Treasury bondwith the specified life.
Differential Swaps
A differential swap, sometimes referred to as a diff swap, is an interest rate swap where the floatinginterest rate is observed in one currency and applied to a principal in another currency. Supposethat we observe the LIBOR rate for the period between tf; and t;,; in currency Y and apply it toaprincipal in currency X with the payment taking place at time 1;,,. Define F, as the forward interestrate between t; and t;,, in currency Y and G; as the forward exchange rate for a contract withmaturity t;,,; (expressed as the number of units of currency Y that equal one unit of currency X). Ifthe LIBOR rate in currency Y were applied to a principal in currency Y, we would value the cashflow on the assumption that the LIBOR rate equaled F;. From the analysis in Section 21.8, aquanto adjustment is necessary when it is applied to a principal in currency X. It is correct to valuethe cash flow on the assumption that the LIBOR rate equals
Fi + Fidiog iF pit (25.4)
where of; is the volatility of F;, og; is the volatility of G;, and p; is the correlation between F;and G;.
Example 25.4 Zero rates in both the United States and Britain are flat at 5% per annum withannual compounding. In a three-year diff swap agreement with annual payments, USD 12-monthLIBOR is received and sterling 12-month LIBOR is paid, with both being applied to a principal of£10 million. The volatility of all one-year forward rates in the United States is estimated to be 20%,the volatility of the forward USD-sterling exchange rate (dollars per pound) is 12% for allmaturities, and the correlation between the two is 0.4.
In this case, F; = 0.05, p; = 0.4, og; = 0.12, and og; = 0.2. The floating-rate cash flows dependenton the one-year USD rate observed at time 1; should therefore be calculated on the assumption thatthe rate will be
0.05 + 0.05 x 0.4 x 0.12 x 0.2 x t; = 0.05 + 0.00048t;
This means that the net cash flows from the swap at times one, two, and three years should beassumed to be 0, 4,800, and 9,600 pounds sterling for the purposes of valuation. The value of theswap is therefore
0 4,800 9,600
105 * Tos? * 1.05% = 12647
or £12,647.
25.6 EQUITY SWAPS
 
In an equity swap, one party promises to pay the return on an equity index on a notional principalwhile the other promises to pay a fixed or floating return on a notional principal. Equity swapsenable a fund managers to increase or reduce their exposure to an index without buying and selling
stock. An cauity swan is a convenient wav of nackacine a series of forward contracts on an indexStO°eK. AN Cquity swap is a convenient way or packaging @ Scrics Of torwara contracts On an maex
to meet the needs of the market.The equity index is usually a total return index where dividends are reinvested in the stockscomprising the index. An example of an equity swap is in Table 25.3. In this, the six-month return602 CHAPTER 25
Table 25.3 Extract from confirmation for an hypothetical equity swap
Trade date 4-January-2001
Effective date 11-January-2001
Business day convention (all dates) Modified following business dayHoliday calendar US.
Termination date 11-January-2006
 
Equity amounts
Equity payer MicrosoftEquity index Total return S&P 500 indexEquity payment 100(Z; — 1o)/4o, where 1) is the
index level on the payment dateand fp is the index level on theimmediately preceding paymentdate. In the case of the firstpayment date, fp is the indexlevel on 11-January-2001Equity payment dates Each 11-July and 11-Januarycommencing 11-July-2001 up toand including 1 1-January-2006
 
Floating amounts
Floating-rate payer Intel
Floating-rate notional principal USD 100 million
Floating rate USD 6-month LIBORFloating-rate day count convention Actual/360
Floating-rate payment dates Each 11-July and 11-January
commencing 11-July-2001 up toand including 11-January-2006
eee
on the S&P 500 is exchanged for LIBOR. ‘The principal on either side of the swap is $100 millionand payments are made every six months.
Consider an equity-for-floating swap such as that in Table 25.3. At the start of its life, it is worthzero. This is because a financial institution can arrange to costlessly replicate the cash flows to oneside by borrowing the principal on each payment date at LIBOR and investing it in the index untilthe next payment date with any dividends being reinvested. A similar argument shows that theswap is always worth zero immediately after a payment date.
Appendix 25A shows how to value equity-for-floating swaps between payment dates. An equity-for-fixed swap can be conveniently valued by regarding it as a combination of a swap where thereturn on an equity index is exchanged for floating combined with a plain vanilla interest rate swapwhere floating is exchanged for fixed.
25.6 SWAPS WITH EMBEDDED OPTIONS
 
