- 1. 1. Introduction and Basics of Investments 12/20/2013 1
- 2. The purpose of this paper is to help you learn how to manage your Money so that you will derive the maximum benefit from what you earn. To accomplish you need 1) to learn about investment alternatives that are available today, 2) to develop a way of analyzing and thinking about investments that will remain with you in years to come when new and different opportunities become available. The paper mixes theory, practical, and application of the theories using modern/contemporary tool Microsoft Excel. Evaluation – (Internal -100) and BREAKUP WILL BE TOLD AT LATER STAGE. Classes – 30 classes 1. Introduction and Basics of Investments 12/20/2013 2
- 3. The detailed topics are given separately as a file, but in brief we shall be discussing over following topics a) Investments Basics – Risk and Return Measurement b) Modern Portfolio Theories c) Equity Analysis and Debt Analysis d) Portfolio Optimization e) Portfolio Evaluation References: a) Investment Analysis and Portfolio management by Frank K. Reilly and Keith C. Brown. – Thomson Publication b) Investments by William F. Sharpe, Gordon J. Alexander, and Jeffery V. Bailey. – Prentice Hall Publication c) Class Notes and Handouts. 1. Introduction and Basics of Investments 12/20/2013 3
- 4. Let us Start the session!!! 1. Introduction and Basics of Investments 12/20/2013 4
- 5. Is the current commitment of rupees for a period of time in order to derive future payments that will compensate the investor for a) the time the funds are committed (Pure time value of money or rate of interest) b) the expected rate of inflation, and c) the uncertainty of the future of payments (investment risk so there has to be risk premium) So in short individual does trade a rupee today for some expected future stream of payments that will be greater than the current outlay. Investor invest to earn a return from savings due to their deferred consumption so they require a rate of return that compensates them. 1. Introduction and Basics of Investments 12/20/2013 5
- 6. So we answered following Questions? ◦ Why people invest? ◦ What they want from their investment? And now we will discuss ◦ Where all they can invest and what parameters they adopt to invest? ◦ How they measure risk and return and how they 1. Introduction and Basics of Investments 12/20/2013 6
- 7. Gold Shares Silver Real Estate Bonds Indira Vikas Patra Post Office Deposits Bank Deposits Mutual Funds Debentures PF NSC 1. Introduction and Basics of Investments 12/20/2013 7
- 8. Investments Parameters ◦ Return ◦ Risk ◦ Time Horizon ◦ Tax Considerations ◦ Liquidity ◦ Marketability 1. Introduction and Basics of Investments 12/20/2013 8
- 9. •Derivatives Return •Shares •MFs Equity Fund •Real Estate •MFs Debt Funds •Debentures •NSC, Post-Office Deposit Kisan Vikas Patra •PF •Bonds •Bank Deposit •Gold Risk 1. Introduction and Basics of Investments 12/20/2013 9
- 10. Next How to Measure Return and Risk??? 1. Introduction and Basics of Investments 12/20/2013 10
- 11. Return ◦ Risk ◦ Historical HPR ◦ Expected ◦ Historical HPY Expected 1. Introduction and Basics of Investments 12/20/2013 11
- 12. 12/20/2013 2. Return and Risk 12
- 13. What we did in last class… 12/20/2013 2. Return and Risk 13
- 14. ◦ Why people invest? ◦ What they want from their investment? ◦ Where all they can invest and what parameters they adopt to invest? 2. Return and Risk 12/20/2013 14
- 15. Return ◦ Historical HPR (Holding Period Return) HPY (Holding Period Yield) ◦ Expected Risk ◦ Historical Variance and Standard Deviation Coefficient of Variance ◦ Expected Variance and Standard Deviation Coefficient of Variance 2. Return and Risk 12/20/2013 15
- 16. ◦ HPR - When we invest, we defer current consumption in order to add our wealth so that we can consume more in future, hence return is change in wealth resulting from investment. If you commit Rs 1000 at the beginning of the period and you get back Rs 1200 at the end of the period, return is Holding Period Return (HPR) calculated as follows HPR = (Ending Value of Investment)/(beginning value of Investment) = 1200/1000 = 1.20 ◦ HPY – conversion to percentage return, we calculate this as follows, HPY = HPR-1 = 1.20-1.00 = 0.20 = 20% ◦ Annual HPR = (HPR)1/n = (1.2) ½, = 1.0954, if n is 2 years. ◦ Annual HPY = Annual HPR – 1 = 1.0954 – 1 = 0.0954 = 9.54% 2. Return and Risk 12/20/2013 16
