Managing Bond Portfolios

13.1 INTEREST RATE RISK

 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  Long-term bonds tend to be more price sensitive than short-term bonds

 As maturity increases, price sensitivity increases at a decreasing rate  Price sensitivity is inversely related to a bond ’ s coupon rate  Price sensitivity is inversely related to the yield to maturity at which the bond is selling Bond Pricing Relationships

Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually)

Prices of Zero-Coupon Bond (Semiannually Compounding)

 A measure of the effective maturity of a bond  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

Duration: Calculation

Calculating the Duration of Two Bonds

Price change is proportional to duration and not to maturity D * = modified duration D * = D / (1+y)  P/P = - D * ·  y Duration/Price Relationship

 Duration is a key concept ◦ Effective average maturity ◦ Essential tool to immunizing portfolios from interest rate risk ◦ Measure of interest rate sensitivity of a portfolio

Rules for Duration 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 lower Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

Figure 16.2 Bond Duration versus Bond Maturity

Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)

13.2 CONVEXITY

 Duration is only an approximation  Duration asserts that the percentage price change is directly proportional to the change in the bond ’ s yield  Underestimates the increase in bond prices when yield falls  Overestimates the decline in price when the yield rises

Price Yield Duration Pricing Error from Convexity

)( y yD P P    Modify the pricing equation: Convexity is Equal to:             N t t t t t y CF P )1(y)(1 1 Where: CF t is the cash flow (interest and/or principal) at time t.

Callable Bonds  As rates fall, there is a ceiling on possible prices ◦ The bond cannot be worth more than its call price  Negative convexity  Use effective duration:

Price – Yield Curve for a Callable Bond

Mortgage-Backed Securities  Among the most successful examples of financial engineering  Pass-through securities  Subject to negative convexity ◦ When mortgage rates go down, the homeowner has right to prepay the loan. ◦ MBS best viewed as a portfolio of callable amortizing loans  Often sell for more than their principal balance ◦ Homeowners do not refinance their loans as soon as interest rates drop

Price -Yield Curve for a Mortgage- Backed Security

Mortgage-Backed Securities  They have given rise to many derivatives including the CMO (collateralized mortgage obligation) ◦ Use of tranches ◦ Redirect the cash flow stream of the MBS to several classes of derivative securities called tranches. ◦ Tranches may be designed to allocate interest rate risk to investors most willing to bear that risk

 Example ◦ The underlying mortgage pool is divided into three tranches ◦ Original pool has 10 million of 15-year-maturity mortgages, interest rate of 10.5% ◦ Subdivided into three thanches  A 4 million, short-pay  B 3 million, intermediate-pay  C 3 million, long-pay ◦ Suppose, each year, 8% of outstanding loans in the pool prepay

Panel A: Cash Flows to Whole Mortgage Pool; Panels B – D Cash Flows to Three Tranches

13.3 PASSIVE BOND MANAGEMENT

 Bond-Index Funds  Immunization of interest rate risk: ◦ Net worth immunization Duration of assets = Duration of liabilities ◦ Target date immunization Holding Period matches Duration Passive Management

 Bond-Index Funds ◦ Recreate a portfolio that mirrors the composition of an index ◦ Government/corporate/mortgage- backed/Yankee bond ◦ Maturities greater than 1 year  Difficulty ◦ Difficult to purchase each security in the index ◦ Bonds dropped from the index and added ◦ Interest income reinvestment  Sampling Passive Management

Stratification of Bonds into Cells

 Immunization ◦ To insulate their portfolios from interest rate risk ◦ Strategies used by such investors to shield their overall financial status from exposure to interest rate fluctuations  Banks or thrift ◦ Protecting the current net worth of the firm against interest fluctuations  Pension funds ◦ Face an obligation to make payments after a given number of years

 Banks ◦ L: deposits, shorter term, low duration ◦ A: commercial and consumer loans or mortgages, longer duration  Pension funds ◦ L: promise to make payments to retirees, a future fixed obligation ◦ A: the fund, value fluctuated  The idea behind immunization is that duration- matched assets and liabilities let the asset portfolio meet the firm ’ s obligations despite interest rate movements

