# Hedging runoff risk; severe runoff doesn't happen often, but often enough to teach dramatic lessons.

Severe runoff doesn't happen often, but often enough to teach
dramatic lessons. Hedging tools for managing the risk of widespread
prepayments should be carefully examined because they are not all
created equally.

As result of last year's significant decline in interest rates and the corresponding avalanche of refinancings, mortgage servicers and servicing investors were confronted again with the often-ignored problem of managing the prepayment risk of servicing portfolios.

Consider that as 30-year fixed mortgage rates fell from 9.41 percent in March 1991 to 8.57 percent in March 1992 - a drop of only 84 basis points-mortgage servicers witnessed the conditional prepayment rate of moderately seasoned, Fannie Mae 9s jump fivefold, from 7.3 percent (120 PSA) to 36.1 percent (600 PSA). At a runoff rate of 600 PSA, the servicing portfolio shows negligible returns that are well below the opportunity cost of capital of any servicing operation.

To put it in dollars and cents, we have calculated that a typical current-coupon servicing portfolio loses, in a single quarter, $1 million to $2 million for every $1 billion of principal serviced - or 7 percent to 15 percent or more of its economic value - as a flood of prepayments rushes in. Such a portfolio would end up losing $5 million or more for every $1 billion of principal serviced - or one-third or more of its value - before prepayments subsided.

However, if the same portfolio were properly hedged, total losses could be limited to less than $1 million, avoiding a loss of $4 million on every $1 billion of principal, or more than 25 percent of the total portfolio value.

Industry publications have given this timely subject considerable coverage. In this magazine alone, two lengthy articles described the difficulty of predicting prepayments and the evolution of some of the servicing prepayment-risk management tools available (see "Dodging the Prepayment Bullet" by Howard Schneider and "Perfecting the Prepayment Hedge" by Robert Stowe England, Mortgage Banking, October and September 1991, respectively). The two articles paint a picture of even the most forward-looking mortgage servicers not going beyond static scenario analyses of prepayment risk in their portfolios.

We believe, however, that by borrowing a few tools from the analytical arsenal of mortgage-backed securities investors, mortgage servicers can develop, with incremental effort, a substantially richer understanding of the prepayment risk embedded in their servicing portfolios. Only on the basis of this effort can mortgage servicers consciously chart a prepayment-risk management course to avoid the disastrous servicing losses we have seen periodically through the last decade.

In this article we describe how option-adjusted analysis technology can be used to accurately measure the prepayment risk of mortgage-servicing portfolios. We describe the effects of prepayment-risk hedging and prepayment insurance on the performance of a typical portfolio. The article also shows how well the prepayment-risk hedging strategy actually performed in a five-year period ending October 1989, a period in which current-coupon mortgages prepaid at rates as high, or even higher, than those experienced in 1992.

Model description

At MSI Group, we have developed a set of proprietary analytical models that can be used to simulate the prepayment risk of any mortgage-based asset, such as mortgage servicing, and to hedge the risk using a variety of instruments. The models generate a probability distribution of prepayment rates, calculate the cash flows from the mortgage-based asset on the basis of these rates and show the effects of various hedging strategies on these cash flows. The model components include: * A Treasury-rate simulator; * A mortgage-rate simulator; * A prepayment-rate simulator; * A calculator for valuing mortgage-based

assets; * An options-based hedge model; * A futures-based hedge model; * A mortgage derivative-based hedge

model; * A swap-based hedge model; * A simulation summary.

The models are based on the historical behavior of fixed-rate, long-term, agency, mortgage-backed securities - a segment that accounts for approximately nine-tenths of the $1.3 trillion in investor-owned, mortgage-backed securities outstanding. The models also can be modified to simulate the prepayment behavior of adjustable-rate and nonconforming mortgages.

For ease of discussion, we will assume in this article that we are helping a mortgage servicer who is faced with the challenge of measuring and managing the prepayment risk of a $1 billion portfolio of seasoned, nationally diversified, 30-year, 8 percent coupon, Fannie Mae mortgages. We will also assume that his (or her) analysis will cover the next five years.

Simulating Treasury and

mortgage rates

Our analysis of long-term mortgage rates shows that they are closely correlated with 10-year Treasury rates. Our model, therefore, starts with a simulation of 10-year Treasury yields. We estimate these yields in a regression model, using 10-plus years of monthly yield data. The model accounts for the historical movements in the yields in terms of a mean-reversion target, the recent trends in yields and an error term. Each simulation is performed by introducing randomly generated error terms into the Treasury regression model.

Next, we derive the 30-year, fixed-rate, single-family mortgage rates that may result from the Treasury rates just simulated. We do that by modeling the mortgage rates through another regression analysis as a function of recent Treasury rates, mortgage rates and an error term. The model is estimated, again, on 10-plus years of monthly mortgage-and Treasury-rate data. As before, we perform each simulation by introducing randomly generated error terms into the mortgage regression model.

Simulating prepayment rates

Our prepayment model for Fannie Mae securities is a non-linear regression model estimated on 10 years of monthly prepayment data. The model accounts for the prepayment rates of a generic mortgage-backed security in terms of its relative coupon (a measure of how much in-, at- or out-of-the-money is the borrower's option to prepay the mortgage); its age; its burnout (a measure of its previous exposure to mortgage-rate environments when the prepayment option was in-the-money); its seasonality; its recent prepayment experience; the recent trends in mortgage rates; and, again, an error term. The real world predictive capability of this regression, as measured by its out-of-sample r-squared of 87 percent, is very strong.

