The Black-Karasinski Model

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The Black-Karasinski Model

^a

• The BK model stipulates that the short rate follows d ln r = κ(t)(θ(t) − ln r) dt + σ(t) dW.

• This explicitly mean-reverting model depends on time through κ( · ), θ( · ), and σ( · ).

• The BK model hence has one more degree of freedom than the BDT model.

• The speed of mean reversion κ(t) and the short rate volatility σ(t) are independent.

aBlack and Karasinski (1991).

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The Black-Karasinski Model: Discrete Time

• The discrete-time version of the BK model has the same representation as the BDT model.

• To maintain a combining binomial tree, however, requires some manipulations.

• The next plot illustrates the ideas in which t₂ ≡ t1 + Δt₁,

t3 ≡ t2 + Δt2.

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ln rd(t2)

ln r(t1) ln rdu(t3) = ln rud(t3)

ln ru(t2)

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The Black-Karasinski Model: Discrete Time (continued)

• Note that

lnr_d(t2) = lnr(t1) +κ(t1)(θ(t1) − ln r(t1)) Δt1 − σ(t1)

Δt1 , lnru(t2) = lnr(t1) +κ(t1)(θ(t1) − ln r(t1)) Δt1 + σ(t1)

Δt1 .

• To make sure an up move followed by a down move coincides with a down move followed by an up move,

ln rd(t2) + κ(t2)(θ(t2) − ln rd(t2)) Δt2 + σ(t2)√

Δt2 ,

= ln ru(t2) + κ(t2)(θ(t2) − ln ru(t2)) Δt2 − σ(t2)√

Δt2 .

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The Black-Karasinski Model: Discrete Time (continued)

• They imply

κ(t₂) = 1 − (σ(t2)/σ(t₁))

Δt₂/Δt₁

Δt₂ .

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• So from Δt₀, we can calculate the Δt₁ that satisﬁes the combining condition and then iterate.

– t0 → Δt0 → t1 → Δt1 → t2 → Δt2 → · · · → T (roughly).^a

aAs κ(t), θ(t), σ(t) are independent of r, the Δtis will not depend on r.

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The Black-Karasinski Model: Discrete Time (concluded)

• Unequal durations Δti are often necessary to ensure a combining tree.^a

aAmin (1991); Chen (R98922127) (2011); Lok (D99922028) and Lyuu (2016).

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Problems with Lognormal Models in General

• Lognormal models such as BDT and BK share the problem that E^π[ M (t) ] = ∞ for any ﬁnite t if they model the continuously compounded rate.^a

• So periodically compounded rates should be modeled.^b

• Another issue is computational.

• Lognormal models usually do not admit of analytical solutions to even basic ﬁxed-income securities.

• As a result, to price short-dated derivatives on long-term bonds, the tree has to be built over the life of the

underlying asset instead of the life of the derivative.

aHogan and Weintraub (1993).

bSandmann and Sondermann (1993).

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Problems with Lognormal Models in General (concluded)

• This problem can be somewhat mitigated by adopting diﬀerent time steps.^a

– Use a ﬁne time step up to the maturity of the short-dated derivative.

– Use a coarse time step beyond the maturity.

• A down side of this procedure is that it has to be tailor-made for each derivative.

• Finally, empirically, interest rates do not follow the lognormal distribution.

aHull and White (1993).

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The Extended Vasicek Model

^a

• Hull and White proposed models that extend the Vasicek model and the CIR model.

• They are called the extended Vasicek model and the extended CIR model.

• The extended Vasicek model adds time dependence to the original Vasicek model,

dr = (θ(t) − a(t) r) dt + σ(t) dW.

• Like the Ho-Lee model, this is a normal model.

• The inclusion of θ(t) allows for an exact ﬁt to the current spot rate curve.

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The Extended Vasicek Model (concluded)

• Function σ(t) deﬁnes the short rate volatility, and a(t) determines the shape of the volatility structure.

