> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-iii-asset-markets/chapter_13_housing_mbs.md). # Chapter 13: Housing Finance and Mortgage Securities *Part III: Asset Markets — Financial Economics: Claims, Prices, and Holders* *** ## Opening Episode: The Summer of 2003 On June 13, 2003 the ten-year Treasury yield closed at 3.11 percent, the lowest reading since the Eisenhower administration. Freddie Mac's weekly survey put the average thirty-year fixed mortgage rate at 5.21 percent that same week, a record at the time, and the Mortgage Bankers Association's refinance index had touched an all-time high a fortnight earlier. Nearly every American mortgage in existence was refinanceable, and a large share of them were being refinanced. Then the market turned. On June 25 the Federal Open Market Committee cut the funds target by twenty-five basis points to one percent rather than the fifty some traders had priced, and the statement said less about unconventional easing than the market had come to expect. On July 15 Alan Greenspan's semiannual testimony discouraged the idea that the Federal Reserve was preparing to buy long-dated Treasuries outright. Those are ordinary repricings of the policy path, and on their own they explain a move of perhaps forty or fifty basis points. The move was closer to a hundred and forty. By the first days of August the ten-year yield was near 4.5 percent, and by early September it had reached roughly 4.6 percent — one of the sharpest ten-week selloffs in the intermediate Treasury market since 1994. Implied volatility on Treasury options reached the highest levels the standard index had recorded. The refinance index fell by roughly four-fifths between late May and early September. What happened in between is this chapter's subject. The holders of American mortgage-backed securities own a claim whose maturity is decided by borrowers. When rates fell through the spring, borrowers refinanced, the securities paid down fast, and their effective maturity was short. When rates rose in July, expected refinancing collapsed, the securities stopped paying down, and their effective maturity lengthened — by contemporaneous estimates the duration of the agency MBS index went from something like one year in June to something like four by September. Investors who had promised themselves a fixed interest-rate exposure suddenly held far more of it than they wanted. To get back to target they sold Treasuries and paid fixed in interest-rate swaps. They were selling duration into a market that was already falling, which pushed yields higher, which extended duration further, which required more selling. Put a number on the aggregate. Agency MBS outstanding stood at roughly three and a half trillion dollars. A three-year extension across that stock is about ten and a half trillion dollars-years of interest-rate exposure appearing from nowhere; expressed in ten-year notes, whose modified duration is near eight, that is roughly 1.3 trillion dollars of equivalent selling if every holder hedged. If only a third of the market hedges, it is still on the order of four to five hundred billion dollars of ten-year equivalents — into a Treasury market that at the time turned over a fraction of that in a day. Perli and Sack (2003), writing from the Federal Reserve Board as it happened, estimated that mortgage hedging materially amplified the move and raised the volatility of long rates; Duarte (2008) and Hanson (2014) later confirmed the channel with better identification, Hanson showing that the aggregate negative convexity of the mortgage universe forecasts the volatility and the term premium of long-dated Treasuries. The thesis is stated in that arithmetic. The American mortgage market is large enough, and its holders' risk management mechanical enough, that a change in *who has to hedge what* moves the entire rates complex. This chapter builds the instrument (§13.1), the securitization that makes it tradable (§13.2), the pricing of the option inside it (§13.3), and the one adjacent asset class that shares its discount-rate logic (§13.4). Section 13.5 asks the question this book requires and this market answers unusually well: who holds the mortgage, and what does their hedging do to rates? *** ## 13.1 The Instrument The claim at the center of this chapter is a contract that exists almost nowhere else in the world in the form Americans take for granted: a **thirty-year, fixed-rate, level-payment, freely prepayable** loan on a single home. Three of those adjectives are unusual on their own and the combination is close to unique. The rate is fixed for thirty years, so the borrower bears no interest-rate risk at all. The payment is level, so the amortization schedule is decided at origination. And the loan may be repaid in full at any time, at par, without penalty. That last feature is the whole chapter. **The borrower holds a call option on her own debt**, struck at the outstanding balance, exercisable at any moment for thirty years, and carrying no penalty at exercise. Costless to exercise is not the same as free: the borrower pays for the option in the note rate, and §13.3 measures what she pays as the gap between the security's static spread and its option-adjusted spread. ### The arithmetic Write $$\mathrm{Bal}\_0$$ for the original balance, $$r$$ for the monthly note rate (the annual rate divided by twelve), and $$n$$ for the number of monthly payments. The level payment $$\mathrm{PMT}$$ is the annuity that exhausts the balance: $$ \mathrm{PMT} = \mathrm{Bal}\_0 \cdot \frac{r}{1 - (1+r)^{-n}} $$ which is Chapter 3 §3.1's present-value formula solved for the coupon. Take a loan of four hundred thousand dollars at a 6.5 percent annual rate over thirty years, so $$r = 0.0054167$$ and $$n = 360$$. The payment is 2,528.27 dollars a month. Table 13.1 shows what that payment does. **Table 13.1: Amortization of a thirty-year level-payment loan (four hundred thousand dollars, 6.5 percent)** | Month | Payment | Interest | Principal | Ending balance | | ----- | -------- | -------- | --------- | -------------- | | 1 | 2,528.27 | 2,166.67 | 361.61 | 399,638.39 | | 12 | 2,528.27 | 2,142.47 | 385.80 | 395,529.10 | | 60 | 2,528.27 | 2,032.34 | 495.93 | 374,443.91 | | 120 | 2,528.27 | 1,844.24 | 684.03 | 339,104.51 | | 180 | 2,528.27 | 1,584.29 | 943.98 | 290,236.56 | | 360 | 2,528.27 | 13.62 | 2,514.65 | 0.00 | *Source: Author's calculation. Total payments over the life of the loan are 910,177.95, of which 510,177.95 is interest — more than the amount borrowed. The final balance of zero is the arithmetic check.* Read the second and third columns down. In the first month, 86 percent of the payment is interest; over the first five years the borrower pays about 126,100 in interest against about 25,600 of principal, retiring less than seven percent of the loan. Amortization is back-loaded because interest accrues on a balance that starts at its maximum. Scheduled principal is a small and predictable trickle, so almost all of the variation in what an MBS holder receives comes from *unscheduled* principal — prepayment. And the borrower's option is deep in the money for a long time in present-value terms: at the same balance, a rate that falls by a point or two is worth thousands to tens of thousands of dollars, which is why Chapter 14 §14.4 can treat failure to refinance as household finance's most expensive documented mistake. ### Why this contract, and where else Mortgage lending differs across markets. In the United Kingdom, Australia, Canada, and much of continental Europe the dominant contract is a floating-rate loan, or one fixed for two to five years and then reset, and prepayment either carries a penalty or is capped. Denmark is the intermediate case: long fixed-rate loans are freely prepayable, but the borrower may also buy back her own loan in the bond market at its market price, which removes the one-way character of the American option. Adjustable-rate mortgages exist in the United States and their share rises when fixed rates are high, but the thirty-year fixed remains the default, sustained by a securitization apparatus that exists to make it fundable. The contract does not eliminate risk; it assigns it. A floating-rate borrower bears interest-rate risk in her own cash flow. An American fixed-rate borrower bears none, and cannot lose from a rate move: if rates fall she refinances, and if rates rise she keeps a below-market loan. Somebody must be on the other side of that free option, and this chapter is about who, and at what price. Campbell (2013) offers a systematic comparison of designs and of who ends up bearing what under each. *** ## 13.2 Securitization, Agency-Style A thirty-year fixed loan on one house in one county is not an investable claim. Securitization turns thousands of them into one. ### The guarantee A pool of conforming loans is assembled and a **pass-through** security is issued against it: holders receive their pro-rata share of interest and principal, scheduled and unscheduled, as it arrives. Between the borrower and the holder sit two deductions. The servicer keeps a **servicing fee**, conventionally about twenty-five basis points, for collecting payments and managing delinquency. The guarantor keeps a **guarantee fee** — on recent vintages roughly half a percentage point, on the Federal Housing Finance Agency's annual reporting — in exchange for making the holder whole on any loan that defaults. A pool with a weighted-average coupon of 6.25 percent therefore pays its investors something near 5.50 percent, and the two fees are the price of the transformation. What the guarantee buys is precise: **the holder of an agency MBS does not bear credit risk.