As in previous simulations, we generate a 60-month series of prepayment rates by introducing randomly generated error terms into the prepayment regression model. The results of a dozen such simulations show that prepayment rates start out at 13 percent, their level in May 1992, but go as low as 2 percent and as high as 51 percent (850 PSA) over five years.

Calculating the value of

mortgage servicing

Given the future path of prepayment rates, the mortgage-based asset can be valued. Significant ingenuity, creativity and flexibility may be introduced when valuing some of the more esoteric mortgage-backed assets, such as some REMIC securities.

In the case of mortgage servicing, fortunately, the valuation is relatively straightforward and hinges on the value of the future stream of servicing-related income that will be lost when a borrower prepays principal. This is accomplished through a discounted cash-flow model. We will assume that the servicer wants to lock in a cumulative prepayment rate over the five-year term and be compensated, through a hedge, for the economic value of lost income whenever the cumulative prepayment rate exceeds this insured rate (see Table 1). For example, for every $1 million of prepaid principal, the servicer loses $15,000, which is the discounted cash-flow value of future servicing income lost. This approach is sometimes referred to as a "book value" hedge, on the expectation that the servicer has booked the economic value of his or her servicing as an asset on the balance sheet.

In Figure 1, we show the hedge payoffs needed over 60 months for one of the simulated prepayment paths. Notice that in a single quarter (ending month 27), when the prepayment rates hit 51 percent (850 PSA), the servicer loses $1,000,000, or about 7 percent of the value of the portfolio.

With the recent deluge of refinancing-related prepayments, mortgage servicers have witnessed losses as high as this - or even higher. In fact, the cumulative losses suffered by the servicer under this prepayment path amount to $5,100,000, or more than one-third of the value of the portfolio. To put it another way, at such astronomical levels of prepayment, the rates of return on this and other typical portfolios approach zero and are dramatically below the cost of capital of any for-profit business.

The result is pure and simple destruction of value for the owners of the servicing assets. This is indeed the Achilles heel of servicing, a business traditionally but mistakenly viewed as a high-return, low-risk business, in actuality it's a business where the embedded prepayment risk can kill you.

Hedging prepayment risk

Theoretically speaking, to hedge prepayment risk the servicer can use a variety of alternative instruments including futures, options on futures and mortgage derivatives such as principal-only (PO) securities and interest-rate swaps. Or the servicer may choose to depend on Wall Street's wisdom and transfer the risk by buying from an investment bank a hedge presumably correlated to the portfolio's particular prepayment risk. From a practical perspective, some instruments work better than others in hedging prepayment risk.

We will illustrate an approach in which the servicer uses three-month call options on 10-year Treasury note futures to manage his prepayment risk. The options are priced in our model through a Black-Scholes option price calculator. As might be expected, the calculated prices proved to be highly accurate when we compared them with actual market-maker price data.

To hedge with three-month call options, the servicer has to figure out how many options must be bought every quarter. The servicer does so by first calculating the incremental servicing losses that would be suffered if interest rates were to decline by x percent. The servicer then calculates the incremental payoff from the option written on 10-year T-note futures. Dividing the incremental loss by the incremental payoff, the servicer derives the "raw" number of options that must be bought to guard against an x percent decline in interest rates. The servicer further modifies this "raw" number to reflect the mean-reversion effect in interest rate paths and buys the modified number of T-note futures options to manage his prepayment risk for the next three months. The process is repeated every three months during the five-year period.

Figure 2 shows the number of options bought to manage the prepayment risk implied by one of the simulations just generated. For this interest rate path, the servicer will need to buy as many as 97 options, or as few as none, in any given three-month period.

In hedging the prepayment risk, our primary objective is to truncate the extreme prepayment-related losses in the servicing portfolio. We accomplish that by reducing the variability of such losses, as measured by their standard deviation. The cost of this benefit is an increase in the average loss.

A successful hedge will significantly truncate the extreme servicing losses generated in high-prepayment scenarios. Table 2 shows what the losses would be with and without the benefit of option-based hedges in one such case of extremely high prepayments. Without the hedges, the discounted value of the servicer's prepayment-related losses would be $4.2 million (42 bp); with the benefit of the hedge, it is $3 million (30 bp) - a saving of $1.2 million, or roughly 30 percent of the unhedged losses.

While the hedge insulates the servicer from extreme risks, most of the time it will cost the servicer something, demonstrating the truism that there is no free lunch. In return for paying marginally more most of the time, the servicer expects to get significant protection from extreme risks. Table 3 shows a less extreme interest rate path where the losses incurred with the hedge are greater than they would have been if the servicer went unhedged. Without the hedge, the servicer would lose $0.4 million (4 bp); with it, the loss is $0.5 million (5 bp). Therefore, the marginal cost of hedging in this example is $100,000.

Hedging through a

prepayment swap

One of the more innovative swaps being offered by a few Wall Street investment banks covers the servicer against prepayment-related losses. Such losses are calculated typically on the basis of the prepayment rate of an index, or reference portfolio. In its simplest form, the servicer pays a fixed, upfront swap fee; the investment bank writing the swap makes variable payments contingent on the cumulative prepayment rate of the reference portfolio exceeding the insured rate.

Let us assume that the mortgage servicer is considering buying such a prepayment rate-based swap from an investment bank. The key aspects of the swap are described in Table 4. Notice that, for this insurance policy, the investment bank will charge the servicer an upfront fee of $3.5 million, or roughly one-fourth of the portfolio value.