 
Some swaps contain embedded options. In this section we consider some commonly encounteredexamples.Swaps Revisited 603
Accrual Swaps
Accrual swaps are swaps where the interest on one side accrues only when the floating referencerate is within a certain range. Sometimes the range remains fixed during the entire life of the swap;sometimes it is reset periodically.
As a simple example of an accrual swap, consider one where a fixed rate, Q, is exchanged forthree-month LIBOR every quarter. We suppose that the fixed rate accrues only on days whenthree-month LIBOR is below 8% per annum. Suppose that the principal is L. In a normal swapthe fixed-rate payer would pay QLn,/n on each payment date, where , is the number of days inthe preceding quarter and nz is the number of days in the year. (This assumes that the day count isactual/actual.) In an accrual swap this is changed to QLn3/n , whore vz is the number of days inthe preceding quarter that the three-month LIBOR was below 8%.° The fixed-rate payer savesQL/ny on each day when three-month LIBOR is above 8%. The fixed-rate payer’s position cantherefore be considered equivalent to a regular swap plus a series of binary options, one for eachday of the life of the swap. The binary options pay off QL/n, when the three-month LIBOR isabove 8%.
To generalize, we suppose that the LIBOR cutoff rate (8% in the case just considered) is Rx andthat payments are exchanged every 7 years. Consider day i during the life of the swap and supposethat t; is the time until day i. Suppose that the t-year LIBOR rate on day i is R; so that interestaccrues when R,; < Rx. Define F, as the forward value of R; and o; as the volatility of F;, (Thelatter is estimated from spot caplet volatilities.) Using the usual lognormal assumption, theprobability that LIBOR is greater than Rx in a world that is forward risk neutral with respect
to a Zcro-coupon bond maturing at time ijt is Nao), wicie
In(Fi/Rx) — oF t:/2CN
The wave Fans van Linge: woealizoed tha wee net data Fallacies ds i Wenit payor from the oinary option is realized at tne swap payment date following Gay i. We
suppose that this is at time s;. The probability that LIBOR is greater than Rx in a world that isforward risk neutral with respect to a zero-coupon bond maturing at time s; is given by N(d>),where d is calculated using the same formula as d,, but with a small timing adjustment to F;reflecting the difference between time t; +7 and time 5;.
The value of the binary option corresponding to day i is
a P(0. 5, N(43)
a=
The total value of the binary options is obtained by summing this expression for every day in thelife of the swap. The timing adjustment (causing d, to be replaced by dj) is so small that, inpractice, it is frequently ignored.
Cancelable Swap
A cancelable swap is a plain vanilla interest rate swap where one side has the option to terminateon one or more payment dates. Terminating a swap is the same as entering into the offsetting(opposite) swap. Consider a swap between Microsoft and Citibank. If Microsoft has the option to
3 The usual convention is that, if a day is a holiday, the applicable rate is assumed to be the rate on the immediatelypreceding business day.604 CHAPTER 25
 