- 17. Over a number of years, a single investments will likely to give high rates of return during some years and low rates of return, or possibly negative rates of return, during others. We can summarised the returns by computing the mean annual rate of return for this investment over some period of time. There are two measures of mean, Arithmetic Mean and Geometric Mean. Arithmetic Mean = ∑HPY/n Geometric Mean = [{(HPR1) X (HPR2) X (HPR3)}1/n -1] 2. Return and Risk 12/20/2013 17
- 18. Year Beginning Value Ending Value HPR HPY 1 1000 1150 1.15 0.15 2 1150 1380 1.2 0.2 3 1380 1104 0.8 -0.2 AM = [(0.15) + (0.20) + (-0.20)]/3 = 5% GM = [(1.15) X (1.20) X (0.80)] 12/20/2013 1/3 – 1 = 3.35% 2. Return and Risk 18
- 19. Year Beginning Value Ending Value HPR HPY 1 100 200 2.0 1.0 2 200 100 0.5 -0.5 AM = [(1.0) + (-0.50)]/2 = 0.50/2 = 0.25 = 25% GM = [(2.0) X (0.50)] 12/20/2013 1/2 – 1 = 0.00% 2. Return and Risk 19
- 20. Expected Return = ∑RiPi, • where i varies from 0 to n • R denotes return from the security in i outcome • P denotes probability of occurrence of i outcome Economy Growth Deep Recession 5% Mild Recession 20% Average Economy 50% Mild Boom 20% Strong Boom 12/20/2013 Probability of Occurrence 5% 2. Return and Risk 20
- 21. Economy Growth T-Bills Corporate Bonds Equity A Equity B 5% 8% 12% -3% -2% 20% 8% 10% 6% 9% 50% 8% 9% 11% 12% Mild Boom 20% 8% 8.50% 14% 15% Strong Boom 5% 8% 8% 19% 26% 8.00% 9.20% 10.30% 12.00% Deep Recession Mild Recession Average Economy Probability of Occurrence 100% Expected Rate of Return 12/20/2013 2. Return and Risk 21
- 22. Probability Distribution of Equity "A" 60% Probability 50% 40% 30% Series1 20% 10% 0% Series1 -13.300% -4.300% 0.700% 3.700% 8.700% 5% 20% 50% 20% 5% Dispersion from Expected Return 2. Return and Risk 12/20/2013 22
- 23. 2. Return and Risk 12/20/2013 23
- 24. 12/20/2013 2. Return and Risk 24
- 25. Webster define it as a hazard; as a peril ; as a exposure to loss or injury. Chinese definition – Means its a threat but at the same time its an opportunity So what is in practice risk means to us? 2. Return and Risk 12/20/2013 25
- 26. Actual return can vary from our expected return, i.e. we can earn either more than our expected return or less than our expected return or no deviation from our expected return. Risk relates to the probability of earning a return less than the expected return, and probability distribution provide the foundation for risk measurement. 2. Return and Risk 12/20/2013 26
- 27. Variance – is a measure of the dispersion of actual outcomes around the mean, larger the variance, the greater the dispersion. Variance = ∑(HPYi – AM)2 / (n) where i varies from 1 to n. Variance is measured in the same units as the outcomes. Standard Deviation – larger the S.D, the greater the dispersion and hence greater the risk. Coefficient of Variation – risk per unit of return, = S.D/Mean Return 2. Return and Risk 12/20/2013 27
- 28. Variance – is a measure of the dispersion of possible outcomes around the expected value, larger the variance, the greater the dispersion. Variance = ∑(ki – k)2 (Pi) where i varies from 1 to n. Variance is measured in the same units as the outcomes. Standard Deviation – larger the S.D, the greater the dispersion and hence greater stand alone risk. Coefficient of Variation – risk per unit of return, = S.D/Expected Return 2. Return and Risk 12/20/2013 28
- 29. Expected Return or Risk Measure T-Bills Corporate Bonds Expected return 8% 9.20% 10.30% 12.00% Variance 0% 0.71% 19.31% 23.20% Standard Deviation 0% 0.84% 4.39% 4.82% Coefficient of Variation 0% 0.09% 0.43% 0.40% Semi variance 0.00% 0.19% 12.54% 11.60% 12/20/2013 2. Return and Risk Equity A Equity B 29