Our swap-based hedge model can be used to evaluate the effectiveness of the proposed swap in managing the prepayment risk of the servicer's portfolio. Table 5 shows the outcomes of a dozen simulations. In the first simulation, for example, the servicing losses that are incurred whenever the cumulative prepayment rate on the reference portfolio exceeds the insured rate of 15 percent (200 PSA) amount to approximately $4.2 million, in net-present-value terms. This is precisely the discounted value of the payments the servicer would receive over five years from the investment bank. Netting out the upfront swap fee of $3.5 million, we see that for this particularly severely declining (mortgage rates drop to 6.5 percent) path of interest rates, the swap generates a net benefit of $0.7 million. However, for the remaining 11 simulations, which range from moderately declining to stable to moderately or severely inclining (mortgage rates go up to 11.3 percent) interest rates, the swap fee exceeds, mostly by a significant margin, the payments the servicer would get from the investment bank.

Simulation summaries

Of course, the whole point is to use the power of the computer to generate not a dozen or a few dozen simulations, but thousands of simulations and to consider the distribution of outcomes. Going well beyond what may be derived from a static scenario analysis of prepayment risks, our approach allows the sharp-penciled servicer to evaluate the probability distribution of prepayment risks in his or her portfolio.

Just knowing this distribution will make servicers better risk managers. The servicer no longer has to make an explicit or, much worse, implicit guess as to the course of interest and mortgage rates. He or she can clearly see the prepayment-related risk for all possible interest-rate paths. Servicers can evaluate the prepayment-risk distribution in terms of their tolerance for risk. Then servicers can make informed decisions on whether or not and how to hedge their prepayment risk. Servicers may, as we just suggested, hedge on their own or buy insurance from someone else. In either case, they will be substantially better informed, and substantially superior competitors.

In Figure 3 we show the probability distribution of the cumulative prepayment rates for our mortgage servicer's Fannie Mae portfolio. This distribution is based on 1,000 simulations. The median prepayment rate is about 10.5 percent (175 PSA), and the mean is 12 percent (200 PSA). However, there is a 10 percent chance that the prepayment rate will be 15 percent (250 PSA) or more, and a 2 percent chance that it will be 20 percent (333 PSA) or more. The 20 occurrences of prepayment rates in excess of 20 percent included an instance where the prepayment rate went all the way up to 39 percent (650 PSA). One only has to remember that these are compound prepayment rates over a five-year period to understand the force behind the severe servicing losses that would ensue.

Table 6 shows the results of 1,000 simulations where the servicer went unhedged, or chose to transfer the prepayment risk by buying a prepayment swap or decided to manage the portfolio's prepayment risk by using options on Tnote futures.

Let us first see what happens if the servicer goes unhedged. Notice that while this unhedged servicer may expect to lose on average $500,000 (5 bp), a decision to go unhedged results in severe, even disastrous, losses at any significant risk level. For example, there is a 10 percent chance that the unhedged servicing losses will exceed $1,179,000 (12 bp), and a 2 percent chance that they will be greater than $2,088,000 (21 bp). The servicer's maximum unhedged losses can turn out to be as high as $4,368,000 (44 bp), or roughly 30 percent of the value of the portfolio.

Next, let us consider what happens if our servicer elects to buy a prepayment rate-based swap which works exactly as described in the previous section and in Table 4. To refresh your memory, the swap writer will pay the servicer 1.5 percent of any principal prepaid in excess of a cumulative runoff rate of 12 percent in the reference portfolio. On average, the losses will be $3 million (30 bp), the $3.5 million swap fee he pays upfront less the $0.5 million in pay receive from the investment bank. Notice also that as a hedging device, the swap fails to reduce the variability of the servicer's losses: the standard deviation (i.e., the square root of the variance of losses) remains unchanged at $537,000.

The swap puts a cap, albeit a costly one, on this servicer's losses. But the servicer must pay $3.5 million for the cap, and assuming that management does its homework, the servicer knows that the unhedged losses will exceed this with only a 0.1 percent probability. In other words, there is a 0.1 percent chance that the swap will pay out more to the servicer than what the servicer pays for the swap upfront. In that 1-in-1,000 case, the swap generates a net benefit of $868,000 ($4.368 million less $3.5 million) for the servicer.

If the servicer buys the prepayment swap, then at least there is certainty that net losses will be limited to $3.5 million - naturally, the most the servicer can lose is the swap fee that is paid upfront. By the same logic, the servicer's losses will be limited to $3.48 million with a 90 percent probability. Notice that the swap has a fairly high, relatively fixed cost. For example, there is a 10 percent probability that the servicing losses with the swap will be between $3.48 million and $3.50 million. Increasing the probability significantly, and, therefore, taking a significantly higher risk, does not lower the range of losses significantly. For instance, there is a 50 percent probability that the losses will be between $3.16 million and $3.50 million - a high probability of net losses being in a fairly narrow but high range. In other words, if he buys the swap the servicer should expect his net cost to be close to the $3.5 million he pays upfront.

We can also see in Table 6 that, in the vast majority of cases, our servicer is better off hedging prepayment risk through the use of options than through the prepayment swap. For instance, on average, the options-based hedging will cost the servicer $892,000 (9 bp). This is $2.1 million (21 bp) cheaper than the average cost of the prepayment swap - a savings equal to one-seventh of the portfolio value. At virtually all risk levels, it is cheaper to hedge with options using our example. For instance, our servicer can expect with a 98 percent probability that the losses will be limited to $1,985,000 (20 bp); at that risk level, the losses with the swap are $3.5 million, or 75 percent higher. Note also that the options-based hedge is more effective because it reduces the variability of this servicer's losses to $455,000.

Another characteristic of a good hedge is its ability to "not hurt too much." For instance, the average cost of the options-based hedge is 9 bp, only 4 bp more than the average loss from unhedged positions. At the higher end of the distribution, the options-based hedge becomes even less expensive, and at the extreme, as it truncates losses dramatically, the hedged position becomes cheaper than going unhedged.