 Insurance company issues a GIC ◦ $10000, 5-year, zero-coupon, 8% ◦ Its obligation:  If fund with of 8% annual coupon bonds, selling at par, 6 years to maturity

 If interest rate stays at 8%, fully funded the obligation  If interest rates change, two offsetting influences will affect the ability of the fund to grow to the targeted value of ◦ Price risk: if interest rates rise, capital loss, the bonds will be worth less in 5 years ◦ Reinvestment rate risk: higher interest rate, reinvested coupons will grow at a faster rate  If the portfolio duration is chosen appropriately, the two effects will cancel out exactly  If portfolio duration is set equal to the investor’s horizon date, price risk and reinvestment risk exactly cancel out

Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)

Growth of Invested Funds

Immunization

Market Value Balance Sheet

 Rebalancing immunized portfolios  Example  L: payment of in 7 years, 10%  A: 3-year zero, and perpetuities ◦ Calculate the duration of L ◦ Calculate the duration of the asset portfolio ◦ Find the asset mix that sets the duration of A equal to duration of L ◦ Fully fund the obligation  One year later, if interest rate remain. Zero ’ s duration is 2 years, perpetuity ’ s duration remains at 11 years. The weight of the portfolio should be changed to satisfy the 6-year duration of the obligation.

Cash Flow Matching and Dedication  Automatically immunize the portfolio from interest rate movement ◦ Cash flow and obligation exactly offset each other  i.e. Zero-coupon bond  Not widely used because of constraints associated with bond choices  Sometimes it simply is not possible to do

13.4 ACTIVE BOND MANAGEMENT

 Sources of potential profit ◦ Anticipate movements across the entire spectrum of the fixed-income market ◦ Identification of relative mispricing within the fixed-income market  Generate abnormal returns only if the information or insight is superior to the market

 Substitution swap ◦ Exchange of one bond for a nearly identical substitute ◦ Mispriced, discrepancy between the prices represents a profit ◦ Example, sale of 20-year, 8% coupon, YTM 8.05%; purchase of 20-year, 8% coupon, YTM 8.15%. If the two has same credit rating. Active Management: Swapping Strategies

 Inter-market swap ◦ Yield spread between two sectors ◦ Example, if spread between corporate and government bonds is too wide and is expected to narrow, shift from government into corporate. ◦ Spread wider, whether it is the default premium increased, increase in credit risk Active Management: Swapping Strategies

 Rate anticipation swap ◦ Interest rate forecasting ◦ If investors believe rates will fall, then swap into bonds of longer duration  New bond has the same lack of credit risk, but longer duration Active Management: Swapping Strategies

 Pure yield pickup ◦ Not in response to perceived mispricing, but a means of increasing return by holding higher-yield bond ◦ If yield curve is upward-sloping, move into longer-term bonds to earn an expected term premium in higher-yield bonds  Tax swap ◦ Exploit some tax advantage Active Management: Swapping Strategies

Horizon analysis  Select a particular holding period and predict the yield curve 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 ◦ Total return on the bond over the holding period: add the coupon income and prospective capital gain of the bond

 Example ◦ 20-year, coupon rate 10% (annually), YTM-9% ◦ A portfolio manager with a 2-year horizon needs to forecast the total return on the bond over the coming 2 years ◦ In 2 years, the bond will have an 18-year maturity, will sell at YTM of 8%. Coupon payments reinvested in short-term securities over the coming 2 years at 7% ◦ Calculate the 2-year return

 Example  Current price=?  Forecast price=?  Future value of reinvested coupon=?  2-year holding period return=?  Annualized rate of return over the 2-year period=?

Contingent Immunization  A combination of active and passive management  Allow the managers to actively manage until the bond portfolio falls to a threshold level  Once the floor rate or trigger rate is reached, the portfolio is immunized  Active with a floor loss level

 Example ◦ The portfolio value $10 million now, interest rate 10%. Future value will be $12.1 million in 2 years via conventional immunization ◦ If wish to pursue active management, willing to risk losses, minimum acceptable terminal value is $11 million ◦ Only reaching the trigger, immunization initiated; if not, active management

Contingent Immunization