** Ginnie Mae securities carry the full faith and credit of the United States; Fannie Mae and Freddie Mac carry a corporate guarantee that has been backed since September 2008 by Treasury support agreements under conservatorship, which the market prices as very nearly sovereign. A borrower who defaults is bought out of the pool at par, and the holder experiences the default as an early return of principal — that is, as a prepayment. Chapter 10's machinery for default, recovery, and tranching therefore does not apply to this claim. What remains is timing, and timing is the whole risk. ### TBA The second institution is a trading convention. Agency pass-throughs trade overwhelmingly **to-be-announced**: buyer and seller agree on issuer, coupon, maturity, price, face amount, and settlement date, and the specific pools are not identified until two days before delivery. Fungibility across pools makes the market forward, homogeneous, and deep — by trading volume, the TBA market is the second most liquid fixed-income market in the United States after Treasuries. Vickery and Wright (2013) is the standard account. The convention does two things. It makes a heterogeneous asset liquid, which lowers the yield lenders require and therefore the rate borrowers pay. And it lets a lender sell a mortgage *before it has been made*: a bank that promises a borrower a rate today, with the loan closing in forty-five days, hedges that promise by selling the forward TBA. The American rate lock — a free option for the borrower to take the quoted rate if rates rise and walk away if they fall — is fundable only because the TBA market exists to hedge it. The cost is a cheapest-to-deliver option running to the seller, who delivers the worst pools that satisfy the guidelines, and the price of a TBA reflects it. Two derived markets follow from the convention and are worth naming, because both appear in the holder section. Pools with characteristics that predict slow prepayment — low balances, particular geographies, loans made under specific programs — trade as **specified pools** at a premium to TBA, and that premium is a direct market price for the prepayment behavior §13.3 models. And because TBA settlement can be rolled forward a month at a quoted price difference, the **dollar roll** is a financing market: a holder who sells the front month and buys the back month has borrowed against the position, which is how levered holders of agency MBS fund themselves without a repo counterparty. ### The private-label side, in one paragraph Loans too large or too weak to be guaranteed are securitized privately, and there the holder does bear credit risk, allocated by seniority rather than stripped by a guarantee. The pricing machinery for that structure is Chapter 10 §10.8's: attachment and detachment points, a tranche as a call spread on the pool's loss distribution, and a senior tranche whose rating is an assumption about default correlation. How that pipeline was built and how it failed — origination and underwriting standards, the warehouse chain, the ratings, the failure of loan modification after the fact — is narrated in the companion volume on the 2008 crisis and is not retold here. What is worth doing once, and takes an afternoon, is reading a single deal's prospectus supplement — the Washington Mutual asset-backed series of 2007 will do — where the loss waterfall, the credit-support levels, and the representations and warranties are set out at a level of detail no summary preserves. This chapter takes the guarantee as given precisely so that prepayment, and not default, is the priced risk. > **Box 13.1 — One deal's prospectus supplement** > > Section 13.2 tells the reader to go and find a private-label prospectus supplement from 2006 or 2007 and read it. This box says what to look for, because the document is long and only four parts of it matter. > > **The waterfall.** Near the front is a description of how collections are applied: which classes receive interest, in what order, and how principal is allocated between senior and subordinate classes over time. Read it as a set of rules for assigning cash, not as a description of risk. The risk is what those rules imply when the pool underperforms. > > **The credit support levels.** A table gives each class's subordination — the fraction of the pool that must be written off before that class takes a dollar of loss — along with any overcollateralization and excess spread. These are the numbers the rating was assigned against. Compare a senior class's subordination with the pool's eventual cumulative loss and the arithmetic of the vintage is finished in one line. > > **The representations and warranties.** The originator or sponsor represents that the loans conform to stated underwriting criteria, and agrees to repurchase loans that breach. This is the only credit enhancement in the document that depends on a counterparty remaining solvent and willing, which is what made the repurchase litigation of the following decade both large and slow. > > **The servicer's discretion.** Buried in the servicing section is the servicer's authority to modify loans, and its limits. It is usually permitted to modify within stated bounds if it judges the modification to increase recoveries. Who holds that discretion, and whose interests it is exercised in when senior and subordinate classes disagree, is a governance question inside a security. > > Chapter 10 §10.8 supplies the pricing framework — pooling, tranching, and the correlation assumption that makes a senior tranche safe on paper. What this box adds is that the assumption sits inside a contract, drafted by somebody, with a party on the other side of every promise. *** ## 13.3 Prepayment and Negative Convexity ### Why borrowers prepay Four reasons, in descending order of price relevance. **Refinancing.** When the prevailing mortgage rate falls far enough below the borrower's note rate, refinancing is worth more than its transaction cost. This is the option, and it is the source of nearly all the variation in prepayment speeds. **Turnover.** People move. Roughly six to eight percent of a seasoned pool prepays each year for reasons having nothing to do with rates — job changes, divorce, upsizing — which puts a floor under prepayment speeds even when the option is far out of the money. **Curtailment.** Borrowers pay extra, or pay the loan off from savings. **Default.** In an agency pool a default is a buyout at par, which is a prepayment. Its price effect is the opposite of the refinancing effect: it is *more* likely when the borrower is under stress, which is not when rates are low. Prepayment is measured as the **conditional prepayment rate** $$\mathrm{CPR}$$, the annualized fraction of the surviving balance that prepays, and its monthly equivalent the **single monthly mortality** rate, $$\mathrm{SMM} = 1 - (1-\mathrm{CPR})^{1/12}$$. A six percent CPR is an SMM of 0.5143 percent. The industry's seasoning benchmark, 100 PSA, ramps the CPR from 0.2 percent in the first month by 0.2 percent per month to six percent at month thirty and holds it there; speeds are then quoted as multiples of it. Plot CPR against the borrower's refinancing incentive — the note rate minus the currently available rate — and the relation is an **S-curve**, which is Figure 13.1. Far out of the money it is flat at the turnover floor. As the incentive crosses roughly a quarter to a half point it steepens sharply, because the population of borrowers for whom refinancing clears its fixed cost expands quickly. And it flattens again at high incentive, well below one hundred percent, because some borrowers will not refinance at any incentive: they cannot qualify, they do not have the equity, they do not expect to stay, or they simply do not act. ![Figure 13.1: The prepayment S-curve](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-33a152b6e97e478e40a228704d143ebd1b7c2e9b%2Ffig_13_01_prepayment_s_curve.png?alt=media) **Figure 13.1: The prepayment S-curve.