- 30. • Variance and Standard Deviation The spread of the actual returns around the expected return; The greater the deviation of the actual returns from expected returns, the greater the varian • Skewness The biasness towards positive or negative returns; • Kurtosis The shape of the tails of the distribution ; fatter tails lead to higher kurtosis 2. Return and Risk 12/20/2013 30
- 31. 2. Return and Risk 12/20/2013 31
- 32. 12/20/2013 2. Return and Risk 32
- 33. Thank You!!! 12/20/2013 2. Return and Risk 33
- 34. 12/20/2013 2. Return and Risk 34
- 35. What we did in last class… 12/20/2013 2. Return and Risk 35
- 36. ◦ How do we calculate Risk and Return of a single Security? ◦ Historical and Expected Risk and Return ◦ Concept of Price Adjustments - Bonus, Stock Split, and Demerger 2. Return and Risk 12/20/2013 36
- 37. 12/20/2013 2. Return and Risk 37
- 38. Markowitz Portfolio Market Model/Index Model 2. Return and Risk 12/20/2013 38
- 39. ◦ Measure of Return – Probability Distribution and its Weighted Average Mean. ◦ Measure Risk – Standard Deviation (Variability) of Expected Return of a Portfolio? ◦ Investors do not like risk and like return. ◦ Nonsatiation – always prefer higher levels of terminal wealth to lower levels of terminal wealth. ◦ Risk Aversion – investor choose the portfolio with smaller S.D. ( not like Fair Gamble). ◦ Investors get positive utility with return as they help them in maximising wealth and vice-versa with Risk. 2. Return and Risk 12/20/2013 39
- 40. Utility U2 U1 Wealth 2. Return and Risk 12/20/2013 40
- 41. Expected Return of Portfolio S.D. of Portfolio 2. Return and Risk 12/20/2013 41
- 42. Expected Return of Portfolio Risk Averse Risk Taker S.D. of Portfolio 2. Return and Risk 12/20/2013 42
- 43. All the portfolios on a given indifference curve provide same level of utility. They Never Intersect Each Other otherwise they will violate law of transitivity. An investor has an infinite number Indifference Curves. A risk-averse investor will find any portfolio that is lying on an indifference curve that is “farther north-west” to be more desirable than any portfolio lying on an indifference curve that is “not as far northwest”. 2. Return and Risk 12/20/2013 43
- 44. Every investor has an indifference map representing his/her preferences for expected returns and standard deviations. An investor should determine the expected return and standard deviation for each potential portfolio. The two assumptions of Nonsatiation and risk aversion cause indifference curves to be positively sloped and convex. The degree of risk aversion will decide the extent of positiveness in slope of indifference curves. More Flat is the indifference curves of an individual – higher risk aversion and vice-versa. 2. Return and Risk 12/20/2013 44
- 45. 12/20/2013 2. Return and Risk 45
- 46. Expected return of Portfolio = ∑Xiki Xi is the fraction of the portfolio in the ith asset, n is the number of assets in the portfolio. Here i range from 0 to n. 2. Return and Risk 12/20/2013 46
- 47. Probability Possible Returns 0.35 0.3 0.2 0.08 0.1 0.12 0.15 Expected Return 12/20/2013 0.028 0.03 0.024 0.14 0.021 10.30% 2. Return and Risk 47
- 48. Weight Expected Returns of Securities 0.2 0.3 0.3 0.2 Expected Return of Portfolio 12/20/2013 2. Return and Risk 0.1 0.11 0.12 0.13 0.02 0.033 0.036 0.026 0.115 48
- 49. 12/20/2013 2. Return and Risk 49
- 50. Expected Return Year Stock A Stock B Portfolio AB 2001 -10% 40% 15% 35% -5% 15% 2004 -5% 35% 15% 2005 12/20/2013 15% 2003 S.D. -10% 2002 Avg Return 40% 15% 15% 15% 15% 15% 15% 22.64% 22.64% 0.00% 2. Return and Risk 50
- 51. 50% 40% 30% Series1 20% Series2 10% Series3 0% -10% 2001 2002 2003 2004 2005 -20% 2. Return and Risk 12/20/2013 51
- 52. Expected Return Year Stock A Stock B Portfolio AB 50% 40% 2001 40% -10% 15% 30% 2002 -10% 40% 15% 2003 35% -5% 15% 2004 -5% 35% 15% 0% 2005 15% 15% 15% -10% Avg Retur n S.D. 20% 10% 2001 2002 2003 2004 2005 -20% 15% 15% 15% 22.64 % 22.64 % 0.00% Correlation Coefficient = -1.0 2. Return and Risk 12/20/2013 52
- 53. Expected Return Year Stock A Stock B Portfolio AB 2001 40% 40% 40% -5% -5% -5% 2004 35% 35% 35% 2005 12/20/2013 -10% 2003 S.D. -10% 2002 Avg Return -10% 15% 15% 15% 15% 15% 15% 22.64% 22.64% 22.64% 2. Return and Risk 53
