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

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

t tt

w CF y ice ( )1 Pr

twtDT

t

1

CF CashFlow for period tt

Price change is proportional to duration and not to maturity

D* = modified duration

D* = D / (1+y)P/P = - D* · y

(1 )

1

P yD

P y

*P

D yP

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

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

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

)(21 2yConvexityyD

P

P

Modify the pricing equation:

Convexity is Equal to:

N

tt

t tty

CFP 1

22

)1(y)(1

1

Where: CFt is the cash flow (interest and/or principal) at time t.

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:

/Effective Duration =

P P

r

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

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

13.3 PASSIVE BOND MANAGEMENT

Bond-Index Funds Immunization of interest rate risk:

◦Net worth immunizationDuration of assets = Duration of

liabilities◦Target date immunization

Holding Period matches Duration

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

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 10000 of 8% annual coupon bonds, selling at par, 6 years to maturity

510000* 1.08 14693.28

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 14693.28◦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

Rebalancing immunized portfolios Example

L: payment of 19487 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.

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.

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

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

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

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=?

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