** Conditional prepayment rate against refinancing incentive, with the two bounds the section names: a turnover floor at six percent, which people moving and divorcing and upsizing put there whatever rates do, and a ceiling at forty percent, well below one hundred, because some borrowers will not refinance at any incentive. Between them the curve steepens sharply as the incentive crosses roughly a quarter to a half point, because the pool of borrowers for whom refinancing clears its fixed cost expands quickly. The two shape parameters — a steepness of 1.5 per percentage point and a midpoint at one point — are not chosen for the drawing: they are the pair that returns every conditional prepayment rate in Table 13.2, and the seven marked points are that table's rate shifts placed on this curve. *Source: Author's calculation. Curve parameters recovered from Table 13.2's CPR column.* The holder therefore faces the risk in two directions and the market names them separately. **Contraction risk** is the risk that rates fall, prepayments accelerate, and the holder's principal comes back to be reinvested at the new lower rate — the reason a premium pass-through cannot rally like an ordinary bond. **Extension risk** is the risk that rates rise, prepayments stop, and the holder is locked into a below-market coupon for years longer than she expected. They are the two halves of one option position, and the reason a holder cannot escape by choosing a coupon: a discount pass-through simply trades one risk for the other. ![Figure 13.3: The refi wave](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-ee37aaaecdfec1e00b62110b6436985164d5c198%2Ffig_13_03_the_refi_wave.png?alt=media) **Figure 13.3: The refi wave.** The prepayment option, exercised, at the frequency it is exercised at: quarterly mortgage originations against the thirty-year fixed rate since 2003. The three waves are marked, and the fact that matters is the rate at each — 6.0 percent in 2003, 3.4 in 2012, 3.0 in 2021. Each wave required a lower rate than the last, because the households who could gain from refinancing at six percent had already done so. That is the burnout of §13.3, visible as a sequence rather than as a parameter. The asymmetry is the other half. Volume triples when rates fall and does not fall below a floor when they rise, because a borrower with a three-percent loan does not refinance into seven — which is why a mortgage book's duration shortens in a rally and extends in a selloff, and why the 2022 rate rise produced the smallest origination volumes of the sample rather than a symmetric wave in the other direction. One measurement caution: the bars are all first-lien originations, refinancing and purchase together, because the Consumer Credit Panel does not separate them. A refinancing wave is visible in the total because it is large relative to purchase volume, not because the series isolates it. *Source: Federal Reserve Bank of New York Consumer Credit Panel / Equifax, Household Debt and Credit Report; Freddie Mac primary mortgage market survey via FRED. Author's calculations.* ### The MBS as a bond minus a call The compact statement is this. A mortgage pass-through is a fixed-coupon bond, less a call option written to the borrower, struck at par and continuously exercisable. Chapter 8's machinery prices such an option in the abstract; what makes this one hard is that the exercise rule is not optimal exercise by a rational agent but the empirical behavior of several million households, and the "underlying" is not a traded price but a term structure. The consequence for the holder is immediate and follows from Chapter 9 §9.1's definitions. **Effective duration** $$D\_{\mathrm{eff}}$$ and **effective convexity** $$C\_{\mathrm{vx}}$$ must be computed by repricing the security at shifted rate levels *allowing the cash flows to respond*, because the cash flows are what respond: $$ D\_{\mathrm{eff}} = \frac{p(y-h) - p(y+h)}{2hp(y)}, \qquad C\_{\mathrm{vx}} = \frac{p(y-h) + p(y+h) - 2p(y)}{p(y)h^{2}} $$ Chapter 9 showed that an ordinary bond has $$C\_{\mathrm{vx}} > 0$$: a hundred basis point rally is worth more than a hundred basis point selloff costs. A mortgage security reverses the sign. ### A worked illustration Take a thirty-year pool with a weighted-average coupon of 6.25 percent paying investors 5.50 percent after fees, and let the mortgage rate available to borrowers be the benchmark yield plus a hundred and fifty basis points, so that at a benchmark of 4.75 percent the pool is exactly at the money. Let the CPR follow a logistic S-curve running from a six percent turnover floor to a forty percent ceiling, and discount at the benchmark plus fifty basis points. Table 13.2 reprices it. **Table 13.2: A mortgage pass-through and a matched non-callable claim under parallel rate shifts** | Shift in benchmark | CPR | Weighted-average life (years) | MBS price | MBS change | Non-callable change | MBS effective duration | | ------------------ | ----- | ----------------------------- | --------- | ---------- | ------------------- | ---------------------- | | −200 bp | 33.8% | 1.76 | 103.985 | +3.03% | +7.00% | 0.95 | | −100 bp | 23.0% | 2.43 | 103.042 | +2.09% | +3.42% | 1.14 | | −50 bp | 16.9% | 3.03 | 102.277 | +1.33% | +1.69% | 2.00 | | 0 | 12.2% | 3.73 | 100.931 | — | — | 3.35 | | +50 bp | 9.2% | 4.35 | 98.921 | −1.99% | −1.66% | 4.58 | | +100 bp | 7.6% | 4.76 | 96.478 | −4.41% | −3.28% | 5.27 | | +200 bp | 6.4% | 5.06 | 91.384 | −9.46% | −6.42% | 5.39 | *Source: Author's calculation. Thirty-year monthly-pay pool, weighted-average coupon 6.25 percent, net pass-through coupon 5.50 percent, borrower rate equal to the benchmark plus 150 basis points, logistic prepayment S-curve from a 6 percent floor to a 40 percent ceiling, cash flows discounted at the benchmark plus 50 basis points. The non-callable comparator is a zero-coupon claim whose modified duration equals the pass-through's effective duration of 3.35 at the base level. Effective duration and convexity use a 25 basis point central difference; effective convexity at the base level is −272.* ![Figure 13.2: Negative convexity](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-5e24f4be80e09523282532a8cd059a063175362e%2Ffig_13_02_negative_convexity.png?alt=media) **Figure 13.2: Negative convexity.** Table 13.2, drawn. Panel (a): the pass-through's price against the rate level, against a duration-matched non-callable claim that starts at the same price and the same duration. Rates fall to the right. The non-callable curve bends upward, which is what positive convexity looks like: the rally is worth more than the selloff costs. The pass-through's does the opposite — it flattens as rates fall, because falling rates put the borrower's option into the money and the principal comes back to be reinvested at the new lower rate. The shaded wedge is the rally the holder does not get. Panel (b): the effective duration that produces it, falling from 5.39 years at plus two hundred basis points to 0.95 at minus two hundred. That is the mechanism §13.5 needs. A holder who hedges this duration sells into a selloff and buys into a rally, which amplifies the rate move before any household has refinanced anything. *Source: Author's calculation; prices and effective durations are Table 13.2's.* Read the two change columns against each other. On a hundred basis point rally the non-callable claim gains 3.42 percent and the mortgage security gains 2.09 percent — the borrower took the rest, by refinancing. On a hundred basis point selloff the non-callable claim loses 3.28 percent and the mortgage security loses 4.41 percent. The loss on the selloff is more than twice the gain on the rally. Set that beside Chapter 9's Table 9.2, where an ordinary ten-year bond gained 8.37 percent on a rally against a 7.58 percent loss on a selloff, and the reversal is exact: positive convexity means the gain exceeds the loss, and negative convexity means the loss exceeds the gain. The last column is where the money is lost in practice. Effective duration is 3.35 at the base, 1.14 after a hundred basis point rally, and 5.27 after a hundred basis point selloff. **The holder's interest-rate exposure moves against her in both directions**: she owns least duration when duration has just performed and most when it has just cost her. A holder who targets a fixed exposure must therefore buy duration into rallies and sell it into selloffs — mechanically, and in the same direction as the market. That is the feedback loop the opening episode describes, and Section 13.5 gives it a size. ### Option-adjusted spread A quoted yield spread over Treasuries is uninformative for this claim, because the cash flows it discounts are one scenario out of many and the wrong ones. The market's answer is the **option-adjusted spread**. Simulate many paths of the term structure under a no-arbitrage model calibrated to the current curve and to interest-rate volatility; on each path, run a prepayment model to generate cash flows; discount each path's cash flows at that path's short rates plus a constant increment; and solve for the increment that makes the average present value equal the observed price. That increment is the OAS. Its interpretation is exactly what its name says: the compensation the holder earns *after* paying for the option she wrote, and therefore the number comparable across a callable and a non-callable claim. The difference between the static spread and the OAS is the market's price of the prepayment option, which for a current-coupon pass-through has commonly run in the tens of basis points and widens with rate volatility, since the borrower's option is long volatility and the holder is short it. The number carries a specific fragility. An OAS is a residual computed *given* a prepayment model and *given* a volatility assumption, so it is a joint statement about the price and the model. Two desks with the same price and different prepayment models report different OAS, and the difference is not measurement error — it is a disagreement about household behavior. When every large holder calibrates against similar models, and those models are what determine hedging demand, the models help produce the very prepayment environment they forecast: a candidate for the performativity Chapter 7 §7.6 describes, and a reason to read a reported OAS as one desk's view rather than as a property of the security. ### Burnout, media, and boundedly rational borrowers A prepayment model that treated borrowers as optimizing option-holders would fail on its first prediction, and the ways it fails are informative. **Burnout.