Thursday, November 28, 2019

What Solutions Are Possible to the Free Rider Problem, Both Inside and Outside of Government free essay sample

Outline of the Chapter †¢ Bond pricing and sensitivity of bond pricing to interest rate changes †¢ Duration analysis – What is duration? – What determines duration? †¢ Convexity †¢ Passive bond management – Immunization †¢ Active bond management 16-2 Interest Rate Risk †¢ There is an inverse relationship between interest rates (yields) and price of the bonds. †¢ The changes in interest rates cause capital gains or losses. †¢ This makes fixed-income investments risky. 16-3 Interest Rate Risk (Continued) 16-4 Interest Rate Risk (Continued) What factors affect the sensitivity of the bonds to interest rate fluctuations? †¢ Malkiel’s (1962) bond-pricing relationships – Bond prices and yields are inversely related. – An increase in a bond’s YTM results in a smaller price change than a decrease in yield of equal magnitude. – Prices of long-term bonds tend to be more sensitive to interest rat e changes than prices of short-term bonds. 16-5 Interest Rate Risk (Continued) – The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. We will write a custom essay sample on What Solutions Are Possible to the Free Rider Problem, Both Inside and Outside of Government or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page – Interest rate risk is inversely related to the bond’s coupon rate. Homer and Liebowitz’s (1972) bond-pricing relationship – The sensitivity of a bond’s price to change in its yield is inversely related to the YTM at which the bond currently is selling. 16-6 Interest Rate Risk (Continued) †¢ Why and how different bond characteristics affect interest rate sensitivity? 16-7 Interest Rate Risk (Continued) †¢ Duration – Macaulay’s duration: the weighted average of the times to each coupon or principal payment made by the bond. †¢ Weight applied to each payment is the present value of the payment divided by the bond price. wt D CFt /(1 y ) t , Bondprice T T wt t 1 1 t * wt t 1 16-8 Interest Rate Risk (Continued) †¢ Example: 16-9 Interest Rate Risk (Continued) – Duration is shorter than maturity for all bonds except zero coupon bonds. – Duration is equal to maturity for zero coupon bonds. †¢ Why duration is important? – Simple summary statistic of the effective average maturity of the portfolio. – Tool for immunizing portfolios from interest rate risk. – Measure of the interest rate sensitivity of a portfolio. 16-10 Interest Rate Risk (Continued) – The long-term bonds are more sensitive to interest rate movements than are short-term bonds. – By using duration we can quantify this relation. P P D (1 y ) 1 y 16-11 Interest Rate Risk (Continued) – Modified Duration: †¢ Measure of the bond’s exposure to changes in interest rates. †¢ The percentage change in bond prices is just the product of modified duration and the change in the bond’s yield to maturity. †¢ Note that the equations are only approximately valid for large changes in the bond’s yield. D* P P (1 D /(1 D* y) y) y y 16-12 Interest Rate Risk (Continued) †¢ What determines Duration? – The duration of a zero-coupon bond equals its time to maturity. – Holding maturity constant, a bond’s duration is higher when the coupon rate is lower. Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. †¢ For zero-coupon bonds the maturity=the duration †¢ For coupon bonds duration increases by less than a year with a year’s increase in maturity. 16-13 Interest Rate Risk (Continued) – Ho lding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. †¢ At lower yields the more distant payments made by the bond have relatively greater present values and account for a greater share of the bond’s total value. The duration of a level perpetuity is equal to: (1+y) / y †¢ The PV-weighted CFs early on in the life of the perpetuity dominate the computation of duration. 16-14 Interest Rate Risk (Continued) 16-15 Convexity †¢ By employing the duration concept we can analyse the impact of interest rate changes on bond prices. – The percentage change in the value of a bond approximately equals the product of modified duration times the change in the bond’s yield. – However if this formula were exactly correct then the graph of the percentage change in bond prices as a function of the change in ts yield would be a straight line, with a slope D*. 