• Many European-style securities can be evaluated analytically.

• Eﬃcient numerical procedures can be developed for American-style securities.

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The Hull-White Model

• The Hull-White model is the following special case, dr = (θ(t) − ar) dt + σ dW.

• When the current term structure is matched,^a θ(t) = ∂f (0, t)

∂t + af (0, t) + σ² 2a

1 − e^−2at .

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The Extended CIR Model

• In the extended CIR model the short rate follows dr = (θ(t) − a(t) r) dt + σ(t)√

r dW.

• The functions θ(t), a(t), and σ(t) are implied from market observables.

• With constant parameters, there exist analytical solutions to a small set of interest rate-sensitive securities.

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The Hull-White Model: Calibration

^a

• We describe a trinomial forward induction scheme to calibrate the Hull-White model given a and σ.

• As with the Ho-Lee model, the set of achievable short rates is evenly spaced.

• Let r0 be the annualized, continuously compounded short rate at time zero.

• Every short rate on the tree takes on a value r0 + jΔr

for some integer j.

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The Hull-White Model: Calibration (continued)

• Time increments on the tree are also equally spaced at Δt apart.

• Hence nodes are located at times iΔt for i = 0, 1, 2, . . . .

• We shall refer to the node on the tree with t_i ≡ iΔt,

r_j ≡ r₀ + jΔr, as the (i, j) node.

• The short rate at node (i, j), which equals rj, is eﬀective for the time period [ ti, ti+1).

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The Hull-White Model: Calibration (continued)

• Use

μi,j ≡ θ(ti) − arj (152) to denote the drift rate^a of the short rate as seen from node (i, j).

• The three distinct possibilities for node (i, j) with three branches incident from it are displayed on p. 1109.

• The middle branch may be an increase of Δr, no change, or a decrease of Δr.

aOr, the annualized expected change.

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The Hull-White Model: Calibration (continued)

(i, j)

(i + 1, j + 2)

*(i + 1, j + 1)

-_{(i + 1, j)}

(i, j)

*(i + 1, j + 1)

-_{(i + 1, j)} j(i + 1, j − 1)

(i, j) -_{(i + 1, j)} j(i + 1, j − 1)

R(i + 1, j − 2)

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The Hull-White Model: Calibration (continued)

• The upper and the lower branches bracket the middle branch.

• Deﬁne

p1(i, j) ≡ the probability of following the upper branch from node (i, j) p2(i, j) ≡ the probability of following the middle branch from node (i, j) p3(i, j) ≡ the probability of following the lower branch from node (i, j)

• The root of the tree is set to the current short rate r₀.

• Inductively, the drift μ_i,j at node (i, j) is a function of (the still unknown) θ(t_i).

– Both describe the expected change from node (i, j).

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The Hull-White Model: Calibration (continued)

• Once θ(t_i) is available, μ_i,j can be derived via Eq. (152) on p. 1108.

• This in turn determines the branching scheme at every node (i, j) for each j, as we will see shortly.

• The value of θ(ti) must thus be made consistent with the spot rate r(0, t_i+2).^a

aNot r(0, t_i+1)!

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The Hull-White Model: Calibration (continued)

• The branches emanating from node (i, j) with their probabilities^a must be chosen to be consistent with μ_i,j and σ.

• This is done by letting the middle node be as closest to the current short rate r_j plus the drift μ_i,jΔt.^b

ap1(i, j), p2(i, j), and p3(i, j).

bA predecessor to Lyuu and Wu’s (R90723065) (2003, 2005) mean- tracking idea, which is the precursor of the binomial-trinomial tree of Dai (B82506025, R86526008, D8852600) and Lyuu (2006, 2008, 2010).

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The Hull-White Model: Calibration (continued)

• Let k be the number among { j − 1, j, j + 1 } that

makes the short rate reached by the middle branch, r_k, closest to

rj + μi,jΔt.

– But note that μ_i,j is still not computed yet.