** A pool that has already been through a refinancing wave prepays more slowly at the same incentive than a pool that has not. The reason is selection: the fast borrowers left, and what remains is the population that did not act when acting was clearly worth it. Burnout means the pool's prepayment function depends on its own history, which is why models carry a state variable for cumulative exposure to refinancing incentive. **Media effects.** Prepayment responds to *news about* rates and not only to rates. A well-publicized decline to a round-numbered record low produces more refinancing than an equal decline that goes unreported, and prepayment speeds jump when mortgage rates make the front page. This is Chapter 15's attention story arriving in the largest household liability class in the country. Both are the same fact seen from the investor's side that Chapter 14 §14.4 documents from the household's. Roughly a fifth of eligible borrowers fail to refinance when it is clearly optimal, at a median cost in the neighborhood of eleven thousand dollars each; inaction is concentrated among older, poorer, and less educated borrowers. That inaction is not merely a welfare loss. It is an asset to somebody: it is what makes a premium pass-through worth more than a naive option model says, and estimating its magnitude and persistence is a large part of what mortgage analysts are paid for. The same household appears on both sides of this book — as the borrower who fails to exercise in Chapter 14, and as the source of return in the security priced here. *** ## 13.4 Real Estate as an Asset Class Housing finance is one part of real estate's appearance in a portfolio. The other is the income-producing property itself, and the discount-rate logic that prices it is Chapter 3's, applied with a different vocabulary. ### Cap rates The sector's valuation ratio is the **capitalization rate**: net operating income divided by price. Inverted, it is a pricing formula: $$ p = \frac{\mathrm{NOI}}{\mathrm{cap}} $$ A building generating twelve million dollars of net operating income at a five percent cap rate is worth two hundred and forty million; at seven and a half percent it is worth one hundred and sixty million, a decline of a third with no change whatever in the building or its tenants. The cap rate is a discount rate wearing different clothes. Set beside Chapter 3 §3.1's growing perpetuity, $$p = C/(r-g)$$, the identity is $$\mathrm{cap} = r - g$$: the cap rate is the required return on the property less the expected growth rate of its income. A five percent cap rate with two percent expected NOI growth is a seven percent required return, and the difference between a five and a seven-and-a-half percent cap rate is either two hundred and fifty basis points of required return or an equivalent revision of growth — and the ratio alone cannot tell you which. That ambiguity is the same one Chapter 7 §7.4 finds in the dividend-price ratio, in a market with far worse data. ### REITs A **real estate investment trust** is a corporate form, created by US statute in 1960, that pays no entity-level tax provided it holds mostly real estate, earns mostly real-estate income, is widely held, and — the binding condition — **distributes at least ninety percent of its taxable income to shareholders each year**. The design converts illiquid buildings into a listed equity claim, and the distribution requirement sharply constrains internal funding — up to a tenth of taxable income may be retained, and because depreciation is a noncash charge the cash a REIT keeps can exceed that — so it returns to the capital markets for nearly every acquisition, which makes its cost of capital an unusually direct constraint on its investment. Listed US equity REITs are worth something over a trillion dollars in aggregate. What a REIT index measures has also changed. The listed sector's composition has shifted away from the traditional property types toward infrastructure-like assets — cell towers, data centers, logistics, self-storage — so a REIT index return is increasingly a claim on tenant demand from a handful of technology and distribution businesses rather than on the office and retail buildings the sector was built from. An investor who buys listed real estate as a diversifier from equities, and finds it behaves like equities, has often discovered nothing about correlations and something about what is in the index. The same buildings are also held in private funds and separate accounts, valued not by trading but by periodic appraisal. The two valuations diverge, and the divergence is a measurement artifact before it is anything else. Appraisals anchor on the last transaction and move cautiously, so private real estate indices are smoothed versions of the truth in exactly the sense Chapter 18 §18.2's Getmansky-Lo-Makarov model formalizes: reported volatility is understated, measured beta against public markets is understated by the smoothing weight, and drawdowns arrive late and small. When listed REITs fall thirty percent in a quarter and appraised private portfolios fall three, the honest reading is not that the private assets held up. It is that one series is a price and the other is an estimate. ### The office repricing The clearest recent illustration is the American office market since 2020. Remote and hybrid work reduced space demand persistently rather than cyclically; office vacancy rates have run near or above twenty percent nationally, well beyond anything in the previous four decades. At the same time the discount rate moved: policy rates rose sharply through 2022 and 2023, and cap rates followed. Both terms of $$p = \mathrm{NOI}/\mathrm{cap}$$ moved the wrong way at once, and transaction-based indices of US office values have been reported down by something on the order of a third or more from their 2022 peak, with the largest declines in older buildings in secondary locations. The magnitudes remain uncertain and should be read as such: transaction volumes collapsed, which means the price indices are estimated from a thin and selected sample, and appraisal-based indices continued to lag them by quarters. What is not uncertain is the arithmetic. A one-third decline is what a cap rate moving from five to seven and a half percent delivers on unchanged income, and the income did not stay unchanged. *** ## 13.5 Who Holds the Mortgage, and What Their Constraints Do to Its Price The constraint that matters here is a hedging mandate, and what it does reaches past the mortgage to the level of rates. Chapter 2's Table 2.5 gives the agency row of the master map. Table 13.3 opens it. **Table 13.3: Holders of agency and GSE-backed securities, United States, 2026:Q1 (trillions of dollars)** | Holder | Holding, 2026:Q1 | Hedges convexity? | | ---------------------------------------------------- | ---------------- | ------------------------------- | | Federal Reserve (SOMA) and other government accounts | 2.2 | No | | Banks and depositories | 3.2 | Mostly not | | Funds (mutual, money market) | 1.9 | Some — total-return mandates do | | Insurers and pension funds | 1.0 | Partly | | Rest of world | 1.4 | Rarely | | Households, mortgage REITs, and other | 2.1 | Mortgage REITs intensively | | **Total** | **11.9** | | *Source: Financial Accounts of the United States (Z.1), table L.211, holder rows through their FRED mirrors; data through 2026:Q1, retrieved 25 August 2026. The rows exhaust the table and sum to the published total, and the total includes agency debentures as well as agency MBS. The Z.1 publishes no separate exchange-traded fund row for this instrument, so the fund row is mutual funds and money funds. Cross-checks: the Federal Reserve's H.4.1 carries $2.0 trillion of agency MBS held outright at the same date, against $1.7 trillion in the Z.1's central-bank row, and puts the SOMA peak at $2.7 trillion in April 2022; the Treasury International Capital data put foreign holdings of long-term agency bonds at $1.4 trillion in June 2026, of which $0.5 trillion official and $1.0 trillion non-official — foreign official holders were two thirds of the foreign total at the 2008 peak and are about a third of it now; and the Call Report securities schedules, summed over 4,352 FDIC-insured institutions at 2026:Q1, show $5.8 trillion of securities, of which $3.7 trillion available-for-sale and $2.1 trillion held-to-maturity by difference, with mortgage-backed securities of $3.0 trillion. The hedging column is the author's characterization of typical mandates and accounting treatments, not a measured quantity.