cancel, it can regard the swap as a regular swap plus a long position in an option to enter into theoffsetting swap. If Citibank has the cancclation option, Microsoft has a regular swap plus a shortposition in an option to enter into the swap.
If there is only one termination date, a cancelable swap is the same as a regular swap plus aposition in a European swap option. Consider, for example, a ten-year swap where Microsoft willreceive 6% and pay LIBOR. Suppose that Microsoft has the option to terminate at the end of sixyears. The swap is a regular ten-year swap to receive 6% and pay LIBOR plus long position in asix-year European option to enter into a four-year swap where 6% is paid and LIBOR is received
{the latter is referred to as a 6 x 4 European option). The standard market model for valuing
European swap options is described in Chapter 22.
When the swap can be terminated on a number of different payment dates, it is a regular swapoption plus a Bermudan-style swap option. Consider, for example, the situation where Microsofthas entered into a five-year swap with semiannual payments where 6% is received and LIBOR ispaid. Suppose that the counterparty has the option to terminate on the swap on payment datesbetween year 2 and year 5. The swap is a regular swap plus a short position in a Bermudan-styleswap option, where the Bermudan-style swap option is an option to enter into a swap that maturesin five years and involves a fixed payment at 6% being received and a floating payment at LIBORbeing paid. The swap can be exercised on any payment date between year 2 and year 5. Wediscussed methods for valuing Bermudan swap options in Chapters 23 and 24.
Cancelable Compounding Swaps
Sometimes compounding swaps can be terminated on specified payment dates. On termination\thefloating-rate payer pays the compounded value of the floating amounts up to the timetermination and the fixed-rate payer pays the compounded value of the fixed payments up tothe time of termination.
Some tricks can be used to value cancelable compounding swaps. Suppose first that the floatingrate is LIBOR and it is compounded at LIBOR. We assume that the principal amount of the swapis paid on both the fixed and floating sides of the swap at the end of its life. This is similar tomoving from Table 6.1 to Table 6.2 for a vanilla swap. It does not change the value of the swapand has the effect of ensuring that the value of the floating side is always equals the notionalprincipal on a payment date. To make the cancelation decision, we need look only at the fixed side.
Wo constr torost rate tree as Vinod in Chanter 22. We roll back throu the tree ceWe construct an interest rate tice as outlined in Chapter 23. We roll back through the treo in the
usual way, valuing the fixed side. At each node where the swap can be canceled, we test whether itis optimal to keep the swap or cancel it. Canceling the swap in effect sets the value of the fixed sideequal to par. If we are paying fixed and receiving floating, our objective is to minimize the value ofthe fixed side; if we are receiving fixed and paying floating, our objective is to maximize the value ofthe fixed side.
When the floating side is LIBOR plus a spread compounded at LIBOR, we can subtract cash
flows corresponding to the spread t rate of interest from the fixed side instead of adding them to the
floating side. The option can then be valued as in the case where there is no spread.When the compounding is at LIBOR plus a spread, an approximate approach is as follows:*
1. Calculate the value of the floating side of the swap at each cancelation date assumingforward rates are realized.
+ This approach is not perfectly accurate in that it assumes that the decision to exercise the cancelation option is notinfluenced by future payments being compounded at a rate different from LIBOR.Swaps Revisited 605
2. Calculate the value of the floating side of the swap at each cancelation date assuming that thefloating rate is LIBOR and it is compounded at LIBOR.
3. Define the excess of (1) over (2) as the “value of spreads” on a cancelation date.
4. Treat the option in the way described above. In deciding whether to exercise thecancelation option, subtract the value of the spreads from the values calculated for thefixed side.
25.7 OTHER SWAPS
 
 
This chapter has discussed only a few of the different types of swaps that trade. In practice thenumber of instruments that trade is limited only by the imagination of financial cngincers and theappetite of corporate treasurers for innovative risk management tools.
A swap that was very popular in the United States in the mid-1990s is an index amortizing rateswap (sometimes also called an indexed principal swap). In this, the principal reduces in a waydependent on the level of interest rates. The lower the interest rate, the greater the reduction in theprincipal. The fixed side of an indexed amortizing swap was originally designed to mirror, at leastapproximately, the return obtained by an investor on a mortgage-backed security after prepaymentoptions are taken into account. The swap therefore exchanged the return on a mortgage-backedsecurity for a floating-rate return.
Commodity swaps are now becoming increasingly popular. A company that consumes 100,000bakrels of oil per year could agree to pay $2 million each year for the next ten years and to receivein return 100,000S, where S is the market price of oil per barrel. The agreement would in effect lockin the company’s oil cost at $20 per barrel. An oil producer might agree to the opposite exchange,thereby locking in the price it realized for its oil at $20 per barrel. We will discuss energyderivatives in Chapter 29.
A recent innovation in swap markets is a volatility swap. In this, the payments depend on thevolatility of a stock (or other asset). Suppose that the principal is L. On each payment date, oneside pays Lo, where o is the historical volatility calculated in the usual way by taking dailyobservations on the stock during the immediately preceding accrual period and the other sidepays LK, where K is a constant prespecified volatility level. Variance swaps, correlation swaps, andcovariance swaps are defined similarly.
A number of other types of swaps will be discussed later in this book. Asset swaps are covered inChapter 26, and credit default swaps and total return swaps in Chapter 27.
25.8 BIZARRE DEALS
 