• Then the three nodes following node (i, j) are nodes (i + 1, k + 1), (i + 1, k), (i + 1, k − 1).

• See p. 1114 for a possible geometry.

• The resulting tree combines because of the constant jump sizes to reach k from j.

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* - j

(0, 0)

* - j

(1, 1)

* - j

(1, 0)

*

(1, −1) -

* - j

- j R

* - j

* --

Δt

6

?^Δr

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The Hull-White Model: Calibration (continued)

• The probabilities for moving along these branches are functions of μ_i,j, σ, j, and k:

p1(i, j) = σ²Δt + η²

2(Δr)² + η

2Δr (153)

p₂(i, j) = 1 − σ²Δt + η²

(Δr)² (153)

p3(i, j) = σ²Δt + η²

2(Δr)² − η

2Δr (153) where

η ≡ μ_i,jΔt + (j − k) Δr.

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The Hull-White Model: Calibration (continued)

• As trinomial tree algorithms are but explicit methods in disguise,^a certain relations must hold for Δr and Δt to guarantee stability.

• It can be shown that their values must satisfy σ√

3Δt

2 ≤ Δr ≤ 2σ√ Δt

for the probabilities to lie between zero and one.

– For example, Δr can be set to σ√

3Δt .^b

• Now it only remains to determine θ(t_i).

aRecall p. 761.

bHull and White (1988).

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The Hull-White Model: Calibration (continued)

• At this point at time t_i,

r(0, t₁), r(0, t₂), . . . , r(0, t_i+1) have already been matched.

• Let Q(i, j) be the state price at node (i, j).

• By construction, the state prices Q(i, j) for all j are known by now.

• We begin with state price Q(0, 0) = 1.

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The Hull-White Model: Calibration (continued)

• Let ˆr(i) refer to the short rate value at time t_i.

• The value at time zero of a zero-coupon bond maturing at time t_i+2 is then

e^−r(0,tⁱ⁺²^{)(i+2) Δt}

=

j

Q(i, j) e^−r^j^Δt E^π

e−ˆr(i+1) Δt ˆr(i) = rj

.(154)

• The right-hand side represents the value of $1 obtained by holding a zero-coupon bond until time t_i+1 and then reinvesting the proceeds at that time at the prevailing short rate ˆr(i + 1), which is stochastic.

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The Hull-White Model: Calibration (continued)

• The expectation in Eq. (154) can be approximated by E^π

e−ˆr(i+1) Δt  ˆr(i) = rj

≈ e^−r^j^Δt

1 − μ_i,j(Δt)² + σ²(Δt)³ 2

. (155) – This solves the chicken-egg problem!

• Substitute Eq. (155) into Eq. (154) and replace μi,j

with θ(ti) − arj to obtain

θ(ti) ≈

j Q(i, j) e−2rjΔt 1 +arj(Δt)2 + σ2(Δt)3/2

− e−r(0,ti+2)(i+2) Δt (Δt)2

j Q(i, j) e−2rjΔt .

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The Hull-White Model: Calibration (continued)

• For the Hull-White model, the expectation in Eq. (155) is actually known analytically by Eq. (25) on p. 162:

E^π

e−ˆr(i+1) Δt ˆr(i) = rj

= e^−r^j^Δt+(−θ(tⁱ^)+ar^j^+σ²^Δt/2)(Δt)².

• Therefore, alternatively,

θ(ti) = r(0, t_i+2)(i + 2)

Δt +σ²Δt

2 +ln

j Q(i, j) e^−2r^j^Δt+ar^j^(Δt)²

(Δt)² .

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The Hull-White Model: Calibration (concluded)

• With θ(ti) in hand, we can compute μi,j,^a the

probabilities,^b and ﬁnally the state prices at time ti+1:

Q(i + 1, j)

=

(i, j^∗) is connected to (i + 1, j) with probability p_j∗

pj^∗e^−r^j∗^ΔtQ(i, j^∗).