* The second column is the point of the table, and it is why the outstanding stock is the wrong denominator for the feedback loop of §13.3. What matters is the **hedged share**: the fraction of the market held by someone who both measures duration against a target and acts when it moves. One scope caution before the column is read that way: the table's total covers agency debentures as well as agency MBS, so the holder shares are used here — and in Problem 5 — as an illustrative allocation of MBS-only exposure rather than as a measured distribution of the callable claim. Read the column down. The Federal Reserve does not hedge. Its holdings arose from quantitative easing — SOMA agency MBS peaked around 2.7 trillion dollars in 2022 and has run down since through passive runoff — and it has no duration target, no mark-to-market income statement that a duration mismatch damages, and no counterparty to trade with. Foreign official holders similarly buy for reserve-management reasons and hold. Most bank portfolios are managed against a balance sheet rather than a benchmark, and their accounting treatment, as we shall see, can suppress the signal to hedge entirely. Together those categories are more than half of the market. The hedgers are the rest, and they are specific. **Mortgage servicers** hold the mirror image of the security: a servicing right is a stream of fees that vanishes when the loan prepays, so its value *rises* when rates rise, and a servicer with a large book therefore carries an asset of *negative* duration, which it hedges by adding positive duration — receiving fixed in a swap — sometimes offsetting other hedgers, sometimes not. **Mortgage REITs** hold levered agency MBS financed in repo, so a duration mismatch is an existential rather than a performance question, and they hedge continuously and mechanically. **Total-return bond funds** are measured against an index whose duration they must track, so any extension in their mortgage holdings must be offset the same week. Historically the retained portfolios of Fannie Mae and Freddie Mac were the largest hedgers of all; those portfolios have been shrunk deliberately under conservatorship, which is one reason the modern convexity event is smaller than 2003's relative to the size of the market. Figure 13.4 reads Table 13.3 backwards through time, and the retained portfolios are drawn on their own line because that is where the change is. ![Figure 13.4: Who holds agency MBS](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-f587e3077370bee38f6fdb9bba132502d3428a9b%2Ffig_13_04_who_holds_agency_mbs.png?alt=media) **Figure 13.4: Who holds agency MBS.** Table 13.3's holder rows as shares of agency and GSE-backed securities outstanding, 1990 to the present, with the GSE retained portfolios split out of the table's residual row and drawn separately. The bands are stacked in the order of the table's hedging column, least likely to hedge at the bottom; the dashed line is the top of the three bands the section classes as not hedging convexity, and it sits at 58 percent today. Two movements matter. The Federal Reserve enters in 2009 and is a fifth of the market within three years — a holder with no duration target, acquired by policy rather than by choice. And the GSE retained portfolios, the largest and most mechanical hedgers the market ever had, peak at 21 percent of the stock in the third quarter of 2003, which is the quarter of the opening episode, and are 3 percent of it now. The market they used to hedge grew ninefold over the same window, from 1.3 to 11.9 trillion dollars. The hedging classification is the table's, which is to say the author's characterization of typical mandates and accounting treatments rather than a measured quantity; the shares themselves are measured, and the seven bands sum to the Z.1's published all-sectors total in every quarter. As in Table 13.3, the total covers agency debentures as well as agency MBS. *Source: Financial Accounts of the United States (Z.1), table L.211, holder rows through their FRED mirrors. Author's calculations.* This gives the resolution of the opening episode. In the summer of 2003 the aggregate negative convexity of the mortgage universe was near its historical maximum: nearly the entire stock was refinanceable, so nearly the entire stock had short duration that would extend sharply on any selloff. The hedged share was also high, because the agency retained portfolios were then at their peak and mortgage REITs and total-return funds were growing. A policy repricing worth perhaps forty or fifty basis points hit a market in which several hundred billion dollars of ten-year equivalents had to be sold to restore duration targets, and the selling arrived over weeks rather than at once, so each increment of yield generated the next. Hanson (2014) makes the general statement: aggregate mortgage duration is a state variable for the Treasury market, and it forecasts both the volatility and the term premium of long rates. This is Chapter 9 §9.5's preferred-habitat picture at higher frequency — a quantity of duration that the private market must absorb, moving for reasons unrelated to the expected path of policy, priced by the risk-bearing capacity of whoever must absorb it. The modern case is Silicon Valley Bank, and it is the same mechanism with the hedging switched off. At the end of 2022 SVB held roughly ninety-one billion dollars of securities — overwhelmingly agency MBS and collateralized mortgage obligations — classified as **held to maturity**, against total assets near two hundred and twelve billion. HTM classification permits carrying at amortized cost, so the securities did not move with rates on the balance sheet. Economically they did: through 2022 rates rose by several hundred basis points, prepayments stopped, duration extended exactly as Table 13.2 predicts, and the unrealized loss on that book reached something on the order of fifteen billion dollars against tangible common equity near sixteen billion. The bank was, on a mark-to-market basis, close to insolvent while reporting itself well capitalized, and it had hedged very little of the exposure. The accounting did not prevent the loss; it deferred the *recognition* of the loss until an event forced the securities to be sold. That event was a run — Chapter 19 §19.2's Diamond-Dybvig coordination failure, with the guarantee removed above the insurance threshold and the depositor base unusually concentrated — and roughly forty-two billion dollars of withdrawal requests arrived on March 9, 2023. The bank failed the next day. Chapter 9's Box 9.2 reads the same failure off §9.1's duration arithmetic and the depositor base; the full banking story belongs to Chapter 19 and to the companion volume on the 2008 crisis. What belongs here is the claim. The same negative convexity that made 2003's hedgers sell Treasuries into a falling market made 2023's non-hedger insolvent in silence, and the difference between the two outcomes was not the security. It was who held it, under what accounting, with what funding. In Chapter 1 §1.2's terms, a convexity event is a flow with a constraint behind it and never news about a borrower: duration that hedgers must sell because their mandates say so, into a market whose remaining holders have finite capacity to absorb it. > **Box 13.2 — The conservatorship's portfolio caps** > > Section 13.5 says in one clause that the retained portfolios of Fannie Mae and Freddie Mac "have been shrunk deliberately under conservatorship." That clause is the reason the mortgage market's largest and most mechanical convexity hedger no longer exists, and it is worth opening. > > In September 2008 both enterprises were placed into conservatorship by their regulator, and the Treasury entered into Senior Preferred Stock Purchase Agreements with each. Those agreements did three things. They committed Treasury to fund any negative net worth, in exchange for senior preferred stock and warrants. They capped the size of each enterprise's retained mortgage portfolio. And the caps declined over time on a stated schedule, so that portfolios which had been measured in hundreds of billions of dollars were required to run down year after year toward a small fraction of their former size. > > The retained portfolio is the object that matters here. An enterprise that guarantees a pool bears credit risk; an enterprise that *holds* mortgages on its own balance sheet bears the