 
Some swaps involve the exchange of payments being calculated in quite bizarre ways. An exampleis provided by the so-called “5/30” swap entered into between Bankers Trust (BT) and Procter andGamble (P&G) on November 2, 1993.° This was a five-year swap with semiannual exchanges ofpayment. The notional principal was $200 million. BT paid P&G 5.30% per annum. P&G paid BTthe average 30-day commercial paper rate minus 75 basis points plus a spread. The average
* See D. J. Smith, “Aggressive Corporate Finance: A Close Look at (he Procter and Gamble-Bankers TrustLeveraged Swap,” Journal of Derivatives, 4, no. 4 (Summer 1997), 67-79.606 CHAPTER 25
commercial paper rate was calculated by observing the 30-day commercial paper rate each dayduring the preceding accrual period and averaging them.
The spread is zero for the settlement at the end of the first payment date (May 2, 1994). For theremaining nine payment dates, it is
 
98.5 x [(S-year CMT %)/5.78%] — (30-year TSY price)max|{ 0,( i )In this, 5-year CMT is the constant maturity Treasury yield (ie., the yield on a 5-year Treasurynote, as reported by the Federal Reserve). The 30-year TSY price is the midpoint of the bid andoffer cash bond prices for the 6.25% Treasury bond maturing on August 2023.
P&G were hoping that the spread would be zero and the deal would enable them to exchangefixed-rate funding at 5.30% for funding at 75 basis points less than the CP rate. As it happened,interest rates rose sharply in early 1994, bond prices fell, and the swap proved very, very expensive.We will discuss this example further in Chapter 30 (see also Problem 25.10).
SUMMARY
 
 
Swaps have proved to be very versatile financial instruments. Many swaps can be valued using thesimple assumption that LIBOR (or some other floating reference rate) will equal its forward value.These include plain vanilla interest swaps, most types of currency swaps, swaps where the ronchanges in a predetermined way, swaps where the payment dates are different on each side, andcompounding swaps.
Some swaps require adjustments to the forward rates when they are valued. These adjustmentsare termed convexity, timing, or quanto adjustments. Among the swaps that require adjustmentsare LIBOR-in-arrears swaps, CMS/CMT swaps, and differential swaps.
Equity swaps involve the return on an equity index being exchanged for a fixed or floating rate ofinterest. They are usually designed so that they are worth zero immediately after a payment date.However, some care must be exercised in valuing them between payment dates.
Some swaps involve embedded options. An accrual swap turns out to be a regular swap plus alarge portfolio of binary options (one for each day of the life of the swap). A cancelable swap turns
out to be a regular swap plus a Bermudan swap option.
SUGGESTIONS FOR FURTHER READING
 
 
Chance, D., and D. Rich, “The Pricing of Equity Swap and Swaptions,” Journal of Derivatives, 5, no. 4(Summer 1998), 19-31.
Demeterfi, K., E. Derman, M. Kamal, and J. Zou, ‘A Guide to Volatility and Variance Swaps,”Journal of Derivatives, 6, no. 4 (Summer 1999), 9-32.
Smith, D. J., “Aggressive Corporate Finance: A Close Look at the Procter and Gamble-Bankers TrustLeveraged Swap,” Journal of Derivatives, 4, no. 4 (Summer 1997), 67-79.Swaps Revisited 607
QUESTIONS AND PROBLEMS
 
25.1.
25.2.
25.4.
25.8.
(ANSWERS IN SOLUTIONS MANUAL)
Calculate all the fixed cash flows and their exact timing for the swap in Table 25.1. Assume thatthe day count conventions are applicd using target payment datcs rather than actual paymentdates.
Suppose that a swap specifies that a fixed rate is exchanged for twice the LIBOR rate. Can theswap be valued using the ‘“‘assume forward rates are realized” rule?
. What is the value of a two-year fixed-for-floating compound swap where the principal is
$100 million and payments are made semiannually. Fixed interest is received and floating ispaid. The fixed rate is 8% and it is compounded at 8.3% (both semiannually compounded). Thefloating rate is LIBOR plus 10 basis points and it is compounded at LIBOR plus 20 basis points.The LIBOR zero curve is flat at 8% with semiannual compounding.
What is the value of a five-year swap where LIBOR is paid in the usual way and in returnLIBOR compounded at LIBOR is received on the other side. The principal on both sides is$100 million. Payment dates on the pay side and compounding dates on the receive side are everysix months and the yield curve is flat at 5% with semiannual compounding.
.5. Explain carefully why a bank might choose to discount cash flows on a currency swap at a rate
slightly different from LIBOR.
. Calculate the total convexity/timing adjustment in Example 25.3 of Section 25.4 if all cap
volatilities are 18% instead of 20% and volatilities for all options on five-year swaps are 13%instead of 15%. What should the five-year swap rate in three years’ time be assumed for thepurpose of valuing the swap? What is the value of the swap?
. Explain why a plain vanilla interest rate swap and the compounding swap in Section 25.2 can be
valued using the “assume forward rates are realized” rule, but a LIBOR-in-arrears swap inSection 25.4 cannot.
Jn the accrual swap discussed in the text, the fixed side accrues only when the floating referencerate lies below a certain level. Discuss how the analysis can be extended to cope with a situationwhere the fixed side accrues only when the floating reference rate is above one level and belowanother.
ASSIGNMENT QUESTIONS
 
