BKM Chapter 16 Flashcards
(48 cards)
Properties of bond prices (6, aka bond-pricing relationships)
(BKM - 16)
bond prices are:
- inversely related to yields
- more sensitive to decreases in yields than increases
- long-term bond prices are more sensitive to yield changes than short-term bond prices
- sensitivity of bond prices to yield changes increases at a decreasing rate as maturity increases
- prices of low-coupon bonds are more sensitive to change in yield than prices of high-coupon bonds
- 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)
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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*)
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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) (2 formulas)
(BKM - 16)
% change in bond price = - D * (change in y / (1 + y / k))
or = - modified duration * change in y
**works best for small changes in yields
Duration rules (5)
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- for a zero-coupon bond, duration = maturity
- holding maturity constant, duration is lower when coupon rates are higher
- holding coupon rates constant, duration increases as maturity increases (always true for bonds selling at or above par value)
- holding other factors constant, duration increases as YTM decreases
- duration of a perpetuity (w/maturity = infinity) = (1 + y) / y
PV(perpetuity)
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PV(perpetuity) = perpetuity payment / y
Duration and coupon rate relationship for perpetuities
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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 price change 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
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because convex bonds increase more in price when yields decrease than they decrease in price when yields increase ( = attractive asymmetry)
“Price” for greater convexity
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bond prices are more expensive and they tend to have lower yields
Convexity of callable bonds and duration approximation
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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
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cannot use the normal effective duration b/c future CFs are unknown (b/c the bond can be terminated)
effective duration = ((max price - min price) / current price) / (max rate - min rate)
Differences between the effective duration for callable bonds and the normal Macaulay duration (2)
(BKM - 16)
- uses the change in interest rate b/c maturity is variable
- relies on option pricing methodology that accounts for interest rate variability
Interpretation of the effective duration for callable bonds
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bond price changes by the effective duration % for a r percentage point change in market interest rates around current values
Mortgage-backed securities (aka pass-throughs)
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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
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total interest = total outstanding loan balance from the prior period * rate / frequency of payment
Total principal for tranche payments
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total principal = total payment - total interest
Total outstanding balance for tranche payments
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total outstanding loan balance = total outstanding loan balance from the prior period - total principal paid
Primary use case for tranches
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to allocate interest rate risk (and credit risk, if applicable) across classes
(low/no credit risk with agency-sponsored MBS)