• There are at most 5 choices for j^∗ (why?).

• The total running time is O(n²).

• The space requirement is O(n) (why?).

aSee Eq. (152) on p. 1108.

bSee Eqs. (153) on p. 1115.

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Comments on the Hull-White Model

• One can try diﬀerent values of a and σ for each option.

• Or have an a value common to all options but use a diﬀerent σ value for each option.

• Either approach can match all the option prices exactly.

• But suppose the demand is for a single set of parameters that replicate all option prices.

• Then the Hull-White model can be calibrated to all the observed option prices by choosing a and σ that

minimize the mean-squared pricing error.^a

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The Hull-White Model: Calibration with Irregular Trinomial Trees

• The previous calibration algorithm is quite general.

• For example, it can be modiﬁed to apply to cases where the diﬀusion term has the form σr^b.

• But it has at least two shortcomings.

• First, the resulting trinomial tree is irregular (p. 1114).

– So it is harder to program (for nonprogrammers).

• The second shortcoming is again a consequence of the tree’s irregular shape.

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The Hull-White Model: Calibration with Irregular Trinomial Trees (concluded)

• Recall that the algorithm ﬁgured out θ(t_i) that matches the spot rate r(0, t_i+2) in order to determine the

branching schemes for the nodes at time t_i.

• But without those branches, the tree was not speciﬁed, and backward induction on the tree was not possible.

• To avoid this chicken-egg dilemma, the algorithm turned to the continuous-time model to evaluate Eq. (154) on p. 1118 that helps derive θ(t_i).

• The resulting θ(ti) hence might not yield a tree that matches the spot rates exactly.

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The Hull-White Model: Calibration with Regular Trinomial Trees

^a

• The next, simpler algorithm exploits the fact that the Hull-White model has a constant diﬀusion term σ.

• The resulting trinomial tree will be regular.

• All the θ(ti) terms can be chosen by backward induction to match the spot rates exactly.

• The tree is constructed in two phases.

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The Hull-White Model: Calibration with Regular Trinomial Trees (continued)

• In the ﬁrst phase, a tree is built for the θ(t) = 0 case, which is an Ornstein-Uhlenbeck process:

dr = −ar dt + σ dW, r(0) = 0.

– The tree is dagger-shaped (preview p. 1127).

– The number of nodes above the r₀-line is j_max, and that below the line is j_min.

– They will be picked so that the probabilities (153) on p. 1115 are positive for all nodes.

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* - j

(0, 0) r0

* - j

(1, 1)

* - j

(1, 0)

* -

(1, −1) j

* - j

* -

- j R

* - j

* -

- j R

* - j

* -

- j R

* - j

* - -

Δt

6?^Δr

The short rate at node (0, 0) equals r0 = 0; here jmax = 3 and jmin = 2.

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The Hull-White Model: Calibration with Regular Trinomial Trees (concluded)

• The tree’s branches and probabilities are now in place.

• Phase two ﬁts the term structure.

– Backward induction is applied to calculate the βi to add to the short rates on the tree at time t_i so that the spot rate r(0, t_i+1) is matched.^a

aContrast this with the previous algorithm, where it was the spot rate r(0, ti+2) that is matched!

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The Hull-White Model: Calibration

• Set Δr = σ√

3Δt and assume that a > 0.

• Node (i, j) is a top node if j = jmax and a bottom node if j = −jmin.

• Because the root of the tree has a short rate of r₀ = 0, phase one adopts rj = jΔr.

• Hence the probabilities in Eqs. (153) on p. 1115 use η ≡ −ajΔrΔt + (j − k) Δr.

• Recall that k denotes the middle branch.

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The Hull-White Model: Calibration (continued)

• The probabilities become

p1(i, j)

= 1

6

+ a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 − ajΔt + (j − k)

2 , (156)

p2(i, j)

= 2

3 −

a2 j2(Δt)2 − 2ajΔt(j − k) + (j − k)2

, (157)

p3(i, j)

= 1 6

+ a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 + ajΔt − (j − k)

2 . (158)

• p₁: up move; p₂: ﬂat move; p₃: down move.