25.9.
25.10.
25.11.
LIBOR zero rates are flat at 5% in the U.S and flat at 10% in Australia (both annuallycompounded). In a four-year swap, Australian LIBOR is received and 9% is paid with bothbeing applied to a USD principal of $10 million. Payments are exchanged annually. Thevolatility of all one-year forward rates in Australia is estimated to be 25%, the volatility of theforward USD-AUD exchange rate (AUD per USD) is 15% for all maturities, and thecorrelation between the two is 0.3. What is the value of the swap?
Estimate the interest rate paid by P&G on the 5/30 swap in Section 25.8 if (a) the CP rate is 6.5%and the Treasury yield curve is flat at 6% and (b) the CP rate is 7.5% and the Treasury yieldcurve is flat at 7% with semiannual compounding.
Suppose that you are trading a LIBOR-in-arrears swap with an unsophisticated counterpartywho does not make convexity adjustments. To take advantage of the situation, should you be608
CHAPTER 25
 
25.12.
paying fixed or receiving fixed? How should you try to structure the swap as far as its life andpayment frequencies.
Consider the situation where the yield curve is flat at 10% per annum with annualcompounding. All cap volatilities are 18%. Estimate the difference between the way asophisticated trader and an unsophisticated trader would value a LIBOR-in-arrears swap wherepayments are made annually and the life of the swap is (a) 5 years, (b) 10 years, and (c) 20 years.Assume a notional principai of $i million.
Suppose that the LIBOR zero rate is flat at 5% with annual compounding. In a five-year swap,company X pays a fixed rate of 6% once a year and receives LIBOR. The volatility of the two-year swap rate in three years is 20%. The principal is $100.
a. What is the value of the swap?
b. Use DerivaGem to calculate the value of the swap if company X has the option to cancel
after three years.
ce rive Gam caloulUse DerivaGem to calcul:
°
cancel after three years.d. What is the value of the swap if either side can cancel at the end of three years?Swaps Revisited 609
APPENDIX 25A
Valuation of an Equity Swap between Payment Dates
To value an equity swap between two payment dates, we define:
Ro: Floating rate applicable to the next payment date (determined at the last payment date)L: Principalt: Time between last payment date and next payment datet: Time between now and next payment date£y: Value of the equity index at the last reset dateE+ Current value of the equity indexR: LIBOR rate for the period between now and the next payment dateIf we borrowE—LEoat rate R for time rt and invest it in the index, we create an exchange ofE; E—L for —L(l+R 25A.1E BMI + Re) Q25A.1)
at the next payment date. Since this exchange can be created costlessly, it is worth zero. Theexchange that will actually take place at the next payment date is
EG- )t for RoLt
Adding the principal L to both sides, we see the actual exchange is equivalent toety for L(t + Rot) (25A.2)0
Comparing equation (25A.1) with equation (25A.2), see that value of the swap to the partyreceiving floating is the present value of
EL(t + Rot) — L—(1 + Rr)EyThis is1+ Rot _,£
L1+Rt EoSimilarly, the value of the swap to the party receiving the equity return is
E_ i+ RoteEy i+RrCHAPTER 26CREDIT RISK
 