Actual performance of

options-based hedge

If all of this sounds academic or highly theoretical, let us consider what would have happened to our servicer between October 1984 and October 1989 - for many servicers a period as difficult as 1992. Let's assume that in October 1984 our servicer held the servicing rights on a $1 billion portfolio of Fannie Mae 12s, which, back then, were current-coupon mortgages.

Figure 4 shows the actual path of interest, mortgage and prepayment rates for the next five years. Starting at 12.2 percent, the 10-year T-note yields drop by 510 bp going to 7.1 percent in January 1987, then recover to 9.5 percent in October 1987 and recede to 8 percent by October 1989. The servicer's concern with the prepayment risk of the Fannie Mae 12s is quite understandable when we look at what mortgage rates do in such a volatile environment. Starting at 13.1 percent (which means that the servicing portfolio with a weighted average coupon of 12.8 percent is basically at-the-money), mortgage rates plummet down to 9 percent by April 1987 (which means that the portfolio is deep-in-the-money), recover to 10.8 percent by May 1989 and end up at 10.1 percent by October 1989.

The prepayment-rate path shown in Figure 4 illustrates the consequent deluge of prepayments. Starting out at a very modest 5 percent (80 PSA), prepayments climb rapidly to 61 percent (1020 PSA) by August 1986, a 13-fold increase. Notice that most of the time during this five-year period, prepayment rates oscillate above 20 percent (330 PSA) and close out at that level. For many of those months prepayment rates hover around 50 percent (820 PSA). After mortgage rates bottom out in April 1987, prepayment rates hit their second peak at 57 percent (950 PSA) and stay at similarly extreme levels for several more months as the portfolio burns out.

Unhedged servicing losses prove to be catastrophic at the actual five-year, compound prepayment rate of 27 percent (450 PSA), almost six times the initial prepayment rate. At this runoff rate, only 20 percent of the portfolio, or $200 million, remains outstanding by October 1989. Virtually all of this principal loss is due to prepayments because at the normal rate of amortization, 96 percent of the principal would still be outstanding after five years.

Reliving the 1986 refi boom

with a hedge

Let us suppose that in October 1984, the servicer puts in place a hedge strategy that will effectively guarantee that the servicing portfolio will prepay at 9 percent (150 PSA) during the next five years. At the targeted prepayment rate of 9 percent, the servicer would still have 60 percent, or three times the actual remaining balance outstanding at the end of five years. As a result, he would look to the hedge to generate sufficient benefits to cover the value of the income lost on 40 percent (60 percent to 20 percent) of principal that is prepaid.

Figure 5 translates the gap between actual and targeted prepayment rates into servicing losses that we would expect the hedge to truncate. Let us assume that the value of prepaid principal is 1.5 percent (150 bp). Notice that most of the losses come in a nine-month period starting in July 1986. In this period, servicing losses amount to $4.7 million, or roughly three-tenths of the value of the servicing portfolio. For the five-year period, the discounted value of the servicing losses amounts to $5.2 million, or more than one-third of the portfolio value.

Remember that the primary objective of hedging is to truncate the extreme prepayment-related losses. Let us assume that we help the servicer create two alternative hedging strategies with different powers of truncation. Strategy A is somewhat conservative and aims to truncate high losses by about 30 percent. This strategy would lead our servicer to buy as many as 130 options in a three-month period. Strategy B, by comparison, is more aggressive in hedging and aims to recover 80 percent or more of the losses through the benefits of the hedge. This strategy would lead the servicer to buy as many as 360 options in a three-month period.

Both strategies prove to be effective in averting disaster in servicing losses during this period. Table 7 shows the actual results of these two hedging strategies against the actual prepayment rates. The unhedged losses amount to $5.2 million, in net-present-value terms. Strategy A curtails the losses by $1.5 million, or 27 percent, and limits them to $3.8 million, or 25 percent of the portfolio value. Strategy B, by comparison, is more powerful in its truncation. It reduces the losses by $4.3 million (83 percent) and limits them to $0.9 million, or only 6 percent of the portfolio value.

Guesswork or homework?

This article has presented a reliable analytical method by which mortgage servicers and servicing investors can take the guesswork out of prepayment risk and substitute, in its place, informed risk management decision making. Such risk management is built upon a rich understanding of the prepayment risk inherent in all servicing portfolios, regardless of coupon, age or other characteristics. This discussion has also demonstrated - and history has periodically reminded the industry of this - that prepayments can ruin the profitability of an otherwise stellar business. The good news is that prepayment risk can be managed - if you do the homework.

As result of last year's significant decline in interest rates and the corresponding avalanche of refinancings, mortgage servicers and servicing investors were confronted again with the often-ignored problem of managing the prepayment risk of servicing portfolios.

Consider that as 30-year fixed mortgage rates fell from 9.41 percent in March 1991 to 8.57 percent in March 1992 - a drop of only 84 basis points-mortgage servicers witnessed the conditional prepayment rate of moderately seasoned, Fannie Mae 9s jump fivefold, from 7.3 percent (120 PSA) to 36.1 percent (600 PSA). At a runoff rate of 600 PSA, the servicing portfolio shows negligible returns that are well below the opportunity cost of capital of any servicing operation.

To put it in dollars and cents, we have calculated that a typical current-coupon servicing portfolio loses, in a single quarter, $1 million to $2 million for every $1 billion of principal serviced - or 7 percent to 15 percent or more of its economic value - as a flood of prepayments rushes in. Such a portfolio would end up losing $5 million or more for every $1 billion of principal serviced - or one-third or more of its value - before prepayments subsided.