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The Hull-White Model: Calibration (continued)

• The dagger shape dictates this:

– Let k = j − 1 if node (i, j) is a top node.

– Let k = j + 1 if node (i, j) is a bottom node.

– Let k = j for the rest of the nodes.

• Note that the probabilities are identical for nodes (i, j) with the same j.

• Furthermore, p₁(i, j) = p₃(i, −j).

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The Hull-White Model: Calibration (continued)

• The inequalities

3 − √ 6

3 < jaΔt <

2

3 (159)

ensure that all the branching probabilities are positive in the upper half of the tree, that is, j > 0 (verify this).

• Similarly, the inequalities

− 2

3 < jaΔt < −3 − √ 6 3

ensure that the probabilities are positive in the lower half of the tree, that is, j < 0.

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The Hull-White Model: Calibration (continued)

• To further make the tree symmetric across the r0-line, we let j_min = j_max.

• As

3 − √ 6

3 ≈ 0.184, a good choice is

j_max = 0.184/(aΔt) .

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The Hull-White Model: Calibration (continued)

• Phase two computes the β_is to ﬁt the spot rates.

• We begin with state price Q(0, 0) = 1.

• Inductively, suppose that spot rates

r(0, t1), r(0, t2), . . . , r(0, ti) have already been matched as of time ti.

• By construction, the state prices Q(i, j) for all j are known by now.

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The Hull-White Model: Calibration (continued)

• The value of a zero-coupon bond maturing at time t_i+1 equals

e^−r(0,tⁱ⁺¹^{)(i+1) Δt} =

j

Q(i, j) e^−(βⁱ^+r^j^)Δt by risk-neutral valuation.

• Hence

βi = r(0, t_i+1)(i + 1) Δt + ln

j Q(i, j) e^−r^j^Δt

Δt .

(160)

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The Hull-White Model: Calibration (concluded)

• The short rate at node (i, j) now equals β_i + r_j.

• The state prices at time t_i+1,

Q(i + 1, j), − min(i + 1, jmax) ≤ j ≤ min(i + 1, jmax), can now be calculated as before.

• The total running time is O(nj_max).

• The space requirement is O(n).

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A Numerical Example

• Assume a = 0.1, σ = 0.01, and Δt = 1 (year).

• Immediately, Δr = 0.0173205 and jmax = 2.

• The plot on p. 1138 illustrates the 3-period trinomial tree after phase one.

• For example, the branching probabilities for node E are calculated by Eqs. (156)–(158) on p. 1130 with j = 2 and k = 1.

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* - j

A

* - j

B *

- j

C *

- j

D

- j R

E *

- j

F *

- j

G *

- j

H

* -

I

Node A, C, G B, F E D, H I

r (%) 0.00000 1.73205 3.46410 −1.73205 −3.46410 p1 0.16667 0.12167 0.88667 0.22167 0.08667 p2 0.66667 0.65667 0.02667 0.65667 0.02667 p3 0.16667 0.22167 0.08667 0.12167 0.88667

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A Numerical Example (continued)

• Suppose that phase two is to ﬁt the spot rate curve 0.08 − 0.05 × e^−0.18×t.

• The annualized continuously compounded spot rates are r(0, 1) = 3.82365%, r(0, 2) = 4.51162%, r(0, 3) = 5.08626%.

• Start with state price Q(0, 0) = 1 at node A.

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A Numerical Example (continued)

• Now, by Eq. (160) on p. 1135,

β₀ = r(0, 1) + ln Q(0, 0) e^−r⁰ = r(0, 1) = 3.82365%.

• Hence the short rate at node A equals β₀ + r₀ = 3.82365%.