interest-rate and prepayment risk §13.3 is about — and hedges it, continuously, in the Treasury and swap markets, because a levered portfolio of negatively convex assets is otherwise unmanageable. Two institutions running that hedge on portfolios of that size were, through the 1990s and early 2000s, the largest single source of the convexity-driven flows the opening episode describes. > > A 2012 amendment to the agreements replaced the fixed dividend on the senior preferred with a sweep of substantially all net worth to Treasury, and tightened the wind-down further. The combined effect is visible in Figure 13.4: the enterprises' share of the agency market peaks in the third quarter of 2003 — the quarter of the opening episode — at about a fifth of the market, and is a few percent of it now. > > The point for this chapter is not that convexity hedging has stopped. It is that the hedgers have changed identity, from two very large institutions with a public mandate and a stable rule to a dispersed set of mortgage REITs, servicers and total-return funds with their own leverage and their own drawdowns. Chapter 20 would call that a change in the composition of a demand curve, and Chapter 19 §19.5 would say what it does to the price of the claim. *** ## Elsewhere in the Series * **The originate-to-distribute pipeline, underwriting standards and their collapse, the ratings apparatus, and the failure of loan modification** — the companion volume on the 2008 crisis, which owns the industry narrative end to end. Section 13.2 takes the agency guarantee as given precisely so this chapter can price prepayment rather than retell that story. * **Foreign official holdings of agency securities and the reserve-management demand behind them** — *International Finance*, whose chapters on the dollar's international role own the monetary economics; Table 13.3 uses the holding as a fact. * **This book.** The agency/MBS row of the master holdings table: Chapter 2, Table 2.5. Option payoffs, replication, and the machinery that prices a call: Chapter 8. Duration, convexity, and the price-yield expansion this chapter's negative convexity inverts: Chapter 9 §9.1, whose forward pointer this chapter answers; preferred habitat and the price of duration: Chapter 9 §9.5. Credit risk, tranching, and the loss distribution of a private-label deal: Chapter 10, especially §10.8. Housing in the household portfolio, mortgage leverage, and refinancing inertia from the borrower's side: Chapter 14 §14.4. Attention and inaction as behavioral regularities: Chapter 15. Smoothed marks and what appraisal-based indices measure: Chapter 18 §18.2. Runs, deposit insurance, and bank balance sheets: Chapter 19 §§19.2-19.3. Models as market infrastructure: Chapter 7 §7.6. *** ## Summary 1. **The American mortgage is a globally unusual contract.** Thirty years, fixed rate, level payment, freely prepayable at par: the borrower holds a free call option on her own debt and bears no interest-rate risk in either direction. Table 13.1 shows the amortization that makes scheduled principal a small, predictable trickle, so that nearly all variation in an MBS holder's cash flow is unscheduled. 2. **The agency guarantee strips credit risk and leaves timing.** Servicing and guarantee fees of roughly seventy-five basis points in total buy the holder a claim on which a default arrives as a prepayment at par. Chapter 10's default machinery therefore does not apply to agency MBS; the private-label side, where it does, points to Chapter 10 §10.8 and to the 2008 volume. 3. **TBA trading is what makes the contract fundable.** Fungibility across pools produces the second most liquid fixed-income market in the United States, and it is the mechanism by which a lender hedges a rate lock on a loan that has not yet closed. 4. **Prepayment follows an S-curve, not an optimal exercise rule.** A turnover floor of six to eight percent CPR, a steep rise once the refinancing incentive clears transaction costs, and a ceiling well below one hundred percent because some borrowers never act. 5. **Negative convexity is the pricing core.** In Table 13.2 a mortgage pass-through gains 2.09 percent on a hundred basis point rally against a matched non-callable claim's 3.42 percent, and loses 4.41 percent on a hundred basis point selloff against the comparator's 3.28 percent. The loss exceeds the gain, which reverses Chapter 9 §9.1's positive-convexity result exactly. 6. **Duration moves the wrong way.** Effective duration in the same table runs from 1.14 after a rally to 5.27 after a selloff. A holder targeting a fixed exposure must buy duration into rallies and sell it into selloffs, procyclically and mechanically. 7. **OAS is the spread after paying for the option — and is a joint statement about price and model.** Two desks with the same price and different prepayment models report different OAS. Burnout and media effects are the evidence that borrower behavior is boundedly rational, and they are the investor's side of Chapter 14 §14.4's refinancing inertia. 8. **Cap rates are discount rates.** $$p = \mathrm{NOI}/\mathrm{cap}$$, with $$\mathrm{cap} = r - g$$, so a move from five to seven and a half percent takes a third off the value of unchanged income. REITs' ninety-percent distribution rule forces them back to the capital markets for growth; appraisal-based private indices are the smoothed series of Chapter 18 §18.2, and the office repricing since 2020 moved both terms of the ratio at once. 9. **The hedged share, not the outstanding stock, drives the feedback.** Table 13.3 shows that the Fed, foreign officials, and most bank portfolios do not hedge convexity, while servicers, mortgage REITs, and total-return funds do. In 2003 aggregate negative convexity was near its maximum and the hedged share was high; a policy repricing worth forty or fifty basis points became a move of roughly a hundred and forty. 10. **The same convexity failed a bank in 2023.** Silicon Valley Bank held roughly ninety-one billion dollars of mortgage securities at amortized cost under held-to-maturity accounting, extended duration through 2022 without hedging, and carried an unrealized loss near its entire equity. The accounting deferred recognition until a run forced the sale. The claim was the same; the holder, the accounting, and the funding were not. *** ## Key Terms * **Prepayment**: Repayment of mortgage principal ahead of schedule — from refinancing, moving, curtailment, or (in an agency pool) default, which is bought out at par; measured by the conditional prepayment rate $$\mathrm{CPR}$$ and its monthly equivalent $$\mathrm{SMM}$$ * **S-curve**: The empirical relation between refinancing incentive and prepayment speed — flat at a turnover floor, steep through the money, flattening below one hundred percent because some borrowers never act * **Negative convexity**: $$C\_{\mathrm{vx}} < 0$$; the property of a claim whose duration shortens as rates fall and extends as they rise, so that a rally is worth less than an equal selloff costs and the holder is a procyclical trader of duration * **Option-adjusted spread** ($$\mathrm{OAS}$$): The constant increment to path short rates that equates a simulated average present value to the observed price; the holder's compensation after paying for the option she wrote, and a joint statement about the price and the prepayment model * **TBA**: The to-be-announced forward market in agency pass-throughs, in which specific pools are identified only at delivery; the source of the market's liquidity and the mechanism behind the rate lock * **Agency guarantee**: The undertaking by Ginnie Mae, Fannie Mae, or Freddie Mac to make MBS holders whole on defaulted loans, paid for by the guarantee fee, which removes credit risk from the claim and leaves prepayment timing * **Cap rate**: Net operating income divided by price; a discount rate net of expected income growth, $$\mathrm{cap} = r - g$$ * **REIT**: A tax-transparent corporate form for holding income-producing real estate, conditional on asset and income tests and on distributing at least ninety percent of taxable income * **Held to maturity (HTM)**: An accounting classification permitting debt securities to be carried at amortized cost rather than fair value, which suppresses mark-to-market volatility and defers loss recognition until a sale * **Convexity hedging**: Trading duration — selling Treasuries and paying fixed in swaps as rates rise, and the reverse as they fall — to restore a target exposure after a mortgage portfolio's duration has moved *** ## Readings ### Required * Fabozzi, F. J., ed. *The Handbook of Mortgage-Backed Securities*, 7th ed. Oxford University Press, 2016. *The technical reference for everything in §§13.1-13.3: pool conventions, prepayment modeling, OAS methodology, and structured products; use it as a manual rather than a monograph.* * Vickery, J. and J. Wright (2013). "TBA Trading and Liquidity in the Agency MBS Market." *Federal Reserve Bank of New York Economic Policy Review* 19(1): 1-18. *The clearest account of why the forward market exists, what it does for liquidity and for the rate lock, and what the cheapest-to-deliver option costs; free.