 
Potential defaults by borrowers, counterparties in derivatives transactions, and bond issuers giverise to significant credit risk for banks and other financial institutions. As a result, most financialinstitutions devote considerable resources to the measurement and management of credit risk.Regulators require banks to keep capital to reflect the credit risks they are bearing.!
In this chapter we focus on the quantification of credit risk. We first discuss a number ofdifferent approaches to estimating the probability that a company will default. We explain the keydifference between risk-neutral and real-world probabilities of default. We then move on to discusshow a bank or other financial institution can estimate its loss given that a default by a compaoccurs. The probability of default and the loss given default jointly determine the financialinstitution’s expected loss. The last part of the chapter covers credit ratings migration and defaultcorrelation, and shows how the value-at-risk measure introduced in Chapter 16 can be used forcredit risk as well as market risk.
In Chapter 27 we will use the material in this chapter to analyze credit derivatives. These arederivatives where the payoff is dependent in some way on the creditworthiness of one or morecompanies or sovereign entities.
26.1 BOND PRICES AND THE PROBABILITY OF DEFAULT
 
 
 
Rating agencies such as Moody’s and S&P are in the business of providing ratings describing thecreditworthiness of corporate bonds. Using the Moody’s system, the best rating is Aaa. Bonds withthis rating are considered to have almost no chance of defaulting. The next best rating is Aa.Following that come A, Baa, Ba, B, and Caa. Only bonds with ratings of Baa or above areconsidered to be investment grade. The S&P ratings corresponding to Moody’s Aaa, Aa, A, Baa,Ba, B, and Caa are AAA, AA, A, BBB, BB, B, and CCC, respectively. To create finer ratingmeasures, Moody’s divides the Aa rating category into Aal, Aa2, and Aa3; it divides A into Al,A2 and A3; and so on. Similarly S&P divides its AA rating category into AA+, AA, and AA-; itdivides its A rating category into A+, A, and A—; and so on. (Only the Aaa category for Moody’sand the AAA category for S&P are not subdivided.)
Bond traders have developed procedures for taking credit risk into account when pricingcorporate bonds. They collect market data on actively traded bonds to calculate a genericzero-coupon yield curve for each credit rating category. (See Chapter 5 for a description ofhow zero-coupon yields can be calculated from coupon-bearing bond yields using the bootstrap
' This capital is in addition to the capital, mentioned in Chapter 16, that banks are required to keep for market risk.
610Credit Risk 611
 
A Spread of corporate Baa/BBBbond yield overTreasury yield .AIAAalAAAaa/AAAMaturity (years)
 
 
Figure 26.1 Spread over Treasury zero curve for corporate bond yields
method.) These zero-coupon yield curves are then used to value other bonds. For example, anewly issue A-rated bond would be priced using the zero-coupon yield curve calculated fromother A-rated bonds.”
Figure 26.1 shows a typical pattern for the spread over the Treasury zero curve for the yields oninvestment grade zero-coupon bonds with different maturities and different credit ratings. Thespread increases as the rating declines. It also increases with maturity. Note that the spread tends toincrease faster with maturity for low credit ratings than for high credit ratings. For example, thedifference between the five-year spread and the one-year spread for a BBB-rated bond is greaterthan that fora AAA-rated bond.?
Expected Default Losses on Bonds
The first step in estimating default probabilities from bond prices is to calculate the expecteddefault losses on corporate bonds of different maturities. This involves comparing the price of acorporate bond with the price of a risk-free bond that has the same maturity and pays the samecoupon. The usual assumption is that the present value of the cost of defaults equals the excess ofthe price of the risk-free bond over the price of the corporate bond. This means that the higheryield on a corporate bond is entirely compensation for possible losses from default.*
The price of the risk-free bond is calculated from the risk-free zero curve. A natural choice forthe risk-free zero curve is the Treasury curve and many analysts do compare corporate bond yields
? As we discuss later, there are some theoretical problems with this procedure because it relies on value additivity,
3 This is consistent with the point made during the discussion of the comparative advantage argument for swaps inSection 6.2: that the rollover risk for a BBB issuer is greater than the rollover risk for a AAA issuer. SeeProblem 26.2.
4 This is an approximation. Other factors such as liquidity influence the spread between corporate and risk-freebond yields.612 CHAPTER 26
 
Table 26.1 Yields and expected default losses on bonds issued by acorporation. (All rates continuously compounded.)
Maturity Risk-free Corporate bond Expected default loss(years) zero rate (%) zero rate (%) (% of no-default value)
 
1 5.00 5.25 0.24972 5.00 5.50 0.99503 5.00 5.70 2.078