• The state prices at year one are calculated as Q(1, 1) = p1(0, 0) e^−(β⁰^+r⁰⁾ = 0.160414, Q(1, 0) = p2(0, 0) e^−(β⁰^+r⁰⁾ = 0.641657, Q(1, −1) = p₃(0, 0) e^−(β⁰^+r⁰⁾ = 0.160414.

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A Numerical Example (continued)

• The 2-year rate spot rate r(0, 2) is matched by picking

β1 = r(0, 2)×2+ln

Q(1, 1) e^−Δr + Q(1, 0) + Q(1, −1) e^Δr

= 5.20459%.

• Hence the short rates at nodes B, C, and D equal β₁ + r_j,

where j = 1, 0, −1, respectively.

• They are found to be 6.93664%, 5.20459%, and 3.47254%.

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A Numerical Example (continued)

• The state prices at year two are calculated as

Q(2, 2) = p1(1, 1) e^−(β¹^+r¹⁾Q(1, 1) = 0.018209,

Q(2, 1) = p2(1, 1) e^−(β¹^+r¹⁾Q(1, 1) + p1(1, 0) e^−(β¹^+r⁰⁾Q(1, 0)

= 0.199799,

Q(2, 0) = p3(1, 1) e^−(β¹^+r¹⁾Q(1, 1) + p2(1, 0) e^−(β¹^+r⁰⁾Q(1, 0) +p1(1, −1) e^−(β¹^+r⁻¹⁾Q(1, −1) = 0.473597,

Q(2, −1) = p3(1, 0) e^−(β¹^+r⁰⁾Q(1, 0) + p2(1, −1) e^−(β¹^+r⁻¹⁾Q(1, −1)

= 0.203263,

Q(2, −2) = p3(1, −1) e^−(β¹^+r⁻¹⁾Q(1, −1) = 0.018851.

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A Numerical Example (concluded)

• The 3-year rate spot rate r(0, 3) is matched by picking

β2 = r(0, 3) × 3 + ln

Q(2, 2) e^−2×Δr + Q(2, 1) e^−Δr + Q(2, 0) +Q(2, −1) e^Δr + Q(2, −2) e^2×Δr

= 6.25359%.

• Hence the short rates at nodes E, F, G, H, and I equal β₂ + r_j, where j = 2, 1, 0, −1, −2, respectively.

• They are found to be 9.71769%, 7.98564%, 6.25359%, 4.52154%, and 2.78949%.

• The ﬁgure on p. 1144 plots β_i for i = 0, 1, . . . , 29.

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<HDU +L/

-L+/

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The (Whole) Yield Curve Approach

• We have seen several Markovian short rate models.

• The Markovian approach is computationally eﬃcient.

• But it is diﬃcult to model the behavior of yields and bond prices of diﬀerent maturities.

• The alternative yield curve approach regards the whole term structure as the state of a process and directly speciﬁes how it evolves.

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The Heath-Jarrow-Morton (HJM) Model

^a

• This inﬂuential model is a forward rate model.

• The HJM model speciﬁes the initial forward rate curve and the forward rate volatility structure.

– The volatility structure describes the volatility of each forward rate for a given maturity date.

• Like the Black-Scholes option pricing model, neither risk preference assumptions nor the drifts of forward rates are needed.

aHeath, Jarrow, and Morton (1992).

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Introduction to Mortgage-Backed Securities

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Anyone stupid enough to promise to be responsible for a stranger’s debts deserves to have his own property held to guarantee payment.

— Proverbs 27:13

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Mortgages

• A mortgage is a loan secured by the collateral of real estate property.

• Suppose the borrower (the mortgagor) defaults, that is, fails to make the contractual payments.

• The lender (the mortgagee) can foreclose the loan by seizing the property.

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Mortgage-Backed Securities

• A mortgage-backed security (MBS) is a bond backed by an undivided interest in a pool of mortgages.^a

• MBSs traditionally enjoy high returns, wide ranges of products, high credit quality, and liquidity.

