1. Background:
In this short report we describe how Everysk propagates a macro shock to corporate bonds in a multiasset portfolio.
It is well known that, all else equal, bonds issued by better rated companies have more interest rate than credit spread sensitivity. The worst the issuer’s rating, the more relevant the influence of credit spreads is. It is also well known that for junk bonds, the rate and credit risk factors provide some amount of selfhedge as they are generally negatively correlated.
Despite their sophistication, sellside portfolio systems are not designed to propagate risks within the portfolio (for example to infer the negative correlation between rate and credit mentioned above). They provide a framework for user’s to manually change these variables and retrieve a localized profit and loss (PL) impact from first/second derivatives of the pricing formula (dv01, spread duration, convexity, etc.). Throw an exogenous macro shock, such as oil or equity index for example, and these systems are at a loss as oil and equity are not part of the pricing formulas.
When presented with exogenous macro shocks, less experienced practitioners have regressed bond prices to the macro index being shocked. Despite its simplicity, this approach will produce unreliable results, as the bond prices are not an invariant property (bonds pull to par as they approach maturity).
In what follows we describe how Everysk calculation engine reliably propagates the shocks within multiasset portfolios and addresses the issues above.
In order to illustrate the concepts we will use a portfolio with 2 bonds with different credit ratings, namely:
 A bond issued by Tesla maturing on 08/15/25 with a 4.7% coupon
 A bond issued by 3M maturing on 08/17/25 with a 3% coupon
The properties for these bonds are:
Midprices 
YTM 
Duration(yrs) 
Spread over benchmark 

Tesla 
$80.02 
9.04% 
5.49 
6.19% 
3M 
$97.15 
3.69% 
6.17 
0.83% 
2. Methodology:
Everysk simulates thousands of potential bond prices in the near future to calculate detailed risk/return characteristics. In order to do that, it first builds a list of risk factors per bond, as follows:
2.1. Rate risk factors:
We calculate the duration of each bond to find a blended interest rate exposure, using constant maturity treasuries ranging from 1 month to 30 years:
Duration (yrs) 
5 year CMT 
10 year CMT 

Tesla 
5.49 
80.6% 
19.4% 
3M 
6.17 
76.6% 
23.4% 
Thus, given their durations, both bonds will interpolate between the 5 and 10 year key rates with proportions above.
2.2. Credit spread risk factor:
For credit spreads the calculations are a bit more complex. First we calculate a blended credit rating for each bond using a 20x20 table of default rates published by S&P (the average cumulative issuerweighted global default rates by letter rating from 1983 to 2015). The columns of this table are letter ratings ranging from AAA to CCC and rows are years ranging from 1 to 20. The elements of the table are cumulative probabilities of default.
Thus, in order to use the table we need the maturity of the bonds to interpolate rows and the probability of default of the issuer to interpolate columns and retrieve the blended rating. We use a pointwise structural model with the following inputs to calculate the probability of default:
Debt ($M) 
Market Cap($M) 
Maturity (yrs) 
Stock vol 
Risk free rate 

Tesla 
12,570.34 
50,885.88 
7.025 
43.17% 
2.36% 
3M 
14,519.00 
124,181.22 
7.027 
21.96% 
2.36% 
With the above inputs we can compute firm value and firm volatility as well as probability of default. Details of the model are beyond the scope of this report and can be found in our white paper, chapter 7:
Firm Value ($M) 
Firm volatility 
Probability of Default 

Tesla 
61,429 
36.27% 
9.01% 
3M 
132,884.96 
19.92% 
~0% 
With both maturities in years and probabilities of default above, we can find a blend of ratings for each bond. The interpolated row in the matrix using 7.02 years to maturity is:
AAA 
AA 
BBB 
BB 
B 
CCC 

7 years 
0.139 
0.546 
2.446 
8.346 
28.228 
47.183 
7.02 
0.139 
0.548 
2.456 
8.370 
28.288 
47.277 
8 years 
0.143 
0.652 
2.852 
9.508 
30.648 
50.950 
Thus, by interpolating Tesla’s probability of default of 9.01%, its credit spread risk can be represented by a blend of 96.8% BB and 3.2% B:
BB 
Blended Rating 
B 

