balancing the expected return from a strategy with its risk.

31 Evaluating Risky Payoffs One approach to evaluating risky outcomes is to ask if the expected return is high enough to compensate for the risk. A risk neutral individual compares risky outcomes by comparing expected returns. Risk averse individuals prefer less risky outcomes among those with the same expected return. Consequently, a risk averse individual requires a higher expected return to hold riskier assets. Individuals may have asymmetric preferences for gains and losses (loss aversion) - finding that the cost of a particular loss outweighs the benefit of the same size gain.

32 Measuring Risk Evaluating expected return and variance (or the standard deviation) of returns is a specific way of describing return and risk tradeoffs. Other measures of risk may be more appropriate in some circumstances. These could include tail percentiles (the 95 th or 99 th loss percentiles) or computing the spread of returns weighting losses more heavily than gains.

33 Return and Risk by Asset Type There is substantial variation in average return and risk among asset types. Histograms of 10 Year Treasury, Dow Jones Industrial Average, and MSCI Europe returns.

34 10-Year Treasury Note Histogram Mean Standard Deviation Skewness Monthly Return (Annualized Rate)

35 DJIA Histogram Mean Standard Deviation Skewness Monthly Return (Annualized Rate)

36 MSCI Histogram Mean Standard Deviation Skewness Monthly Return (Annualized Rate)

37 Equity Returns Tend To Be More Risky Than Government Bond Returns Frequency 10-Year Treasury MSCI DJIA MSCI 10-Year DJIA Mean Standard Deviation Skewness Return

38 Portfolios Can Have Lower Risk Than Either Component, Reflecting Diversification Frequency DJIA, MSCI DJIA MSCI DJIA MSCI DJIA-MSCI Mean Standard Deviation Skewness Return

39 Diversification Typically a portfolio of several assets has lower risk than simply the weighted average of the assets individual risk. (Diversification). The portfolio return is less sensitive to particular asset returns. Instead it tends to reflect common movements in the returns of the assets that make up the portfolio. (Non-diversifiable risk). Returns on portfolios with many, small investments in different assets will typically reflect these common sources of return only. Returns on individual assets reflect both non-diversifiable risk and asset-specific (idiosyncratic) risk, which can be eliminated by diversification.

40 Diversification II Typically a portfolio of several assets has lower risk than simply the weighted average of the assets individual risk. (Diversification). The extent of risk reduction depends on how closely the asset returns tend to move together. Correlation is one measure of how closely two asset returns move together. Correlations are between -1 (data fall on a downward line) and 1 (upward sloped line). One exceptional case is when the asset returns move exactly together. (Correlation = 1.) Then there is no diversification benefit. Another case is when an asset is negatively correlated with others in the portfolio. Such an asset can substantially reduce portfolio risk. Examples of Correlation

41 Daily Currency Appreciation: Sterling/Euro vs Yen/Dollar Daily Data, Jan through Aug Pound Sterling/Euro Changes (Daily Percent) 4 0 Correlation = Yen/Dollar Changes (Daily Percent)

42 Daily Returns on 10 year and 5 year Treasury Notes 10 yr Treasury Note Return (Daily Percent) 6 0 Daily Data, Jan through Aug Correlation = Bond return = coupon + capital gains yr Treasury Note Return (Daily Percent)

43 Daily Returns: DJIA and 10 year Treasury Note 25 Daily Data, Jan through Aug DJIA Return (Daily Percent) 0 Correlation = Stock return = Capital Gains yr Treasury Note Return (Daily Percent)

44 Daily Returns: DJIA and 10 year Treasury Note Daily Data, Jan through Aug Correlation = DJIA Return (Daily Percent) yr Treasury Note Return (Daily Percent)

45 Monthly Returns: DJIA and MSCI Europe Monthly Data, Jan through August Correlation = 0.73 DJIA Return (Annualized Percent) MSCI Europe Index Return (Annualized Percent)

46 Mean-Variance Frontier Given a set of assets, which combination with a particular expected return has the lowest variance (or SD) and what is that variance? Example - U.S. stocks, European stocks, and U.S. bond returns. How is the analysis modified if there is a riskless asset? Example - Adding Eurodollar deposits.