16-16 Convexity (Continued) †¢ The duration rule is a good approximation for small changes in bond yields. †¢ The duration approximation always understates the value of the bond. †¢ It underestimates the increase in price when yields fall. †¢ It overestimates the decline in prices when yields rise. †¢Due to the curvature of the true price-yield relationshipconvexity 16-17 Convexity (Continued) †¢ Convexity is the rate of change of the slope of the price-yield curve, expressed as a fraction of the bond price. Higher convexity refers to higher curvature in the price-yield relationship. – The convexity of noncallable bonds are usually positive. – The slope of the cuve that shows the price-yield relation increases at higher yields. Convexity 1 P (1 y ) 2 n t 1 CFt (t 2 t ) (1 y )t 16-18 Convexity (Continued) †¢ We can improve the duration approximation for bond price changes by taking into account for convexity. †¢ The new equation becomes: P P D y 1 [Convexity ( y ) 2 ] 2 †¢ The convexity becomes more important when potential interest rate changes are larger. 16-19 Convexity (Continued) †¢ Why convexity is important? †¢ In the figure bond A is more convex than bond B. †¢The price increases are more in A when interest rates fall. †¢The price decreases are less in A when interest rates rise. 16-20 †¢ Callable Bonds Convexity (Continued) – When interest rates are high the curve is convex. The price-yield curve lies above the tangency line estimated by the duration approximation. – When interest rates are low the curve is negative convex (concave). The priceyield curve lies beolw the tangency line. 16-21 Convexity (Continued) In the region of negative convexity the price-yield curve exhibits an unattractive asymmetry. †¢ Increase in interest rates causes a larger price decline than the price gain due to the decrease in interest rates. †¢ Bondholders are compensated with lower prices and higher yields. – Effective Duration Effectiveduration P/P r 16-22 Convexity (Continued) †¢ Macaulayâ₠¬â„¢s Duration – The weighted average of the time until receipt of each bond payment. †¢ Modified Duration – Macaulay’s duration divided by (1+y). – Percentage change in bond price per change in yield. †¢ Effective Duration Percentage change in bond price per change in market interest rates. 16-23 Convexity (Continued) †¢ Mortgage-Backed securities – In a sense similar to callable bonds-subject to negative convexity. – If mortgage rates decrease then homeowners may decide to take a new loan at lower rate and pay the principal for the first mortgage. – Thus there is a ceiling for the bond price written on these mortgage loans as in callable bonds. 16-24 Passive Bond Management †¢ Passive managers take bond prices as fairly set and try to control only the risk of their fixed-income portfolio. Indexing Strategy – Attempts to replicate the performance of a given bond index. – A bond-index portfolio will h ave the same risk-reward profile as the bond market index to which it is tied. †¢ Immunization Strategy – Designed to shield the overall financial status of the institution from exposure to interest rate fluctuations. – Try to establish a zero-risk profile, in which interest rate movements have no impact on the value of the firm. 16-25 Passive Bond Management (Continued) †¢ Bond-Index Funds – Form a portfolio that mirrors the composition of an index that measures the broad market. The major bond indexes in USA are Lehman Aggregate Bond Index, Salomon Smith Barney Broad Investment Grade (BIG) Index, and Merill Lynch U. S. Broad Market Index. – They are market-value weighted indexes of total return. They include government, corporate, mortgage backed, and Yankee bonds with maturity over a year. 16-26 Passive Bond Management (Continued) – They are not easy to replicate however: †¢ There are more than 5000 securities. †¢ Rebalancing problems †¢ Immunization – Banks and pension funds in general try to protect their portfolios from interest rate risk altogether. Banks try to protect the current net worth (net market value) of the firm against interest rate fluctuations. – Pension funds try to protect the future value of their portfolios since they have an obligation to make payments after several years. 16-27 Passive Bond Management (Continued) – Interest rate exposure of the assets and the liabilites should match so the value of assets will follow the value of liabilities whether rates rise or fall. – Duration-matched assets and liabilities let the asset potfolio meet firm’s obligations despite interest rate movements. 