However, if the same portfolio were properly hedged, total losses could be limited to less than $1 million, avoiding a loss of $4 million on every $1 billion of principal, or more than 25 percent of the total portfolio value.

Industry publications have given this timely subject considerable coverage. In this magazine alone, two lengthy articles described the difficulty of predicting prepayments and the evolution of some of the servicing prepayment-risk management tools available (see "Dodging the Prepayment Bullet" by Howard Schneider and "Perfecting the Prepayment Hedge" by Robert Stowe England, Mortgage Banking, October and September 1991, respectively). The two articles paint a picture of even the most forward-looking mortgage servicers not going beyond static scenario analyses of prepayment risk in their portfolios.

We believe, however, that by borrowing a few tools from the analytical arsenal of mortgage-backed securities investors, mortgage servicers can develop, with incremental effort, a substantially richer understanding of the prepayment risk embedded in their servicing portfolios. Only on the basis of this effort can mortgage servicers consciously chart a prepayment-risk management course to avoid the disastrous servicing losses we have seen periodically through the last decade.

In this article we describe how option-adjusted analysis technology can be used to accurately measure the prepayment risk of mortgage-servicing portfolios. We describe the effects of prepayment-risk hedging and prepayment insurance on the performance of a typical portfolio. The article also shows how well the prepayment-risk hedging strategy actually performed in a five-year period ending October 1989, a period in which current-coupon mortgages prepaid at rates as high, or even higher, than those experienced in 1992.

Model description

At MSI Group, we have developed a set of proprietary analytical models that can be used to simulate the prepayment risk of any mortgage-based asset, such as mortgage servicing, and to hedge the risk using a variety of instruments. The models generate a probability distribution of prepayment rates, calculate the cash flows from the mortgage-based asset on the basis of these rates and show the effects of various hedging strategies on these cash flows. The model components include: * A Treasury-rate simulator; * A mortgage-rate simulator; * A prepayment-rate simulator; * A calculator for valuing mortgage-based

assets; * An options-based hedge model; * A futures-based hedge model; * A mortgage derivative-based hedge

model; * A swap-based hedge model; * A simulation summary.

The models are based on the historical behavior of fixed-rate, long-term, agency, mortgage-backed securities - a segment that accounts for approximately nine-tenths of the $1.3 trillion in investor-owned, mortgage-backed securities outstanding. The models also can be modified to simulate the prepayment behavior of adjustable-rate and nonconforming mortgages.

For ease of discussion, we will assume in this article that we are helping a mortgage servicer who is faced with the challenge of measuring and managing the prepayment risk of a $1 billion portfolio of seasoned, nationally diversified, 30-year, 8 percent coupon, Fannie Mae mortgages. We will also assume that his (or her) analysis will cover the next five years.

Simulating Treasury and

mortgage rates

Our analysis of long-term mortgage rates shows that they are closely correlated with 10-year Treasury rates. Our model, therefore, starts with a simulation of 10-year Treasury yields. We estimate these yields in a regression model, using 10-plus years of monthly yield data. The model accounts for the historical movements in the yields in terms of a mean-reversion target, the recent trends in yields and an error term. Each simulation is performed by introducing randomly generated error terms into the Treasury regression model.

Next, we derive the 30-year, fixed-rate, single-family mortgage rates that may result from the Treasury rates just simulated. We do that by modeling the mortgage rates through another regression analysis as a function of recent Treasury rates, mortgage rates and an error term. The model is estimated, again, on 10-plus years of monthly mortgage-and Treasury-rate data. As before, we perform each simulation by introducing randomly generated error terms into the mortgage regression model.

Simulating prepayment rates

Our prepayment model for Fannie Mae securities is a non-linear regression model estimated on 10 years of monthly prepayment data. The model accounts for the prepayment rates of a generic mortgage-backed security in terms of its relative coupon (a measure of how much in-, at- or out-of-the-money is the borrower's option to prepay the mortgage); its age; its burnout (a measure of its previous exposure to mortgage-rate environments when the prepayment option was in-the-money); its seasonality; its recent prepayment experience; the recent trends in mortgage rates; and, again, an error term. The real world predictive capability of this regression, as measured by its out-of-sample r-squared of 87 percent, is very strong.

As in previous simulations, we generate a 60-month series of prepayment rates by introducing randomly generated error terms into the prepayment regression model. The results of a dozen such simulations show that prepayment rates start out at 13 percent, their level in May 1992, but go as low as 2 percent and as high as 51 percent (850 PSA) over five years.

Calculating the value of

mortgage servicing

Given the future path of prepayment rates, the mortgage-based asset can be valued. Significant ingenuity, creativity and flexibility may be introduced when valuing some of the more esoteric mortgage-backed assets, such as some REMIC securities.

In the case of mortgage servicing, fortunately, the valuation is relatively straightforward and hinges on the value of the future stream of servicing-related income that will be lost when a borrower prepays principal. This is accomplished through a discounted cash-flow model. We will assume that the servicer wants to lock in a cumulative prepayment rate over the five-year term and be compensated, through a hedge, for the economic value of lost income whenever the cumulative prepayment rate exceeds this insured rate (see Table 1). For example, for every $1 million of prepaid principal, the servicer loses $15,000, which is the discounted cash-flow value of future servicing income lost. This approach is sometimes referred to as a "book value" hedge, on the expectation that the servicer has booked the economic value of his or her servicing as an asset on the balance sheet.

TABLE 1 Servicing Portfolio Profile 30-year, fixed-rate, single-family FNMA MBS (prefix CL) Current Balance $1,000,000,000 Pass-through coupon: 8% WAC: 8.78% WARM: 234 months Age: 126 months Target prepayment rate: 12% (200 PSA) Value of prepaid principal: 1.50%

In Figure 1, we show the hedge payoffs needed over 60 months for one of the simulated prepayment paths. Notice that in a single quarter (ending month 27), when the prepayment rates hit 51 percent (850 PSA), the servicer loses $1,000,000, or about 7 percent of the value of the portfolio.