47 Mean-Variance Frontier - topics Preferences toward risk and return Portfolio outcomes with risky and riskless assets. Portfolio choice

48 Mean Return (%) 16 Mean Variance Trade-Off (Annualized Monthly Returns, ) A risk neutral investor prefers outcomes with higher expected return Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

49 Mean Return (%) 16 Mean Variance Trade-Off (Annualized Monthly Returns, ) A risk averse investor prefers higher expected return and lower risk Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

50 Mean Return (%) 16 Mean Variance Trade-Off (Annualized Monthly Returns, ) Indifference curve shows portfolio outcomes of equal value A risk averse investor requires higher return to bear more risk Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

51 Mean Return (%) 16 Mean Variance Trade-Off (Annualized Monthly Returns, ) Preferred combinations of risk and return Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

52 Mean Return (%) Risk-Return Outcomes ( ) Mean Variance Trade-Off (Annualized Monthly Returns, ) 10 Year Treasury Dow Jones 8 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

53 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 8 6 Morgan Stanley Capital International - Europe 4 2 Eurodollar Deposits Combining the Riskless Asset with One Risky Asset - Portfolio Outcomes Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

54 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) % 100% 10 Year Treasury Dow Jones Morgan Stanley Capital International - Europe 4 2 Eurodollar Deposits Expected return proportional to portfolio weights; Standard deviation grows with weight on risky asset Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

55 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 8 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits 2 The Slopes are the Sharpe Ratios for These Assets = Expected Return over Risk Free Rate per Unit of Risk Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

56 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits 2 Outcomes for Portfolios of Risky Assets Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

57 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

58 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) % Dow Jones 8 10 Year Treasury Dow Jones 6 100% 10 yr. Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

59 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) 10 8 Portfolio tradeoff curved because of diversification 10 Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

60 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

61 Mean Return (%) Same investment possibilities here because optimal weight on European stocks is 0% Mean Variance Trade-Off (Annualized Monthly Returns, ) 10 Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

62 Mean Return (%) Mean Variance Trade-Off (Annualized Monthly Returns, ) Little additional benefit to holding European stocks over the whole period. Might have been better in sub-periods or at other times Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

63 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 2 Eurodollar Deposits Which portfolio to select? Depends on the investor s risk-return tradeoff Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

64 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe Eurodollar Deposits Sources: Datstream, DRI, FRBNY Standard Deviation of Returns (%) Combinations of portfolio outcomes with equal value given one investor s tastes for risk and return

65 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits 2 Preferable portfolios Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

66 Mean Return (%) Best risky portfolio - given these preferences on risk-return Mean Variance Trade-Off (Annualized Monthly Returns, ) 10 Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

67 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits 2 Combining the riskless asset with portfolios of risky assets Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

68 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) % Riskless 10 Year Treasury Dow Jones 6 4 Eurodollar Deposits 100% Risky Morgan Stanley Capital International - Europe A Specific Risky Portfolio Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

69 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) 10 A better tradeoff 8 10 Year Treasury Dow Jones 6 4 Eurodollar Deposits Morgan Stanley Capital International - Europe Another Risky Portfolio Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

70 Mean Return (%) The best tradeoff Mean Variance Trade-Off (Annualized Monthly Returns, ) 8 10 Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

71 Mean Return (%) Mean Variance Trade-Off (Annualized Monthly Returns, ) Best portfolio combining risky and riskless assets given these preferences for risk and 10 Year Treasury return Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

72 Mean Return (%) Mean Variance Trade-Off (Annualized Monthly Returns, ) Best portfolio combining risky and riskless assets given these preferences for risk and return 10 Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Best portfolio with only risky assets Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY

73 Preferences toward risk The actual portfolio selected depends on preferences toward risk and return. Many guides to identifying risk tolerance. Quiz (WSJ July 14, 2000) a) Would you put $5,000 of your assets into an investment where you have a 70% chance of doubling your money (to $10,000) and a 30% chance of losing the entire $5,000? b) How about an 80% chance of doubling to $10,000 and a 20% chance of losing the entire $5,000? c) How about a 60% chance of doubling to $10,000 and a 40% chance of losing the entire $5,000?

