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Properties of bond prices (6, aka bond-pricing relationships)

(BKM - 16)

bond prices are:
1. inversely related to yields
2. more sensitive to decreases in yields than increases
3. long-term bond prices are more sensitive to yield changes than short-term bond prices
4. sensitivity of bond prices to yield changes increases at a decreasing rate as maturity increases
5. prices of low-coupon bonds are more sensitive to change in yield than prices of high-coupon bonds
6. sensitivity of bond price to change in yield is inversely related to the yield at which it is currently selling


Reason bond prices for long-term bonds are more sensitive to yield changes

(BKM - 16)

more distant CFs will be more heavily discounted, resulting in a larger price reduction


Effective maturity (aka Macaulay's duration or duration)

(BKM - 16)

measure of interest rate risk/sensitivity

effective maturity = weighted average maturity of all CFs where the weights are the discounted CFs / total bond price


Modified duration (D*)

(BKM - 16)

D* = D / (1 + y / k)

where y = YTM


Linear approximation of the change in bond price from a change in interest rates (% change in bond price)

(BKM - 16)

% change in bond price = - D * (change in y / (1 + y / k))

or = -D* change in y

**works best for small changes in yields


Duration rules (5)

(BKM - 16)

1. for a zero-coupon bond, duration = maturity
2. holding maturity constant, duration is lower when coupon rates are higher
3. holding coupon rates constant, duration increases as maturity increases (always true for bonds selling at or above par value)
4. holding other factors constant, duration increases as YTM decreases
5. duration of a perpetuity (w/maturity = infinity) = (1 + y) / y



(BKM - 16)

PV(perpetuity) = perpetuity payment / y


Duration and coupon rate relationship for perpetuities

(BKM - 16)

duration and coupon rate are independent - ONLY true for perpetuities


Duration linear approximation to change in bond prices and actual change in bond prices

(BKM - 16)

duration approximation always understates bond price (b/c the actual price change is convex)

understates the increase in bond price when yield decreases &
overstates the decrease in bond price when yield increases

*curves are tangent at the initial yield


Convexity adjustment to the duration approximation

(BKM - 16)

% change in bond price = -modified duration * change in y + .5 * convexity * (change in y)^2


Convexity (formula for the convexity variable in the convexity adjustment to the duration approximation)

(BKM - 16)

Convexity = [sum over all times, t, of PV(CF(t)) * (t^2 + t / k)] / (Price * (1 + y / k)^2)


Reason that convexity is desirable

(BKM - 16)

because convex bonds increase more in price when yields decrease than they decrease in price when yields increase ( = attractive asymmetry)


"Price" for greater convexity

(BKM - 16)

bond prices are more expensive and they tend to have lower yields


Convexity of callable bonds

(BKM - 16)

has a region of negative convexity with low interest rates near the call price (max price at y intercept) and positive convexity at higher interest rates

in the region of negative convexity, the duration approximation overstates bond value


Effective duration for callable bonds

(BKM - 16)

cannot use the normal effective duration b/c future CFs are unknown (b/c the bond can be terminated)

effective duration = - (change in price / price) / change in interest rate


Differences between the effective duration for callable bonds and the normal Macaulay duration (2)

(BKM - 16)

1. uses the change in interest rate b/c maturity is variable
2. relies on option pricing methodology that accounts for interest rate variability


Interpretation of the effective duration for callable bonds

(BKM - 16)

bond price changes by the effective duration % for a change in r percentage point change in market interest rates around current values


Mortgage-backed securities (aka pass-throughs)

(BKM - 16)

many mortgages pooled together and sold on the fixed-income market

homeowner > pays lender > pays federal agency > pays purchaser of MBS

early termination of mortgage loan from a homeowners option to refinance is similar to a call provision for a bond


Main difference b/w mortgage-backed securities and callable bonds

(BKM - 16)

"call price" = remaining balance of the mortgage loan, but because homeowners do not always refinance, it's possible for the bond price > principal balance

>> means that the call price is not a firm ceiling on bond price


Collateralized mortgage operations (CMOs) and how payments work

(BKM - 16)

redirection of CF streams from mortgage-backed securities into separate securities called tranches with varying risk exposure based on seniority and duration

each tranche receives share of the total interest paid based on the outstanding balance of the tranche but the principal payments are paid off in the order of seniority


Total interest for tranche payments

(BKM - 16)

total interest = total outstanding loan balance from the prior period * rate / frequency of payment


Total principal for tranche payments

(BKM - 16)

total principal = total payment - total interest


Total outstanding balance for tranche payments

(BKM - 16)

total outstanding loan balance = total outstanding loan balance from the prior period - total principal paid


Primary use case for tranches

(BKM - 16)

to allocate interest rate risk (and credit risk, if applicable) across classes

(low/no credit risk with agency-sponsored MBS)


Reasons highest seniority tranches are less risky (2)

(BKM - 16)

1. less exposure to default risk b/c they receive principal payments first
2. less exposure to interest rate risk b/c they have shorter duration


Classes of passive management (2)

(BKM - 16)

1. indexing
2. immunization


Bond indexing

(BKM - 16)

type of passive management that attempts to replicate the performance of a broad bond index with the same risk-reward profile of the bond index it is replicating



(BKM - 16)

type of passive management that protects the investor from interest rate fluctuations by creating a zero-risk profile

sets duration(asset portfolio) = duration(liability portfolio)


Challenges of constructing an indexed bond portfolio (3)

(BKM - 16)

1. bond indexes can contain thousands of securities, making it difficult to purchase each security in the index in proportion to its market value
2. market value of some bonds may be difficult to determine due to low trade volume
3. difficult to keep the portfolio balanced b/c bonds are constantly dropped and added to the index


Use of stratified sampling in bond indexing (aka a cellular approach)

(BKM - 16)

used to ensure the portfolio has a similar composition to the index for important variables such as maturity, coupon rate, credit risk, etc.

>> used b/c it is not feasible to perfectly replicate a broad bond index