With the recent deluge of refinancing-related prepayments, mortgage servicers have witnessed losses as high as this - or even higher. In fact, the cumulative losses suffered by the servicer under this prepayment path amount to $5,100,000, or more than one-third of the value of the portfolio. To put it another way, at such astronomical levels of prepayment, the rates of return on this and other typical portfolios approach zero and are dramatically below the cost of capital of any for-profit business.

The result is pure and simple destruction of value for the owners of the servicing assets. This is indeed the Achilles heel of servicing, a business traditionally but mistakenly viewed as a high-return, low-risk business, in actuality it's a business where the embedded prepayment risk can kill you.

Hedging prepayment risk

Theoretically speaking, to hedge prepayment risk the servicer can use a variety of alternative instruments including futures, options on futures and mortgage derivatives such as principal-only (PO) securities and interest-rate swaps. Or the servicer may choose to depend on Wall Street's wisdom and transfer the risk by buying from an investment bank a hedge presumably correlated to the portfolio's particular prepayment risk. From a practical perspective, some instruments work better than others in hedging prepayment risk.

We will illustrate an approach in which the servicer uses three-month call options on 10-year Treasury note futures to manage his prepayment risk. The options are priced in our model through a Black-Scholes option price calculator. As might be expected, the calculated prices proved to be highly accurate when we compared them with actual market-maker price data.

To hedge with three-month call options, the servicer has to figure out how many options must be bought every quarter. The servicer does so by first calculating the incremental servicing losses that would be suffered if interest rates were to decline by x percent. The servicer then calculates the incremental payoff from the option written on 10-year T-note futures. Dividing the incremental loss by the incremental payoff, the servicer derives the "raw" number of options that must be bought to guard against an x percent decline in interest rates. The servicer further modifies this "raw" number to reflect the mean-reversion effect in interest rate paths and buys the modified number of T-note futures options to manage his prepayment risk for the next three months. The process is repeated every three months during the five-year period.

Figure 2 shows the number of options bought to manage the prepayment risk implied by one of the simulations just generated. For this interest rate path, the servicer will need to buy as many as 97 options, or as few as none, in any given three-month period.

In hedging the prepayment risk, our primary objective is to truncate the extreme prepayment-related losses in the servicing portfolio. We accomplish that by reducing the variability of such losses, as measured by their standard deviation. The cost of this benefit is an increase in the average loss.

A successful hedge will significantly truncate the extreme servicing losses generated in high-prepayment scenarios. Table 2 shows what the losses would be with and without the benefit of option-based hedges in one such case of extremely high prepayments. Without the hedges, the discounted value of the servicer's prepayment-related losses would be $4.2 million (42 bp); with the benefit of the hedge, it is $3 million (30 bp) - a saving of $1.2 million, or roughly 30 percent of the unhedged losses.

TABLE 2 Truncating Servicing Losses with Payoffs from Option-Based Hedges Compound Prepayment Rate: 26.0% Beginning Treasury Yield: 7.4% Maximum Treasury Yield: 7.4% Minimum Treasury Yield: 4.5% delta (Treasury Yield): 2.9% NPV (Unhedged Servicing Losses) $4.2 million NPV (Hedged Losses) $3.0 million Hedge Benefit $1.2 million

While the hedge insulates the servicer from extreme risks, most of the time it will cost the servicer something, demonstrating the truism that there is no free lunch. In return for paying marginally more most of the time, the servicer expects to get significant protection from extreme risks. Table 3 shows a less extreme interest rate path where the losses incurred with the hedge are greater than they would have been if the servicer went unhedged. Without the hedge, the servicer would lose $0.4 million (4 bp); with it, the loss is $0.5 million (5 bp). Therefore, the marginal cost of hedging in this example is $100,000.

TABLE 3 Hedging with Options against Moderate Prepayment-Related Losses Compound Prepayment Rate: 10.1% Beginning Treasury Yield: 7.4% Maximum Treasury Yield: 10.0% Minimum Treasury Yield: 6.9% delta (Treasury Yield): 3.1% NPV (Unhedged Servicing Losses) $0.4 million NPV (Hedged Losses) $0.5 million Hedge Cost $0.1 million

Hedging through a

prepayment swap

One of the more innovative swaps being offered by a few Wall Street investment banks covers the servicer against prepayment-related losses. Such losses are calculated typically on the basis of the prepayment rate of an index, or reference portfolio. In its simplest form, the servicer pays a fixed, upfront swap fee; the investment bank writing the swap makes variable payments contingent on the cumulative prepayment rate of the reference portfolio exceeding the insured rate.

Let us assume that the mortgage servicer is considering buying such a prepayment rate-based swap from an investment bank. The key aspects of the swap are described in Table 4. Notice that, for this insurance policy, the investment bank will charge the servicer an upfront fee of $3.5 million, or roughly one-fourth of the portfolio value.

