Given a fixed recovery rate and exposure amount, the unexpected loss on the loan is:
)
To make this number comparable with the VARnumber calculated under CreditMetrics for the normal distribution, we can see that the one standard deviation loss of value (VAR) on the loan is $2.99 million versus $2.07 mil- lion under the DM approach.29This difference occurs partly because the MTM approach allows an upside as well as a downside to the loan’s value, and the DM approach fixes the maximum upside value of the loan to its book or face value of $100 million. Thus, economic capital under the DM approach is more closely related to book value accounting concepts than to the market value accounting concepts used in the MTM approach.
SUMMARY
In this chapter, we outlined the VARapproach to calculating the capital re- quirement on a loan or a bond. We used one application of theVAR methodology—CreditMetrics—to illustrate the approach and raise the technical issues involved. Its key characteristics are: (1) it involves a full valuation or MTM approach in which both an upside and a downside to loan values are considered, and (2) the analyst can consider the actual dis- tribution of estimated future loan values in calculating a capital require- ment on a loan. We will revisit VARmethodology and CreditMetrics again in Chapter 11, when we consider calculating the VARand capital require- ments for a loan portfolio.
APPENDIX 6.1
CALCULATING THE FORWARD ZERO CURVE
current yield curve, denoted CYCRF, on risk-free (U.S. Treasury) coupon- bearing instruments, (2) Decompose CYCRFinto a zero yield curve, denoted ZYCRF, using a no arbitrage condition, (3) Solve for a one-year forward zero risk-free yield curve, FYCRF, and finally (4) Add fixed credit spreads obtained from historical loss experience in order to obtain the one-year for- ward zero risky debt yield curve, FYCR.30 Figure 6.6 illustrates data input into the CreditMetrics approach.
The Current Yield Curve on Risk-Free (U.S.
Treasury) Coupon-Bearing Instruments
From the current yield curve (CYCRF) for risk-free coupon bonds, shown in Figure 6.6, a zero-coupon yield curve for risk-free bonds (ZYCRF) can be derived using “no arbitrage” pricing relationships between coupon bonds and zero-coupon bonds, and solving by successive substitution.
FIGURE 6.6 The current yield curve on risk-free U.S. Treasury coupon- bearing instruments.
6.47%
Yield to Maturity p.a.
6 Months
1 Year
2 Years
3 Years
Maturity CYCRF
2.5 Years 1.5
Years 6.25%
6.09%
5.98%
5.511%
5.322%
Calculation of the Current Zero Risk-Free Curve Using No Arbitrage
U.S. Treasury notes and bonds carry semiannual coupon payments; there- fore all yields are halved to reflect semiannual rates.31We utilize the double subscript notation introduced in Chapter 5, with the exception that the semiannual, rather than annual periods are numbered consecutively [i.e., 0r1 is the spot (current) rate on the risk-free U.S. Treasury security maturing in 6 months, 0r2is the spot (current) rate on the risk-free U.S. Treasury secu- rity maturing in one year, 2r1is the one-year forward rate on a six-month U.S. Treasury security, and so on]. Thus:
)
Therefore, the six-month zero risk-free rate is: oz1=5.322 percent per annum:
Therefore, the one-year zero risk-free rate is: 0z2=5.5136 percent per annum. And so on to trace out the zero-coupon yield curve for risk-free U.S. Treasury securities—shown as ZYCRFin Figure 6.7. The next step is to trace out the risk-free forward yield curve, denoted FYCRF,usingZYCRF.
One-year zero: 100
1
= + + +
(
+)
= + +(
++)
=
+
+
+
+
=
+
+
+ C
r
C F r
C z
C F
1 1 1 1 z
100
5 511 2
1 05511
2
100 5 511 2 1 05511
2 5 11
2
1 05322
2
100
0 2 0 2
2
0 0 2
2
2
. .
.
. .
.
5 5 511
2 1 0 55136
2
2
.
.
+
Six-month zero: 100 =
1 1
C F r
C F z +
+ = +
+ =
+
+
1 1
100 5 322 2
1 05322
2
0 0
. .
Derivation of the One-Year Forward Government Yield Curve Using the Current Risk-Free Zero Yield Curve
We can use the expectations hypothesis to derive the risk-free ZYC ex- pected next year, or the risk-free one year forward zero yield curve, FYCRF shown in Figure 6.8. But first we derive a series of six-month forward rates using the rates on the ZYCRFcurve.32
Therefore, the rate for six-months forward delivery of six-month maturity U.S. Treasury securities is expected to be: 1z1=5.7054 percent p.a.