• The mortgage market has witnessed tremendous innovations in product design.

aThey can be traced to 1880s (Levy (2012)).

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Mortgage-Backed Securities (concluded)

• The complexity of the products and the prepayment

option require advanced models and software techniques.

– In fact, the mortgage market probably could not have operated eﬃciently without them.^a

• They also consume lots of computing power.

• Our focus will be on residential mortgages.

• But the underlying principles are applicable to other types of assets.

aMerton (1994).

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Types of MBSs

• An MBS is issued with pools of mortgage loans as the collateral.

• The cash ﬂows of the mortgages making up the pool naturally reﬂect upon those of the MBS.

• There are three basic types of MBSs:

1. Mortgage pass-through security (MPTS).

2. Collateralized mortgage obligation (CMO).

3. Stripped mortgage-backed security (SMBS).

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Problems Investing in Mortgages

• The mortgage sector is one of the largest in the debt market (see p. 3 of the textbook).^a

• Individual mortgages are unattractive for many investors.

• Often at hundreds of thousands of U.S. dollars or more, they demand too much investment.

• Most investors lack the resources and knowledge to assess the credit risk involved.

aThe outstanding balance was US$8.1 trillion as of 2012 vs. the US Treasury’s US$10.9 trillion according to SIFMA.

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Problems Investing in Mortgages (concluded)

• Recall that a traditional mortgage is ﬁxed rate, level payment, and fully amortized.

• So the percentage of principal and interest (P&I) varying from month to month, creating accounting headaches.

• Prepayment levels ﬂuctuate with a host of factors.

• That makes the size and the timing of the cash ﬂows unpredictable.

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Mortgage Pass-Throughs

^a

• The simplest kind of MBS.

• Payments from the underlying mortgages are passed

from the mortgage holders through the servicing agency, after a fee is subtracted.

• They are distributed to the security holder on a pro rata basis.

– The holder of a $25,000 certiﬁcate from a $1 million pool is entitled to 21/2% (or 1/40th) of the cash ﬂow.

• Because of higher marketability, a pass-through is easier to sell than its individual loans.

aFirst issued by Ginnie Mae in 1970.

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Rule for distribution of cash flows: pro rata Loan 2

Loan 10 Loan 1

Pass-through: $1 million par pooled mortgage loans

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Collateralized Mortgage Obligations (CMOs)

• A pass-through exposes the investor to the total prepayment risk.

• Such risk is undesirable from an asset/liability perspective.

• To deal with prepayment uncertainty, CMOs were created.^a

• Mortgage pass-throughs have a single maturity and are backed by individual mortgages.

aIn June 1983 by Freddie Mac with the help of First Boston, which was acquired by Credit Suisse in 1990.

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Collateralized Mortgage Obligations (CMOs) (continued)

• CMOs are multiple-maturity, multiclass debt

instruments collateralized by pass-throughs, stripped mortgage-backed securities, and whole loans.

• The total prepayment risk is now divided among classes of bonds called classes or tranches.^a

• The principal, scheduled and prepaid, is allocated on a prioritized basis so as to redistribute the prepayment risk among the tranches in an unequal way.

aTranche is a French word for “slice.”

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Collateralized Mortgage Obligations (CMOs) (concluded)

• CMOs were the ﬁrst successful attempt to alter

mortgage cash ﬂows in a security form that attracts a wide range of investors

– The outstanding balance of agency CMOs was US$1.1 trillion as of the ﬁrst quarter of 2015.^a

aSIFMA (2015).

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Sequential Tranche Paydown

• In the sequential tranche paydown structure, Class A receives principal paydown and prepayments before

Class B, which in turn does it before Class C, and so on.

• Each tranche thus has a diﬀerent eﬀective maturity.

• Each tranche may even have a diﬀerent coupon rate.

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An Example

• Consider a two-tranche sequential-pay CMO backed by

$1,000,000 of mortgages with a 12% coupon and 6 months to maturity.