- 54. 50% 40% 30% Series1 20% Series2 10% Series3 0% -10% 2001 2002 2003 2004 2005 -20% 2. Return and Risk 12/20/2013 54
- 55. Expected Return Stock A Year Stock B 50% Portfolio AB 40% 30% 2001 -10% -10% -10% 2002 40% 40% 40% 2003 -5% -5% -5% 2004 35% 35% 35% 0% 15% -10% 2005 Avg Retur n S.D. 15% 15% 20% 10% 2001 2002 2003 2004 2005 -20% 15% 15% 22.64 % 22.64 % 15% Correlation Coefficient = +1.0 22.64% 2. Return and Risk 12/20/2013 55
- 56. So Risk is not a simple weighted average of risk with securities like we did in measuring Expected Return………..we need to know following things to measure risk of a Portfolio. Covariance between two securities Correlation Coefficient between two securities Variance of securities Standard Deviation of Securities 2. Return and Risk 12/20/2013 56
- 57. Standard deviation of Portfolio =( ∑ ∑Xi Xj σij)1/2 where i and j vary from 0 to n, and σij is covariance between i and j securities. σij = ρijσi σj, where σi & σj is standard deviation of i and j respectively. 2. Return and Risk 12/20/2013 57
- 58. Thank You!!! 12/20/2013 2. Return and Risk 58
- 59. 12/20/2013 2. Return and Risk 59
- 60. 12/20/2013 2. Return and Risk 60
- 61. What we did in last class… 12/20/2013 2. Return and Risk 61
- 62. ◦ How do we calculate Risk and Return of a single Security? ◦ Historical and Expected Risk and Return ◦ Concept of Price Adjustments - Bonus, Stock Split, and Demerger 2. Return and Risk 12/20/2013 62
- 63. ER 0.103 0.12 Variance SD Coefficient of Variation 0.0019310 0.04394315 0.42663248 0.00232 0.048166 0.401386 Covariance Correlation Coefficient Risk Tolerance 0.00202 0.95436882 0.5 -0.75 12/20/2013 2. Return and Risk 63
- 64. Portfolios Proportion in X Proportion in Y Return A 1 0 5.00% B 0.8 0.2 7.00% C 0.75 0.25 7.50% D 0.5 0.5 10.00% E 0.25 0.75 12.50% F 0.2 0.8 13.00% G 0 1 15.00% 12/20/2013 2. Return and Risk 64
- 65. Portfolios Lower Bound Upper Bound No relationship A 20.00% 20.00% 20.00% B 10.00% 23.33% 17.94% C 0.00% 26.67% 18.81% D 10.00% 30.00% 22.36% E 20.00% 33.33% 27.60% F 30.00% 36.67% 33.37% G 40.00% 40.00% 40.00% 12/20/2013 2. Return and Risk 65
- 67. Expected Return Feasible Sets of Portfolios 0.1250 0.1200 0.1150 0.1100 0.1050 0.1000 0 0.01 0.02 0.03 0.04 0.05 0.06 Standard Deviations 2. Return and Risk 12/20/2013 67
- 68. Two Conditions 1) Offer Maximum Return for varying levels of Risk, and 2) Offer Minimum Risk for varying levels of expected return All the feasible sets are not efficient unless it passes through this test 2. Return and Risk 12/20/2013 68
- 69. Expected Return Efficient Sets of Portfolios Standard Deviations 2. Return and Risk 12/20/2013 69
- 70. Thank You!!! 12/20/2013 2. Return and Risk 70
- 71. 12/20/2013 2. Return and Risk 71
- 73. To Identify Investor’s Optimal Portfolio Investor’s needs to estimate ◦ Expected returns ◦ Variances ◦ Covariances ◦ Riskfree Return Investor’s need to identify tangency portfolio The Optimal Portfolio involves an investment in the tangency portfolio along with either riskfree borrowing or lending to get linear efficient portfolio
- 74. Investors think in terms of single period and choose portfolios on the basis of each portfolio’s expected return and standard deviation over that period. Investors can borrow/lend unlimited amount at a given riskfree rate. No restrictions on short sale. Homogenous Expectations. Assets are perfectly divisible and marketable at a going price. Perfect market. Investors are price takers i.e. their buy/sell activity will not affect stock price
- 75. Allows us to change our focus from how an individual should invest to what would happen to securities prices if everyone invested in same manner. Enables us to develop the resulting equilibrium relationship between each security’s risk and return. Everyone would obtain in equilibrium the same tangency portfolio (Homogenous Expectation) Also the linear efficient frontier same for all investors as they face same risk free rate. So only reason investors to have dissimilar portfolios is their different preferences towards risk and return (Indifference Curve). However they will chose the same combination of risky securities.