* ### Recommended * Hanson, S. G. (2014). "Mortgage Convexity." *Journal of Financial Economics* 113(2): 270-299. *Shows that the aggregate duration of the mortgage universe forecasts both the volatility and the term premium of long-dated Treasuries — the formal statement of this chapter's opening episode.* * Boyarchenko, N., A. Fuster and D. Lucca (2019). "Understanding Mortgage Spreads." *Review of Financial Studies* 32(10): 3799-3850. *Decomposes the agency MBS spread and finds that prepayment risk is priced by a segmented set of MBS investors rather than by the marginal investor of a representative-agent model — Chapter 3 §3.7's question, answered in this market.* * Campbell, J. Y. (2013). "Mortgage Market Design." *Review of Finance* 17(1): 1-33. *The comparative treatment of mortgage contracts across countries and of who bears the interest-rate and prepayment risk under each; the natural companion to §13.1's international contrast.* * Chiquier, L. and M. Lea, eds. *Housing Finance Policy in Emerging Markets*. World Bank, 2009. *The cross-country contract comparison of §13.1 at book length, and well beyond the American agency system: covered bonds, mortgage insurance, state housing banks, and lending to borrowers no securitization pipeline reaches.* * Case, K. E., R. J. Shiller and A. N. Weiss (1995). "Mortgage Default Risk and Real Estate Prices: The Use of Index-Based Futures and Options in Real Estate." *The proposal for index-based futures and options on house prices, from the authors of the index. It is the missing answer to the question §13.4 raises and does not settle — why a household cannot hedge the largest asset it owns — and it sends the option machinery back to Chapter 8.* * Gyourko, J. (2009). "Understanding Commercial Real Estate: Just How Different from Housing Is It?" *Journal of Portfolio Management*, Special Real Estate Issue. *Section 13.4 asserts that commercial property is a different asset from owner-occupied housing and moves on; this paper is that assertion argued, on supply elasticity, cash-flow volatility, and the leverage and holder structure of the two sectors. Read it before accepting any statistic that pools them.* * MIT OpenCourseWare, 11.432J *Real Estate Capital Markets* (Spring 2007), readings list. *A full semester on the material §§13.4-13.5 compress into two sections — cap rates, REIT structure, CMBS, and the private-versus-public valuation gap — with a free and stable bibliography.* *** ## Discussion Questions 1. **Why does this contract exist at all?** The thirty-year fixed-rate prepayable mortgage gives the borrower a free option and gives the lender a claim nobody can hedge cheaply. Identify who bears the option in the American system at each link of the chain — borrower, originator, guarantor, MBS holder, convexity hedger — and say where you think it finally rests. Then argue whether the contract survives because it is efficient risk-sharing (households are the least able to bear interest-rate risk) or because a public guarantee and a public secondary market make it fundable at a price that does not reflect its cost. What evidence from the countries in §13.1's contrast would discriminate? 2. **Does the Fed's unhedged position damp the feedback or defer it?** Roughly two and a half trillion dollars of agency securities sit with a holder that never hedges convexity. The damping reading: that stock is removed from the hedged share, so the amplification in §13.5 is smaller than it would otherwise be, and quantitative easing therefore stabilizes the long end twice over. The deferral reading: the duration is not destroyed, only parked, and quantitative tightening returns it to holders who *will* hedge, so the convexity feedback is being handed forward in time rather than eliminated. Which reading does the 2022-2023 experience favor, and what would you measure to tell? 3. **What is an OAS a statement about?** A portfolio manager reports that a pass-through is cheap at an OAS of forty basis points. Write down everything that number depends on besides the price. Under what circumstances is a wide OAS evidence of value, and under what circumstances is it evidence that the prepayment model is wrong? Relate your answer to Chapter 7 §7.6's argument that models used widely enough become part of the environment they describe. 4. **Smoothed marks and the office question.** Between 2022 and 2024 listed REITs specializing in offices repriced violently while appraisal-based private indices moved gradually. An institutional investor concludes that the private vehicles were better managed. Using Chapter 18 §18.2's machinery, state precisely what is wrong with that inference, what a corrected comparison would require, and whether there is any defensible sense in which the private holder was genuinely better off. 5. **Was SVB a mortgage story or a banking story?** The bank failed because uninsured depositors ran. It was vulnerable to the run because a mortgage portfolio had extended in duration and lost value that accounting did not show. Argue for each framing. Then answer the policy question the framings imply: would requiring fair-value accounting for HTM securities have prevented this failure, or would it have changed which banks hold mortgage risk and therefore where the convexity of §13.5 sits? *** ## Problems **Problem 1 — Amortization and the refinancing option.** A borrower takes a thirty-year level-payment loan of three hundred thousand dollars at an annual rate of 7.0 percent, paid monthly. (a) Compute the monthly payment. (b) Compute the split of the first payment between interest and principal, and state the fraction of the payment that is interest. (c) Compute the outstanding balance after five years. (d) Compute total interest paid over the full thirty years if the loan is never prepaid, and express it as a multiple of the amount borrowed. (e) After five years the prevailing rate falls to 5.0 percent. The borrower refinances the outstanding balance over the remaining twenty-five years. Compute the new payment and the monthly saving. Discounting the saving at 5.0 percent over 300 months, compute its present value. If closing costs are three percent of the balance, is refinancing worth it, and by how much? **Problem 2 — Duration extension and the hedge.** A total-return fund holds one trillion dollars of the pass-through in Table 13.2. Its mandate requires it to hold the effective duration of the mortgage position constant at its base level of 3.35. (a) Rates rise by a hundred basis points. Using Table 13.2, state the position's new effective duration and the change in dollar duration (in trillions of dollars-years). Throughout this problem, compute dollar duration at the pre-shock market value of one trillion dollars, so that the change isolates the move in effective duration. (b) The fund hedges by selling ten-year Treasury notes with a modified duration of 8.0. What face amount must it sell? (c) Repeat (a) and (b) for a hundred basis point *fall*. In which direction does the fund trade, and how does that trade interact with the direction the market has just moved? (d) Suppose the whole market held this pool, that agency MBS outstanding is nine trillion dollars, and that the hedged share is thirty percent. Compute the aggregate ten-year-equivalent selling generated by the hundred basis point rise. Compare it with the roughly four to five hundred billion figure the opening episode attributes to 2003, and explain in two sentences why the two need not be similar even if the market has grown. **Problem 3 — Cap-rate valuation.** An office property generates net operating income of 8.4 million dollars a year, expected to grow at 2.5 percent, and the market cap rate is 5.25 percent. (a) Compute the property's value. (b) Compute the implied required return $$r$$. (c) The cap rate rises to 7.0 percent. Compute the new value and the percentage change, holding NOI fixed. (d) At the same time, hybrid work reduces NOI by twelve percent. Compute the value under both changes and the total percentage decline. Decompose the decline into the cap-rate effect and the income effect, and verify that the two combine multiplicatively rather than additively. (e) The building carries sixty percent loan-to-value debt originated at the original valuation. What has happened to the equity, and what does that tell you about who bears a cap-rate move? **Problem 4 — Reading an OAS.** Two dealers price the same current-coupon pass-through at the same price. Dealer A reports a static (zero-volatility) spread of 75 basis points and an OAS of 25 basis points. Dealer B reports the same static spread and an OAS of 45 basis points. (a) What is each dealer's implied cost of the prepayment option, and what does the difference between them tell you? (b) Name three modeling choices that could produce the gap, and state for each which dealer's OAS would be higher. (c) Interest-rate volatility rises sharply with no change in the level of rates or in the security's price. What happens to each dealer's OAS, and why? (d) A portfolio manager buys the security on the strength of Dealer B's number. Describe the position she has actually taken, in terms of what she is long and what she is short. (e) Explain why an OAS computed with a prepayment model that everyone in the market uses is a different object from an OAS computed with a proprietary model that turns out to be right. **Problem 5 — Who bears the convexity.