TABLE 4 Hedging with a Prepayment Swap Hedged Servicing Portfolio: As in Table 1 Reference Portfolio: FNMA 8s Age of Reference Portfolio: 10.5 yrs Insured Prepayment Rate: 12% (200 PSA) Value of Prepaid Principal: 1.5% (150 bp) Swap Fee (paid upfront): $3.5 million (35 bp)

Our swap-based hedge model can be used to evaluate the effectiveness of the proposed swap in managing the prepayment risk of the servicer's portfolio. Table 5 shows the outcomes of a dozen simulations. In the first simulation, for example, the servicing losses that are incurred whenever the cumulative prepayment rate on the reference portfolio exceeds the insured rate of 15 percent (200 PSA) amount to approximately $4.2 million, in net-present-value terms. This is precisely the discounted value of the payments the servicer would receive over five years from the investment bank. Netting out the upfront swap fee of $3.5 million, we see that for this particularly severely declining (mortgage rates drop to 6.5 percent) path of interest rates, the swap generates a net benefit of $0.7 million. However, for the remaining 11 simulations, which range from moderately declining to stable to moderately or severely inclining (mortgage rates go up to 11.3 percent) interest rates, the swap fee exceeds, mostly by a significant margin, the payments the servicer would get from the investment bank.

TABLE 5 Evaluating the Effectiveness of a Prepayment Swap Unhedged Upfront Net Servicing Swap Cost of Simulation Losses Fee Swap ($000) ($000) ($000) 1 $4,194 $3,500 ($694) 2 $1,068 $3,500 $2,432 3 $354 $3,500 $3,146 4 $2,733 $3,500 $767 5 $22 $3,500 $3,478 6 $422 $3,500 $3,078 7 $1,208 $3,500 $2,292 8 $2,681 $3,500 $819 9 $1,140 $3,500 $2,360 10 $351 $3,500 $3,149 11 $0 $3,500 $3,500 12 $1,817 $3,500 $1,683

Simulation summaries

Of course, the whole point is to use the power of the computer to generate not a dozen or a few dozen simulations, but thousands of simulations and to consider the distribution of outcomes. Going well beyond what may be derived from a static scenario analysis of prepayment risks, our approach allows the sharp-penciled servicer to evaluate the probability distribution of prepayment risks in his or her portfolio.

Just knowing this distribution will make servicers better risk managers. The servicer no longer has to make an explicit or, much worse, implicit guess as to the course of interest and mortgage rates. He or she can clearly see the prepayment-related risk for all possible interest-rate paths. Servicers can evaluate the prepayment-risk distribution in terms of their tolerance for risk. Then servicers can make informed decisions on whether or not and how to hedge their prepayment risk. Servicers may, as we just suggested, hedge on their own or buy insurance from someone else. In either case, they will be substantially better informed, and substantially superior competitors.

In Figure 3 we show the probability distribution of the cumulative prepayment rates for our mortgage servicer's Fannie Mae portfolio. This distribution is based on 1,000 simulations. The median prepayment rate is about 10.5 percent (175 PSA), and the mean is 12 percent (200 PSA). However, there is a 10 percent chance that the prepayment rate will be 15 percent (250 PSA) or more, and a 2 percent chance that it will be 20 percent (333 PSA) or more. The 20 occurrences of prepayment rates in excess of 20 percent included an instance where the prepayment rate went all the way up to 39 percent (650 PSA). One only has to remember that these are compound prepayment rates over a five-year period to understand the force behind the severe servicing losses that would ensue.

Table 6 shows the results of 1,000 simulations where the servicer went unhedged, or chose to transfer the prepayment risk by buying a prepayment swap or decided to manage the portfolio's prepayment risk by using options on Tnote futures.

Let us first see what happens if the servicer goes unhedged. Notice that while this unhedged servicer may expect to lose on average $500,000 (5 bp), a decision to go unhedged results in severe, even disastrous, losses at any significant risk level. For example, there is a 10 percent chance that the unhedged servicing losses will exceed $1,179,000 (12 bp), and a 2 percent chance that they will be greater than $2,088,000 (21 bp). The servicer's maximum unhedged losses can turn out to be as high as $4,368,000 (44 bp), or roughly 30 percent of the value of the portfolio.

Next, let us consider what happens if our servicer elects to buy a prepayment rate-based swap which works exactly as described in the previous section and in Table 4. To refresh your memory, the swap writer will pay the servicer 1.5 percent of any principal prepaid in excess of a cumulative runoff rate of 12 percent in the reference portfolio. On average, the losses will be $3 million (30 bp), the $3.5 million swap fee he pays upfront less the $0.5 million in pay receive from the investment bank. Notice also that as a hedging device, the swap fails to reduce the variability of the servicer's losses: the standard deviation (i.e., the square root of the variance of losses) remains unchanged at $537,000.

The swap puts a cap, albeit a costly one, on this servicer's losses. But the servicer must pay $3.5 million for the cap, and assuming that management does its homework, the servicer knows that the unhedged losses will exceed this with only a 0.1 percent probability. In other words, there is a 0.1 percent chance that the swap will pay out more to the servicer than what the servicer pays for the swap upfront. In that 1-in-1,000 case, the swap generates a net benefit of $868,000 ($4.368 million less $3.5 million) for the servicer.

If the servicer buys the prepayment swap, then at least there is certainty that net losses will be limited to $3.5 million - naturally, the most the servicer can lose is the swap fee that is paid upfront. By the same logic, the servicer's losses will be limited to $3.48 million with a 90 percent probability. Notice that the swap has a fairly high, relatively fixed cost. For example, there is a 10 percent probability that the servicing losses with the swap will be between $3.48 million and $3.50 million. Increasing the probability significantly, and, therefore, taking a significantly higher risk, does not lower the range of losses significantly. For instance, there is a 50 percent probability that the losses will be between $3.16 million and $3.50 million - a high probability of net losses being in a fairly narrow but high range. In other words, if he buys the swap the servicer should expect his net cost to be close to the $3.5 million he pays upfront.