1 1 1
1 059961
2 1 055136
2 1
0 3 3
0 2 2
2 1
3 2
2 1
(
+)
= +( ) (
+)
+
= +
(
+)
z z z
. . z
1 1 1
1 055136
2 1 05322
2 1
0 2 2
0 1 1 1
2
1 1
(
+)
= +( ) (
+)
+
= +
+
( )
z z z
. . z
FIGURE 6.7 Zero coupon risk-free U.S. Treasury yield curve.
6.47%
Yield to Maturity p.a.
6 Months
1 Year
2 Years
3 Years
Maturity
ZYCRF CYCRF 2.5 Years 1.5
Years
6.25%
6.09%
5.98%
6.3127%
6.2747%
6.1075%
5.511%
5.9961%
5.5136%
5.322%
Therefore, the rate for one-year forward delivery of six-month maturity U.S.
Treasury securities is expected to be: 2z1=6.9645 percent p.a.
Therefore, the rate for six-month maturity U.S. Treasury securities to be de- livered in 1.5 years is: 3z1=6.4421 percent p.a.
Therefore, the rate for six-month rate maturity U.S. Treasury securities to be delivered in two-years is 4z1=6.9452 percent p.a.
1 1 1
1 062747
2 1 061075
2 1
0 5 5
0 4 4
4 1
5 4
4 1
(
+)
= +( ) (
+)
+
= +
(
+)
z z z
. . z
1 1 1
1 061075
2 1 059961
2 1
0 4 4
0 3 3
3 1
4 3
3 1
(
+)
= +( ) (
+)
+
= +
(
+)
z z z
. . z
FIGURE 6.8 Derivation of the one-year forward risky debt yield curve.
1 Year ForwardFYCR FYCRF
1 Year Forward ZYCRF 7.4645%
6.9645%
5.322% 5.5136%
5.9961%6.1075% 6.2747%6.3127%
6.703% 6.7837% 6.7135%
7.203% 7.2837% 7.2135%
Maturity 6
Months 1 Year
1.5 Years
2 Years
2.5 Years
3 Years Yield to
Maturity p.a.
Now we can use these forward rates on six-month maturity U.S. Trea- sury securities to obtain the one-year forward risk-free yield curve FYCRF shown in Figure 6.8 as follows:
(1+2z2)2=(1+2z1)(1+3z1)
Therefore, the rate for one-year maturity U.S. Treasury securities to be de- livered in one-year is: 2z2=6.703 percent p.a.
(1+2z3)3=(1+2z1)(1+3z1)(1+4z1)
Therefore, the rate for 18-month maturity U.S. Treasury securities to be de- livered in one-year is 2z3=6.7837 percent p.a.
(1+2z4)4=(1+2z1)(1+3z1)(1+4z1)(1+5z1)
Therefore, the rate for two-year maturity U.S. Treasury securities to be de- livered in one-year is: 2z4=6.7135 percent p.a.
Derivation of One-Year Forward Risky Yield Curve—F Y CR
CreditMetrics adds a fixed credit spread (si) to the risk-free forward zero yield curve in order to obtain the risky debt forward yield curve, FYCR, shown in Figure 6.8. Table 6.5 shows credit spreads provided by commer- cial firms such as Bridge Information Systems for different maturities.
TABLE 6.5 Credit Spreads for AAA Bonds Maturity (in Years,
Compounded Annually) Credit Spread, si
2 0.007071
3 0.008660
5 0.011180
10 0.015811
15 0.019365
20 0.022361
Source: Gupton et al., Technical Document, J.P. Morgan, April 2, 1997, p. 164, from Bridge Information Systems, February 15, 1997.
Typically, commercially-provided credit spreads are calculated using histor- ical averages. The one-year forward yield curve for risky debt in Figure 6.8 is illustrated assuming a fixed 50 basis point credit spread.
A Last Methodological Word
The methodology presented in this Appendix has been criticized for, among other reasons, its assumptions of deterministic interest rates (fixed yield curves) and constant credit spreads, si.The second criticism could be addressed by decomposing risky debt yield curves directly rather than de- composing the risk-free U.S. Treasury yield curve and then adding on a fixed credit spread. However, this approach injects noise into valuations if risky debt markets are illiquid and prices subject to error (see discussion in Chapter 5).