74 Determinants of Expected Returns A portfolio of assets eliminates part of the risk in individual assets through diversification. (The diversifiable risk is called idiosyncratic risk.) M e a n R e tu rn (% ) M ean Variance Trade-O ff (Annualized M onthly Returns, ) Diversification Can Create Less Risky Portfolios 10 Year Treasury Dow Jones 6 M organ Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (% ) S o u rc e s : D a ts tre a m, D R I, F R B N Y The remaining risk is called systematic risk. Since systematic risk cannot be eliminated by diversification, investors require compensation for bearing the risk.

75 Determinants of Expected Returns 2 Assets with exposure to systematic risk will generally have higher expected returns. In the mean-variance framework, systematic risk arises from correlation with the benchmark portfolio (tangency portfolio). M e a n R e tu rn (% ) 12 M ean Variance Trade-Off (Annualized Monthly Returns, ) 10 8 Benchmark Portfolio 10 Year Treasury Dow Jones 6 M organ Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (% ) S o u rce s: D a tstre a m, D R I, F R B N Y

76 Determinants of Expected Returns 3 Beta summarizes how the asset return moves with the return on the benchmark portfolio. Expected return on an asset = Expected riskless return + Beta (Expected return on benchmark portfolio - Expected riskless return)

77 Determinants of Expected Returns - Return and Risk Measures Asset Average Return ( ) Standard Deviation ( ) Sharpe Ratio Beta Eurodollar Deposits (Riskless) N/A N/A 10 Year Treasury Dow Jones Morgan Stanley Capital International- Europe Benchmark Portfolio

78 Determinants of Expected Returns - Example Asset Average Return ( ) Beta Excess Return = Return Riskless Return Beta (Benchmark Return Riskless Return) Eurodollar Deposits (Riskless) 4.5 N/A 0 N/A 10 Year Treasury Dow Jones Morgan Stanley Capital International- Europe Benchmark Portfolio

79 Other considerations My discussion of risk and return highlighted market risk, arising from movements in market prices. A practical problem is estimating the prospective tradeoff. (History can be an imperfect guide to the future. How might sources of risk change in the future?) Risk and return and be evaluated in other frameworks. Sources of market risk can be identified: Macroeconomic growth, inflation, exchange rates, resource prices, for example. There are other important sources of risk: Credit risk, Sovereign risk are two examples.

80 Appendix

81 Present Value - Reference Future payments can be converted to current values using the discount rate, r n, for that future period. PDV n = Payment in the n th period / (1+r n ) n ; The conversion or scaling factor, 1 / (1+r n ) n, is called the discount factor for the future period. Discount factors are typically smaller for more distant payments, 1 / (1+r n ) n. Years from Now PDV of $1000 at 10% $1000 $909 $826 $751 PDV of $1000 at 15% $1000 $870 $756 $658 Discount rates are usually not the same for different times in the future. (The single rates were used above just as illustrations.)

82 Illustrations of NPV: Example: The interest rate is 14%. What is the NPV of an investment that costs $100 today and pays $50 for the next three years? Time Cash Flow NPV = /(1+.14) /(1+.14) /(1+.14) 3 = $16.08 Example: The interest rate is 14%. What is the NPV of an investment that costs $200 today and pays $120 next year and $130 two years from now? Time Cash Flow NPV = /(1+.14) + 130/(1+.14) 2 = $5.29 If the cost of funds is 14%, which investment is preferred? (The first.)

83 Illustrations of NPV - Details: NPV of an investment that costs $100 today and pays $50 for the next three years with 14% interest rate? Time Cash Flow PDV NPV = = $16.08 NPV of an investment that costs $200 today and pays $110 next year and $130 two years from now with 14% interest rate? Time Cash Flow PDV NPV = = $5.29

84 Net Present Value Formulas for Reference How to evaluate a stream of payments (for example from an investment)? Time T Cash Flow -I P 1 P 2 P 3 P T Discount Rates 1 r 1 r 2 r 3 r T Compute the Net Present Value of the Payments. The NPV is the sum of present values of each cash flow. NPV = -I + P 1 /(1+r 1 ) 1 + P 2 /(1+r 2 ) P T /(1+r T ) T

85 Related Concepts Internal Rate of Return (IRR): Every project has a break-even discount rate that makes the NPV of an investment or project zero. IRR is the break-even rate, r, that solves: 0 = P 0 + P 1 /(1+r) + P 2 /(1+r) P T /(1+r) T where P 0, P 1, P 2, P T represent the net cash flows at each time period. The problem is nonlinear and is solved using a calculator, spreadsheet, or other approach to solving nonlinear equations.