We can also see in Table 6 that, in the vast majority of cases, our servicer is better off hedging prepayment risk through the use of options than through the prepayment swap. For instance, on average, the options-based hedging will cost the servicer $892,000 (9 bp). This is $2.1 million (21 bp) cheaper than the average cost of the prepayment swap - a savings equal to one-seventh of the portfolio value. At virtually all risk levels, it is cheaper to hedge with options using our example. For instance, our servicer can expect with a 98 percent probability that the losses will be limited to $1,985,000 (20 bp); at that risk level, the losses with the swap are $3.5 million, or 75 percent higher. Note also that the options-based hedge is more effective because it reduces the variability of this servicer's losses to $455,000.

Another characteristic of a good hedge is its ability to "not hurt too much." For instance, the average cost of the options-based hedge is 9 bp, only 4 bp more than the average loss from unhedged positions. At the higher end of the distribution, the options-based hedge becomes even less expensive, and at the extreme, as it truncates losses dramatically, the hedged position becomes cheaper than going unhedged.

Actual performance of

options-based hedge

If all of this sounds academic or highly theoretical, let us consider what would have happened to our servicer between October 1984 and October 1989 - for many servicers a period as difficult as 1992. Let's assume that in October 1984 our servicer held the servicing rights on a $1 billion portfolio of Fannie Mae 12s, which, back then, were current-coupon mortgages.

Figure 4 shows the actual path of interest, mortgage and prepayment rates for the next five years. Starting at 12.2 percent, the 10-year T-note yields drop by 510 bp going to 7.1 percent in January 1987, then recover to 9.5 percent in October 1987 and recede to 8 percent by October 1989. The servicer's concern with the prepayment risk of the Fannie Mae 12s is quite understandable when we look at what mortgage rates do in such a volatile environment. Starting at 13.1 percent (which means that the servicing portfolio with a weighted average coupon of 12.8 percent is basically at-the-money), mortgage rates plummet down to 9 percent by April 1987 (which means that the portfolio is deep-in-the-money), recover to 10.8 percent by May 1989 and end up at 10.1 percent by October 1989.

The prepayment-rate path shown in Figure 4 illustrates the consequent deluge of prepayments. Starting out at a very modest 5 percent (80 PSA), prepayments climb rapidly to 61 percent (1020 PSA) by August 1986, a 13-fold increase. Notice that most of the time during this five-year period, prepayment rates oscillate above 20 percent (330 PSA) and close out at that level. For many of those months prepayment rates hover around 50 percent (820 PSA). After mortgage rates bottom out in April 1987, prepayment rates hit their second peak at 57 percent (950 PSA) and stay at similarly extreme levels for several more months as the portfolio burns out.

Unhedged servicing losses prove to be catastrophic at the actual five-year, compound prepayment rate of 27 percent (450 PSA), almost six times the initial prepayment rate. At this runoff rate, only 20 percent of the portfolio, or $200 million, remains outstanding by October 1989. Virtually all of this principal loss is due to prepayments because at the normal rate of amortization, 96 percent of the principal would still be outstanding after five years.

Reliving the 1986 refi boom

with a hedge

Let us suppose that in October 1984, the servicer puts in place a hedge strategy that will effectively guarantee that the servicing portfolio will prepay at 9 percent (150 PSA) during the next five years. At the targeted prepayment rate of 9 percent, the servicer would still have 60 percent, or three times the actual remaining balance outstanding at the end of five years. As a result, he would look to the hedge to generate sufficient benefits to cover the value of the income lost on 40 percent (60 percent to 20 percent) of principal that is prepaid.

Figure 5 translates the gap between actual and targeted prepayment rates into servicing losses that we would expect the hedge to truncate. Let us assume that the value of prepaid principal is 1.5 percent (150 bp). Notice that most of the losses come in a nine-month period starting in July 1986. In this period, servicing losses amount to $4.7 million, or roughly three-tenths of the value of the servicing portfolio. For the five-year period, the discounted value of the servicing losses amounts to $5.2 million, or more than one-third of the portfolio value.

Remember that the primary objective of hedging is to truncate the extreme prepayment-related losses. Let us assume that we help the servicer create two alternative hedging strategies with different powers of truncation. Strategy A is somewhat conservative and aims to truncate high losses by about 30 percent. This strategy would lead our servicer to buy as many as 130 options in a three-month period. Strategy B, by comparison, is more aggressive in hedging and aims to recover 80 percent or more of the losses through the benefits of the hedge. This strategy would lead the servicer to buy as many as 360 options in a three-month period.

Both strategies prove to be effective in averting disaster in servicing losses during this period. Table 7 shows the actual results of these two hedging strategies against the actual prepayment rates. The unhedged losses amount to $5.2 million, in net-present-value terms. Strategy A curtails the losses by $1.5 million, or 27 percent, and limits them to $3.8 million, or 25 percent of the portfolio value. Strategy B, by comparison, is more powerful in its truncation. It reduces the losses by $4.3 million (83 percent) and limits them to $0.9 million, or only 6 percent of the portfolio value.

Guesswork or homework?

This article has presented a reliable analytical method by which mortgage servicers and servicing investors can take the guesswork out of prepayment risk and substitute, in its place, informed risk management decision making. Such risk management is built upon a rich understanding of the prepayment risk inherent in all servicing portfolios, regardless of coupon, age or other characteristics. This discussion has also demonstrated - and history has periodically reminded the industry of this - that prepayments can ruin the profitability of an otherwise stellar business. The good news is that prepayment risk can be managed - if you do the homework.

Printer friendly Cite/link Email Feedback | |

Author: | Kasaba, Ekmel |
---|---|

Publication: | Mortgage Banking |

Date: | Jan 1, 1993 |

Words: | 4564 |

Previous Article: | The technology gap. |

Next Article: | Bad news for hedgers. |

Topics: |