** Using Table 13.3, suppose agency MBS outstanding is nine trillion dollars and that a hundred basis point selloff extends the average effective duration of the stock by 1.9 years. As §13.5 notes, the table's holder shares span agency debt as well as MBS, so treat them here as an illustrative allocation of MBS-only exposure. Compute dollar duration at the pre-shock value throughout. (a) Compute the total dollar duration created, in trillions of dollars-years. (b) Compute the ten-year-equivalent selling that would be required if every holder hedged, at a ten-year modified duration of 8.0. (c) Now apply Table 13.3's hedging column, taking each holder's share of the table's total as its share of the nine trillion: assume that the Fed, foreign official, and bank holdings do not hedge at all, that insurers and pensions hedge a quarter of their exposure, that funds hedge half, and that the residual category hedges in full. Compute the hedged share and the actual selling. (d) The Federal Reserve completes quantitative tightening and its entire holding passes to total-return funds. Recompute (c). (e) State in three sentences what (c) and (d) together imply for the relationship between the size of the mortgage market and the volatility of long-term interest rates. *** ## Selected Solutions *Solutions to Problems 1 and 3 follow. Solutions to the remainder are in the instructor materials.* **Problem 1.** (a) With $$r = 0.07/12 = 0.00583\overline{3}$$ and $$n = 360$$, $$ \mathrm{PMT} = 300{,}000 \cdot \frac{0.0058333}{1 - (1.0058333)^{-360}} = 1{,}995.91 $$ (b) First-month interest is $$300{,}000 \times 0.0058333 = 1{,}750.00$$, so principal is 245.91. Interest is **87.7 percent** of the first payment — the back-loading of §13.1. (c) Carrying the schedule forward sixty months (equivalently, $$\mathrm{Bal}\_{60} = \mathrm{Bal}\_0(1+r)^{60} - \mathrm{PMT}\left\[((1+r)^{60}-1)/r\right]$$) gives **282,394.77**. After five years of payments the borrower has retired 5.9 percent of the loan. (d) Total payments are $$360 \times 1{,}995.91 = 718{,}526.69$$, so total interest is **418,526.69** — about 1.40 times the amount borrowed. (e) Refinancing 282,394.77 at 5.0 percent over 300 months gives a payment of **1,650.85**, a saving of **345.06** a month. Discounting that saving at 5.0 percent over 300 months gives a present value of **59,025.26**. Closing costs of three percent of the balance are 8,471.84, so the refinancing is worth about **50,553** in present value — roughly seventeen percent of the outstanding balance, and an order of magnitude larger than the cost of acting. That gap is the size of the option in §13.1, and the fact that roughly a fifth of borrowers in this position do not exercise it (Chapter 14 §14.4) is what §13.3's burnout parameter is measuring. *A caution the problem is designed to expose: extending the term back to thirty years instead of twenty-five would give a payment of 1,515.96 and an apparent monthly saving of 479.95, which is larger and worse. Part of it is not a saving at all but a deferral of principal. Comparing payments across different terms is the most common error in retail refinancing arithmetic.* **Problem 3.** (a) $$p = 8.4/0.0525 =$$ **160.0 million dollars**. (b) $$\mathrm{cap} = r - g$$, so $$r = 0.0525 + 0.025 =$$ **7.75 percent**. (c) $$p = 8.4/0.07 =$$ **120.0 million dollars**, a decline of **25.0 percent** on unchanged income. The implied required return has risen to 9.50 percent. (d) NOI falls to $$8.4 \times 0.88 = 7.392$$ million, so $$p = 7.392/0.07 =$$ **105.6 million dollars**, a total decline of **34.0 percent**. The cap-rate effect alone is −25.0 percent and the income effect alone is −12.0 percent; $$(1-0.25)(1-0.12) - 1 = -0.34$$ exactly, because value is the product of income and the reciprocal of the cap rate. Adding the two effects would give −37 percent and overstate the damage. (e) Debt at sixty percent of the original valuation is 96.0 million, against original equity of 64.0 million. At a value of 105.6 million the equity is 9.6 million: an 85 percent loss on a 34 percent decline in the asset. This is the leverage identity of Chapter 16 §16.1 applied to a building, and it is the reason a cap-rate move that is merely uncomfortable for an unlevered holder is terminal for a levered one — and the reason the holders who absorb a repricing are so rarely the holders who owned the asset going in. *** ## Data Exercise: The Mortgage Spread and the QE Footprint All parts run on free data from FRED unless marked. **Part A — The primary-secondary and mortgage-Treasury spreads.** Download `MORTGAGE30US` (Freddie Mac's weekly survey rate on a thirty-year fixed mortgage) and `DGS10` (the ten-year constant-maturity Treasury yield) over the longest common sample, which begins in 1971. 1. Construct the spread of the mortgage rate over the ten-year yield. Plot the two series and the spread on stacked panels. Report the full-sample mean of the spread and its standard deviation. 2. Mark and describe four episodes: the summer of 2003, the fall of 2008, the spring of 2020, and 2022 to 2023. For each, report the spread at its local peak or trough and say in one sentence which term of §13.3's decomposition — the option cost, the OAS, or the primary-secondary origination margin — you think moved, and what you would need to see to be sure. 3. The spread widened dramatically in 2022 to 2023 and stayed wide. Give two candidate explanations from this chapter — one from §13.3 (the option is worth more when rate volatility is high) and one from §13.5 (the largest price-insensitive holder stopped buying) — and describe the additional series you would need to separate them. **Part B — The QE footprint.** Download `WSHOMCB` (the Federal Reserve's holdings of mortgage-backed securities, weekly, from the H.4.1 release) and `MORTGAGE30US`. 4. Plot the Fed's MBS holdings from 2009 to the present. Date the three accumulation phases and the runoff, and report the peak level and the current level. 5. Overlay the mortgage-Treasury spread from Part A. Does the spread narrow when the Fed is buying and widen when it is not? Run the simplest regression that tests this and state clearly why the coefficient is not a causal estimate. 6. Using Table 13.3's logic, compute the Fed's share of agency MBS outstanding at the peak (SIFMA publishes outstanding agency MBS free) and today. Write one paragraph on what that share implies for the hedged share of the market and therefore for the convexity feedback of §13.5. **Part C — The 2003 episode, reconstructed.** 7. Plot `DGS10` daily from January 2003 through December 2003. Mark June 13, June 25, July 15, and September 2. Report the cumulative change from the June low to the early-August level and to the early-September level. 8. Add `MORTGAGE30US` on the same panel and report how far the mortgage rate rose over the same window. Using §13.3's S-curve description, describe qualitatively what happened to the refinancing incentive of the average outstanding loan. 9. From FRED, add a measure of Treasury market volatility if one is available to you, or use the realized volatility of daily changes in `DGS10` in a rolling twenty-day window. Show that realized volatility rose with the level of yields over the summer, and explain why §13.3 predicts precisely that pattern for a market whose holders are short an option. **Part D — Cap rates and the discount rate.** 10. Download `DGS10` and, from the Nareit website (free), the FTSE Nareit All Equity REITs index dividend yield. Construct the spread of the REIT dividend yield over the ten-year yield from 1990 and plot it. 11. Treating the REIT dividend yield as a rough listed-market proxy for a cap rate, compare its behavior in 2022 to 2023 with the appraisal-based NCREIF Property Index total return over the same period (NCREIF publishes headline index returns free). Quantify the lag, and interpret it with Chapter 18 §18.2's smoothing model: what value of the smoothing weight $$\omega$$ would reconcile the two series? **Part E ★ (if you have institutional access).** Using eMBS or an equivalent pool-level database, download monthly CPR by cohort for agency thirty-year pools, together with the coupon and weighted-average loan age of each cohort. Plot CPR against refinancing incentive for a single cohort over its life and fit the S-curve of §13.3. Then plot the same relation separately for the cohort's first and second passes through the same incentive, and measure the burnout: how much slower is the second pass, and does a single state variable for cumulative incentive exposure account for it?