- 76. Return Indifference Curve Linear Efficient Curve M Risky Securities Efficient Curve Risk Free Rate Risk
- 77. So we are saying in brief Separation theorem The Optimal combination of risky assets for an investor can be determined without any knowledge of the investor’s preferences toward risk and return. Now…..
- 78. Second Point of CAPM is Each investor will hold a certain positive amount of each risky security. Current market price of each security will be at a level where total no. of shares demanded equals the no. of shares outstanding. Risk free rate will be at a level where the total no. of money borrowed equals the total amount of money lent. Hence there is an equilibrium or we can say that tangency portfolio which fulfilled above criteria is also termed as market portfolio. And we define market portfolio as given in next slides….
- 79. The Market Portfolio is a portfolio consisting of all securities I which the proportions invested in each security corresponds to its relative market value. The relative market value of a security is simply equal to the aggregate market value of the security divided by the sum the aggregate market values of all the securities.
- 81. Rp = Rf + (Rm- Rf) X σp σm Slope of line is price of risk And Intercept is price of time
- 82. Uses variance as a measure of risk Specifies that a portion of variance can be diversified away, and that is only the non-diversifiable portion that is rewarded. Measures the non-diversifiable risk with beta, which is standardized around one. Translates beta into expected return Expected Return = Riskfree rate + Beta * Risk Premium
- 83. The risk of any asset is the risk that it adds to the market portfolio Statistically, this risk can be measured by how much an asset moves with the market (called the covariance) Beta is a standardized measure of this covariance Beta is a measure of the non-diversifiable risk for any asset can be measured by the covariance of its returns with returns on a market index, which is defined to be the asset's beta. The cost of equity will be the required return, Cost of Equity = Riskfree Rate + Equity Beta * (Expected Mkt Return – Riskfree Rate)
- 84. (A) Risk-free Rate (B) The Expected Market Risk Premium (The Premium Expected For Investing In Risky Assets Over The Riskless Asset) (C) The Beta Of The Asset Being Analyzed.
- 86. Two Conditions 1) 2) Offer Maximum Return for varying levels of Risk, and Offer Minimum Risk for varying levels of expected return All the feasible sets are not efficient unless it passes through this test
- 88. Efficient Sets and Feasible Sets IC 3 IC 2 B D Feasible Sets IC 1 A C
- 91. Expected Return of Portfolio = ∑Xiri, where i range from 0 to n. and X is Proportion of total investment in ith security and ri is expected return of the security. Standard deviation of Portfolio =( ∑ ∑Xi Xj σij)1/2 where i and j vary from 0 to n, and σij is covariance of i and j securities. σij = ρijσi σj, where σi & σj is standard deviation of i and j respectively.
- 92. Portfolios Proportion in X Proportion in Y Return A 1 0 5.00% B 0.83 0.17 6.70% C 0.67 0.33 8.30% D 0.5 0.5 10.00% E 0.33 0.67 11.71% F 0.17 0.83 13.30% G 0 1 15.00%
- 93. Portfolios Lower Bound Upper Bound No relationship A 20.00% 20.00% 20.00% B 10.00% 23.33% 17.94% C 0.00% 26.67% 18.81% D 10.00% 30.00% 22.36% E 20.00% 33.33% 27.60% F 30.00% 36.67% 33.37% G 40.00% 40.00% 40.00%
- 94. Expected Return Upper and Lower Bounds to Portfolios 16.00% 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00% Standard Deviations
- 96. ri = αiI + βiI rI + εiI Where, ri = return on security i for given period αiI = intercept form βiI = slope form rI = return on market index I for the same period εiI =random error
- 97. ri = αiI + βiI rI
- 98. βiI = σiI σI2 σiI = Covariance σI2 = Variance of Market Index
- 99. Security A Security B Intercept 2% -1% Actual Return on the Market index X beta 10% X 2% = 12% 10% X 8% = 8% Actual Return on Security 9% 11% Random Error 9% - (2% + 12%) 11% - (-1% +8%) = = -5% 4%
- 100. Infotech versus S&P 500: 1992-1996 8.00% 6.00% 4.00% 2.00% -15.00% -10.00% 0.00% -5.00% -2.00% 0.00% -4.00% -6.00% 5.00% 10.00% 15.00% 20.00%
- 101. σi2 =βiI2X σI2 + σεi2 Where , σi2 = variance of security i βiI2X σI2 = Market risk of security i σεi2 = Unique risk of security i
- 102. rp = ∑Xi ri Where i range from o to n. and Xi = proportion of investment in security i. ri = expected return of security i. Also, ri = αiI + βiI rI + εiI Hence rp = ∑Xi (αiI + βiI rI + εiI) .....continued