16-28 Passive Bond Management (Continued) – What if interest rates change and the duration of the assets and liabilites do not match? †¢ If interest rates increase the fund (asset) the firm has will suffer a capital loss which can affect its ability to meet the firm’s obligations (liabilities). †¢ But the reinvestment rate for the coupon payments will increase so the reinvested coupons will grow at a faster rate. †¢ There will be two offsetting types of interest rate risk: price risk and reinvestment rate risk. – If the durations match, the price risk and reinvestment risk will cancel out each other. 6-29 Passive Bond Management (Continued) †¢ The solid red curve shows the accumulated value of bonds if interest rates remain at 8%. †¢ If interest rates increase at time=t* then the initial impact is capital loss but this loss is offset later by the faster growth rate of reinvested funds. †¢ At the duration time two effects cancel out each ot her. 16-30 Passive Bond Management (Continued) †¢ Graph of the PV of the bond and the zero-coupon obligation as a function of interest rate. †¢ For small changes in interest rates the changes in assets and liabilities are same. For greater changes the curves diverge. †¢ Why? †¢ Convexity †¢The asset and liability are not duration-matched across different interest rates. 16-31 Passive Bond Management (Continued) †¢ Rebalancing immunized portfolios: – As the duration of the assets and liabilities may change because of interest rate fluctuations the portfolio managers have to rebalance their portfolios to match the durations. – Also duration changes with the passage of time. †¢ As time passes the duration of the assets and liabilities will change at different rates. 16-32 Passive Bond Management (Continued) Cash Flow Matching – Buying a zero coupon bond that provides a payment in an amount exactly sufficient to cover the projected cash outlay. †¢ Dedication Strategy – Cash flow matching on a multiperiod basis. – The manager selects either zero-coupon or coupon bonds that provide total cash flows in each period that matches the series of obligations. – Advantage: It is once-and-for-all approach and eliminates the interest rate risk. 16-33 Passive Bond Management (Continued) †¢ Problems of Immunization – In the general duration formula the weights are computed by discounting CFs with the same YTM. If YTM differs than for each CF a present value has to be calculated by using appropriate rate. – Immunization only makes sense for nominal liabilities. †¢ If your future obligation depends on the future inflation you can not immunize it by buying an asset with fixed-income (not indexed). 16-34 Active Bond Management †¢ Finding mispriced securities, especially underpriced ones †¢ Forecasting interest rates – If the decline in interest rates are anticipated the managers will increase the duration of their asset portfolios. – Homer and Liebowitz (1972) characterize portfolio rebalancing activities. 16-35 Active Bond Management (Continued) †¢ Substitution swap – It is an exchange of one bond for another with equal coupon, maturity, quality, call features, sinking fund provisions, and so on. – Motivated by a belief that the market has temporarily mispriced two bonds. †¢ Intermarket spread swap – It is an exchange of bonds when an investor believes that the yield spread between two sectors of the bond market is temporarily out of line. 16-36 Active Bond Management (Continued) †¢ Rate anticipated swap – The investors swap shorter duration bonds with the longer duration ones since they believe that interest rates will decrease. Pure yield pickup swap – The investors swap longer-term bonds with the shorter-term bond when yield curve is upward sloping. 16-37 Active Bond Management (Continued) †¢ Horizon Analysis – Form of interest rate forecasting – The analyst selects a particular holding period and predicts the yield cur ve at end of period – Given a bond’s time to maturity at the end of the holding period its yield can be read from the predicted yield curve and the end-of-period price can be calculated. – The total return over the period is computed as adding the capital gain and coupon income for the holding period. 16-38 Active Bond Management (Continued) †¢ Contingent Immunization – Mixed passive and active strategy. – You have to calculate the funds required to lock in via immunization. – This value is trigger point. – If and when the actual portfolio value dips to the trigger point, active management will cease. 16-39 Active Bond Management (Continued) †¢ Panel A: the portfolio falls in value and hits the trigger at time t* then the immunization strategy pursued and portfolio rises to planned terminal value. †¢ Panel B: the portfolio never reaches the trigger point and is worth more than the planned terminal value. 16-40

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