• The cash ﬂow pattern for each tranche with zero

prepayment and zero servicing fee is shown on p. 1162.

• The calculation can be carried out ﬁrst for the Total columns, which make up the amortization schedule.

• Then the cash ﬂow is allocated.

• Tranche A is retired after 4 months, and tranche B starts principal paydown at the end of month 4.

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CMO Cash Flows without Prepayments

Interest Principal Remaining principal

Month A B Total A B Total A B Tota

500,000 500,000 1,000,0

1 5,000 5,000 10,000 162,548 0 162,548 337,452 500,000 837,4

2 3,375 5,000 8,375 164,173 0 164,173 173,279 500,000 673,2

3 1,733 5,000 6,733 165,815 0 165,815 7,464 500,000 507,4

4 75 5,000 5,075 7,464 160,009 167,473 0 339,991 339,9

5 0 3,400 3,400 0 169,148 169,148 0 170,843 170,8

6 0 1,708 1,708 0 170,843 170,843 0 0

Total 10,183 25,108 35,291 500,000 500,000 1,000,000

The total monthly payment is $172,548. Month-i numbers reﬂect the ith monthly payment.

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Another Example

• When prepayments are present, the calculation is only slightly more complex.

• Suppose the single monthly mortality (SMM) per month is 5%.

• This means the prepayment amount is 5% of the remaining principal.

• The remaining principal at month i after prepayment then equals the scheduled remaining principal as

computed by Eq. (7) on p. 46 times (0.95)ⁱ.

• This done for all the months, the interest payment at any month is the remaining principal of the previous month times 1%.

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Another Example (continued)

• The prepayment amount equals the remaining principal times 0.05/0.95.

– The division by 0.95 yields the remaining principal before prepayment.

• Page 1165 tabulates the cash ﬂows of the same two-tranche CMO under 5% SMM.

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Another Example (continued)

Interest Principal Remaining principal

Month A B Total A B Total A B Total

500,000 500,000 1,000,00

1 5,000 5,000 10,000 204,421 0 204,421 295,579 500,000 795,57

2 2,956 5,000 7,956 187,946 0 187,946 107,633 500,000 607,63

3 1,076 5,000 6,076 107,633 64,915 172,548 0 435,085 435,08

4 0 4,351 4,351 0 158,163 158,163 0 276,922 276,92

5 0 2,769 2,769 0 144,730 144,730 0 132,192 132,19

6 0 1,322 1,322 0 132,192 132,192 0 0

Total 9,032 23,442 32,474 500,000 500,000 1,000,000

Month-i numbers reﬂect the ith monthly payment.

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Another Example (continued)

• For instance, the total principal payment at month one,

$204,421, can be veriﬁed as follows.

• The scheduled remaining principal is $837,452 from p. 1162.

• The remaining principal is hence

837452 × 0.95 = 795579.

• That makes the total principal payment 1000000 − 795579 = 204421.

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Another Example (concluded)

• As tranche A’s remaining principal is $500,000, all 204,421 dollars go to tranche A.

– Incidentally, the prepayment is

837452 × 5% = 41873.

• Tranche A is retired after 3 months, and tranche B starts principal paydown at the end of month 3.

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Stripped Mortgage-Backed Securities (SMBSs)

^a

• The principal and interest are divided between the PO strip and the IO strip.

• In the scenarios on p. 1161 and p. 1163:

– The IO strip receives all the interest payments under the Interest/Total column.

– The PO strip receives all the principal payments under the Principal/Total column.

aThey were created in February 1987 when Fannie Mae issued its Trust 1 stripped MBS.

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Stripped Mortgage-Backed Securities (SMBSs) (concluded)

• These new instruments allow investors to better exploit anticipated changes in interest rates.^a

• The collateral for an SMBS is a pass-through.

• CMOs and SMBSs are usually called derivative MBSs.

aSee p. 357 of the textbook.