- 103. rp = ∑Xi (αiI + βiI rI + εiI) = ∑Xi αiI + (∑Xi βiI ) rI + ∑XiεiI = αpI + βpI rI + Intercept Slope X independent Variable Where i range from o to n. εpI Random Error
- 104. σ2p =β2pIσ2I + σ2εp Where , β2pI = [∑Xi βiI] 2 ----- Systematic Risk σ2εp = ∑Xi2 σ2εi ----- Unique Risk
- 105. σp Total Risk Unique Risk Market Risk N
- 106. Stock Portfolio Weight Beta Expected Return of Stock Variance of Stock A 0.25 0.5 0.4 0.07 B 0.25 0.5 0.25 0.05 C 0.5 1 0.21 0.07 Variance of Market 0.06
- 107. Residual Variance of each of the stocks? Beta of the portfolio? Variance of the Portfolio? Expected Return on the portfolio? Portfolio Variance on teh basis of Markowitz Variance – Covariance formula. Covariance (A,B) = 0.020 Covariance (A,C) = 0.035 Covariance (B,C) = 0.035
- 108. Duration, Convexity and Portfolio Immunization
- 109. Bondholders have interest rate risk even if coupons are guaranteed - Why? Unless the bondholders hold the bond to maturity, the price of the bond will change as interest rates in the economy change
- 110. The following basic principles are universal for bonds : Changes in the value of a bond are inversely related to changes in the rate of return. The higher the rate of return (i.e., yield to maturity (YTM)), the lower the bond value. Long-term bonds have greater interest rate There is a greater probability that interest rates will rise (increase YTM) and thus negatively affect a bond’s market price, within a longer time period than within a shorter period Low coupon bonds have greater interest rate sensitivity than high coupon bonds In other words, the more cash flow received in the short-term (because of a higher coupon), the faster the cost of the bond will be recovered. The same is true of higher yields. Again, the more a bond yields in today’s dollars, the faster the investor will recover its cost.
- 111. Bond Pricing Relationships Price Inverse relationship between price and yield An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield (convexity) YTM
- 112. Bond Coupon Maturity YTM A B C 12% 12% 3% 5 years 30 years 30 years 10% 10% 10% D 3% 30 years 6% 0 Change in yield to maturity (%) A B C D
- 113. There are three factors that affect the way the price of a bond reacts to changes in interest rates. These three factors are: ◦ The coupon rate. ◦ Term to maturity. ◦ Yield to maturity. Long-term bonds tend to be more price sensitive than shortterm bonds Price sensitivity is inversely related to the yield to maturity at which the bond is selling
- 114. Duration measures the combined effect of all the factors that affect bond’s price sensitivity to changes in interest rates. Duration is a weighted average of the present values of the bond's cash flows, where the weighting factor is the time at which the cash flow is to be received. The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment Duration is shorter than maturity for all bonds except zero coupon bonds Duration is equal to maturity for zero coupon bonds Note: Each time the discount rate changes, the duration must be recomputed to identify the effect of the change. Duration tells us the sensitivity of the bond price to one percent change in interest rates.
- 115. 1200 Cash flow 1000 800 Bond Duration = 5.97 years 600 400 200 0 1 2 Actual cash flows PV of cash flows 3 4 5 6 7 8 Year Area where PV of CF before and after balance out
- 116. CF t (1 wt t y) Price T D t wt t 1 CFt Cash Flow for Period t PV of cash flows as a % of bond price
- 117. An adjusted measure of duration can be used to approximate the price volatility of a bond Modified Duration Macaulay Duration 1 YTM m Where: m = number of payments a year YTM = nominal YTM
- 118. Eg. Coupon = 8%, yield = 10%, years to maturity = 2 Time (years) C1 Payment PV of CF (10%) C4 Weight C1 XC4 .5 40 38.095 .0395 .0198 1 40 36.281 .0376 .0376 1.5 40 34.553 .0358 .0537 2.0 1040 855.611 .8871 1.7742 sum 964.540 1.000 1.8853 DURATION
- 119. 1. 2. 3. 4. It’s a simple summary statistic of the effective average maturity of the portfolio; It is an essential tool in immunizing portfolios from interest rate risk; It is a measure of interest rate risk of a portfolio Equal duration assets are equally sensitive to changes in interest rates
- 120. Price change is proportional to duration and not to maturity P P ( y) D 1 y • Where D = duration D P D * 1 y P * D y D* is the 1st derivative of bond’s price with respect to yield ie. D* = (-1/P)(dP/dY)