86 Discount Rates from Treasury bonds Discount rates can be extracted from Treasury bonds Example (notes on August 29, 2005): Bid Price Asked Price Ask Yield 1 5 / 8, Feb :29 98: / 8, Aug :13 98: / 4, Feb :16 97: Payments: Feb 2006 Aug 2006 Feb / 8, Feb / 8, Aug / 4, Feb

87 Discount Calculations Payments: Ask Price February-06 August-07 February / 8 February / 8 August / 4 February Computing the implicit discount rates (r 0.5,r 1.0,r 1.5 ): 1 5 / 8, February 2006: = / (1+r 0.5 ) 1/2 ; so r 0.5 = / 8, August 2006: = / ( ) 1/ / (1+r 1.0 ) 1.0 ; so r 1.0 = / 4, February 2007: = / ( ) 1/ / ( ) / (1+r 1.5 )1.5 ; so r 1.5 =

88 Discount Rates from Treasury Bonds Treasury Notes on Sept. 10, 2004: Ask Price February-06 August-07 February / 8 February / 8 August / 4 February Discount Rates on Sept. 10: Discount Rates Aug % Feb % Aug % The full set of discount rates is called the U.S. Treasury term structure. These rates could be used to discount future payments. (Preferred to yields.) These rates (specifically, the spread between long and short maturity rates) have been shown to help forecast recessions and inflation.

89 Some Games A simple random event is a coin toss which has two outcomes (Heads and Tails) that are equally likely if the coin is tossed fairly. Suppose I offered you the following choice of playing one of two games (based on an independent coin toss) with me: Game 1 Game 2 Heads $2 $4 Tails -$1 -$.50 Game 2 is preferable because it has at least as high payoff for every outcome as game 1. However, you might not play either game if the Heads payoff was not valuable enough to compensate you for the Tails loss.

90 Expected Return and Standard Deviation In the special case where the outcomes and probabilities are known, computing expected return, variance, and standard deviation are easy. Expected Return = Sum of (Returns in State) x (Probability of State) Variance = Sum of (Return in State - Expected Value) 2 x (Probability of State) Standard Deviation = Square Root of Variance

91 Expectation and Standard Deviation - Example Game 1 Game 2 Prob Heads $2 $4 ½ Tails -$1 -$.50 ½ Expected Value $2 x ½ +-$1 x ½ =$.50 Variance {(2-.5) 2 x ½ +(-1-.5) 2 x ½} = 2.25 Standard Deviation $4 x ½ +-$.50 x ½ = $1.75 {(4-1.75) 2 x ½ +( ) 2 x ½} = Sqrt(2.25)= $1.50 Sqrt(5.0625)= $2.25

92 Diversification Combining assets in a portfolio can raise or lower risk, depending on how the asset returns move together. Typically a portfolio of several assets has lower risk than simply the weighted average of the assets individual risk. This follows from a property of variance: Variance{ w r + (1-w) R } = = w 2 Variance{r} + (1-w) 2 Variance{R} + 2 w (1-w) Covariance{r,R} = w 2 Variance{r} + (1-w) 2 Variance{R} + 2 w (1-w) Correlation{r,R} x SD{r} x SD{R} One exceptional case is when the asset returns move exactly together. (Correlation = 1.) Then there is no diversification benefit. Another case is when an asset is negatively correlated with others in the portfolio. Such an asset can substantially reduce portfolio risk.

93 Diversification II There are benefits to diversification even when two returns are independent (Correlation = 0). Example: Two independent coin tosses with payoff like game 2 Payoff Prob Heads $4 ½ Tails -$.50 ½ EV = $1.75, SD = $2.25 Outcome Payoff Prob. Head, Head $8.00 ¼ Head, Tail or Tail, Head $3.50 ½ Tail, Tail -$1.00 ¼ EV = $3.50; SD = $3.18 Risk (SD) of sum of two games, $3.18, is much less than twice the risk of one game, 2 x $2.25.

94 Mean Return (%) 12 Mean Variance Trade-Off (Annualized Monthly Returns, ) Year Treasury Dow Jones 6 Morgan Stanley Capital International - Europe 4 Eurodollar Deposits Standard Deviation of Returns (%) Sources: Datstream, DRI, FRBNY