- 121. Duration/Price Relationship P P ( y) D 1 y The relative change in the price of the bond is proportional to the absolute change in yield [dY ] where the factor of proportionality [D/(1+Y)] is a function of the bond’s duration. For a given change in yield, longer duration bonds have greater relative price volatility. This implies that anything that causes an increase in a bond's duration serves to raise its interest rate sensitivity, and vice-versa. Therefore, if interest rates are expected to fall, bonds with lower coupons can be expected to appreciate faster than higher coupon bonds of the same maturity
- 122. E.g. 1. What would be the percentage change in the price of a bond with a modified duration of 9, given that interest rates fall 50 basis points (i.e.. 0.5%)? P * D P y = (-9)(-.05%) = 4.5% E.g. 2. What would be the % change in price of a bond with a Macaulay Duration of 10 if interest rates rise by 50 basis points (i.e.. 0.5%) The current YTM is 4%. D D* = 1 = 10/1.04 =9.615 y Therefore , % change in price ΔP P D * Δy = (-9.615)(.5%) = -4.81%
- 123. Rule 1 The duration of a zero-coupon bond equals its time to maturity Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is
- 124. Duration approximates price change but isn’t exact For small changes in yields, duration is close but for larger changes in yields, there can be a large error Duration always underestimates the value of bond price increases when yields fall and overestimates declines in price when yields rise
- 125. Price Pricing error from convexity Yield Duration (approximates a line vs a curve)
- 126. A is more convex than B: If rates inc A’s price falls less than B’s If rates dec A’s price rises more than B’s Convexity is desirable for investors so they will pay for it (ie. A’s yield is probably less than B’s) Bond A 0 Bond B Change in yield to maturity (%)
- 127. Definition of convexity: ◦ The rate of change of the slope of the price/yield curve expressed as a fraction of the bond’s price.
- 128. 1. 2. 3. Inverse relationship between convexity and coupon rate Direct relationship between maturity and convexity Inverse relationship between yield and convexity
- 129. Classical immunization is a passive bond portfolio strategy to shield fixed-income assets from interest rate risk. It is done by setting the duration of a bond portfolio equal to its time horizon. In an immunized bond portfolio the effects of rising rates reducing the capital value of the bonds, and increasing the return on reinvestment of coupon payments, exactly offset each other, and vice-versa. Immunization techniques thus - Reduces interest rate risk to zero - Shields portfolio from interest rate fluctuations
- 130. Type of Risks to Bondholders Price risk / Market risk : An investor who buys a bond with maturity more than his investment horizon is exposed to market risk : if interest rates go up (down) the investor is worse off (better off). D >H The bond exposes the investor to market risk if the duration of the bond exceeds his investment horizon Reinvestment risk: An investor who buys a bond with maturity less than (or equal to) her investment horizon is exposed to reinvestment risk. So, if interest rates go up (down) the investor is better off (worse off). D < H The bond exposes the investor to reinvestment risk if the duration of the D=bond is shorter than his(H) matches Duration (D), the two risks will H If Holding Period investment horizon exactly offset each other – Bond is said to be immunized.
- 131. Banks are concerned with the protection of the current net worth or net market value of the firm ,whereas, pension fund and insurance companies are concerned with protecting the future value of their portfolio. Here I’ll take the example of pension fund which has to pay back pension fund of Rs. 10,000/- to one of its investor, with guaranteed rate of 8% after 5 years. So, it is obligated to pay Rs. 10,000 *(1.08)^=Rs. Rs.14,693.28 in years. So, suppose, pension fund company chooses to fund its obligation with Rs. 10,000 , of 8% annual coupon bond selling at par value with 6 years maturity. So, if interest rate remains at 8% the amount accrued will exactly be equal to the obligation of Rs.14,693.28 in 5 years. Now we consider two scenarios, where interest rate goes down to 7% and in second case it reaches 9%. In 7% scenario, amount accrued will be equal to Rs. 14,694.05 in years and in 9% scenario it will be Rs. 14,696.02 in years. The three scenarios with their accumulated value of invested payments.
- 132. Payment number Yrs. Remaining until obligation If rates remain at 8% Accumulated value of invested payment Formula used Value of formula 1 4 800*(1.08)^4 1088.391168 2 3 800*(1.08)^3 1007.7696 3 2 800*(1.08)^2 933.12 4 1 800*(1.08)^1 864 5 0 800*(1.08)^0 800 sale of bond 0 10800/1.08 10000 14693.28077
- 133. Yrs. Remaining until obligation Payment number Accumulated value of invested payment if rates fall to 7% Formula used Value of formula 1 800*(1.07)^4 1048.636808 2 3 800*(1.07)^3 980.0344 3 2 800*(1.07)^2 915.92 4 1 800*(1.07)^1 856 5 sale of bond 4 0 800*(1.07)^0 800 0 10800/1.07 10093.45794 14694.04915
- 134. Yrs. Remaining until obligation Payment number Accumulated value of invested payment if rates fall to 9% formula used value of formula 1 800*(1.09)^4 1129.265288 2 3 800*(1.09)^3 1036.0232 3 2 800*(1.09)^2 950.48 4 1 800*(1.09)^1 872 5 sale of bond 4 0 800*(1.09)^0 800 0 10800/1.09 9908.256881 14696.02537
- 135. Accumulated value of invested payment
- 136. Rebalancing required as duration declines more slowly than term to maturity Modified duration changes with a change in market interest rates Yield curves shift In practice, we can’t rebalance the portfolio constantly because of transaction costs
- 137. The duration of a bond portfolio is equal to the weighted average of the durations of the bonds in the portfolio The portfolio duration, however, does not change linearly with time. The portfolio needs, therefore, to be rebalanced periodically to maintain target date immunization
- 138. Risk Immunization: elimination of interest rate risk by matching duration of financial assets and liabilities Financial Institutions: Banks especially utilize these techniques Assets of Bank Loans to customers Liabilities of Bank Deposits from Customers Auto CDs Mortgage Bank accounts Student (Bank is Owed this $) (Bank Owes this $)
- 139. Assets of Bank ◦ Duration=15 yr • Liabilities of Bank – Duration=5 yr If interest rates drop, the value of assets increases more than the value of liabilities decreases. - Bank Value Increases. If interest rates increase, the value of the assets decrease more than the value of liabilities increases. - Bank Value Drops. Bank is speculating on interest rates
- 140. Assets of Bank - Duration=15 yr • Liabilities of Bank - Duration=15 yr For a bank to not be speculating on interest rates Duration of Assets = Duration of Liabilities
- 141. Commercial banks borrow money by accepting deposits and use those funds to make loans. The portfolio of deposits and the portfolio of loans may both be viewed as bond portfolios, with the deposit portfolio constituting the liability portfolio and the loan portfolio constituting the asset portfolio. If a bank’s deposits and loans have different maturities, the bank may lose money in the event of an overall change in interest rate levels. To eliminate this risk, banks may wish to immunize their portfolio. A portfolio is immunized if the value of the portfolio is not affected by a change in interest rates. Immunization is accomplished by managing the duration of the portfolio.
- 142. Bank Immunization Case (contd.) Balance Sheet of Simple National Bank Original Position Assets Loan Portfolio Value Portfolio Duration Interest Rate Liabilities $1,000 5 years 10% Deposit Portfolio Value Portfolio Duration Owners' Equity Interest Rate $1,000 1 year $0 10% Following Rise in Rates to 12 Percent Assets Loan Portfolio Value Liabilities $909 Deposit Portfolio Value Owners' Equity $982 - $72 Notice that the duration of the assets is 5 years and the duration of the liabilities is 1 year.
- 143. Bank Immunization Case (contd.) Assume that interest rates rise from 10% to 12% on both deposit and loan portfolios. What is the change in value of the deposit and loan portfolios? Applying the following duration formula: dP i = - D d (1 + r i (1 + r i i ) ) P i Deposit Portfolio dP = -1 (.02/1.10) $1,000 = -$18.18 Loan Portfolio dP = -5 (.02/1.10) $1,000 = - $90.91 So the deposits (liabilities) have decreased in value by $18.18 and the assets have decreased in value by $90.91. The combined effect is equal to a $72 reduction in equity.
- 144. Bank Immunization Case (contd.) Immunized Balance Sheet of Simple National Bank Original Position Assets Loan Portfolio Value Portfolio Duration Interest Rate Liabilities $1,000 3 years 10% Deposit Portfolio Value Portfolio Duration Owners' Equity Interest Rate $1,000 3 years $0 10% Following Rise in Rates to 12 Percent Assets Loan Portfolio Value Liabilities $945 Deposit Portfolio Value Owners' Equity $945 $0
- 145. Bank Immunization Case (contd.) The previous table illustrates the impact of interest rates changes for a bank with immunization. Both the liabilities and assets have a duration of 3 years. Estimate the price change using the duration formula: dP = -3 (.02/1.10) $1,000 = - $54.55 Because the bank is immunized against a change in interest rates, the change in rates have an equal and offsetting effect on the liabilities and assets of the bank leaving the equity position of the bank unchanged.