Default Probabilities for 7.02 years 
8.370 
9.01 
28.288 
3M's probability of default for 7.02 years (~0%) is lower than the corresponding AAA (0.139%), therefore the “blend” is 100% AAA credit.
Once we find these blended ratings, we use corresponding BofA Merrill Lynch US corporate optionadjusted spread indices to model the spread behavior. Thus, a portfolio with both bonds will require a covariance matrix with the following risk factors: 5 YR CMT, 10 YR CMT, BAML AAA Index (for 3M), BAML BB Index (for Tesla) , BAML B Index (for Tesla) , 3M LIBOR. This covariance matrix would need to be augmented with another risk factor used for stress testing, if applicable. For the examples below we are shocking the BAML BBB spread Index.
Simulations of risk factors are generated from this covariance matrix. In our illustrative example the simulation matrix (called “sims” below) has 7 columns (6 underlying risk factors and 1 shock factor) and 50,000 rows (correlated realizations).
Having found the proper blends for rate and spread risk, and the simulations above, Everysk can reprice each bond thousands of times, using the following yield simulations:
 Tesla Yield Simulation = 9.04% + (80.6% x sims[‘CMT5YR’]+19.4% x sims[‘CMT10YR’]) + (96.8% x sims[‘BAMLBB’] + 3.2% x sims[‘BAMLB’])
 3M Yield Simulation = 3.69% + (76.6% x sims[‘CMT5YR’]+23.4% x sims[‘CMT10YR’]) + 100% x sims[‘BAMLAAA’]
Where sims[‘CMT5YR’] is a columnvector with 50,000 entries and simulations are normal, i.e. additive to yield. Finally, these yield simulations are used in a vectorized bond pricer to come up with thousands of potential PL scenarios, providing a granular picture of risk.
3. Results
The graphs below show the PL for each bond subjected to various rates and credit spread shocks.
3.1. Interest Rate shocks:
Starting with the 3M bond and shocks in the 5 year Constant Maturity Treasury ranging from 1% to 1%:
The horizontal axis represents shocks in the 5year Constant Maturity Treasury from 1% to 1%. The vertical axis shows the expected gain/loss in the bond. The black line represents the expectation and red lines represent the 5percentile best/worst outcomes for each shock. Stress testing this bond via its pricing formula would generate a single PL outcome rather than a distribution.
Some results from the plot above are shown in table format:
CMT 5yr : 1% 
CMT 5yr: +1% 

CVaR + (best 5percentile) 
+14.5% 
+0.85% 
Expected PL 
+5.98% 
5.44% 
CVaR  (worst 5percentile) 
0.93% 
11.45% 
We can see that Everysk provides a more realistic expected PL than just using a pricing formula. If we use the duration of the bond, i.e. 6.17 years, we would expect the positive PL for a rate drop of 1% to be 6.17%. The reason our expected PL is lower (+5.98%) is because our calculation engine takes into account that when rates are falling, credit spreads tend to widen (the weekly correlation between AAA and 5 year CMT is 0.3). Therefore rather than computing a localized number, we propagate the shock. The influence of credit spread is limited in the MMM bond, but we will see below that it is important for the TSLA bond:
CMT 5yr : 1% 
CMT 5yr: +1% 

CVaR + (best 5percentile) 
+12.62% 
+6.33% 
Expected PL 
+3.15% 
2.78% 
CVaR  (worst 5percentile) 
5.88% 
11.13% 
Here, the negative correlation between credit spread and rates has a larger impact. The expected PL of +3.15% when rates are down 1% is much smaller than the localized number obtained with duration, which is +5.19%. When treasuries are appreciating (rates down), credit spreads tend to be widening, partially countering the effect of rates.
As a sanity check, if we turned off the credit spread simulations and took into account only rate simulations, we would obtain the following results:
CMT 5yr : 1% 
CMT 5yr: +1% 

CVaR + (best 5percentile) 
+12.56% 
+0.51% 
Expected PL 
+5.28% 
5.22% 
CVaR  (worst 5percentile) 
0.64% 
10.34% 
Now, the expected PLs are closer to duration measures and the first table (with inclusion of credit spread effects) shows a larger negative(positive) skew when rates are down(up). For example: a 1% rate shock results in a worst 5percentile PL of 5.88% when spread risk is included versus 0.64% when it is not
It is impossible to retrieve this level of granularity with a localized stress test based solely on pricing formulas.
3.2 Credit spread shocks
In what follows we shock a BBB spread index and plot the impact on the 2 bonds. First the 3M:
The 2 extreme shocks from the graph above are:
BBB spread : 1% 
BBB spread: +1% 

CVaR + (best 5percentile) 
+14.33% 
+13.54% 
Expected PL 
+0.16% 
+0.32% 
CVaR  (worst 5percentile) 
11.68% 
11.65% 
The same BBB spread shocks show a more pronounced impact for the Tesla bond :
BBB spread : 1% 
BBB spread: +1% 

CVaR + (best 5percentile) 
+29.02% 
+10.51% 
Expected PL 
+8.95% 
7.62% 
CVaR  (worst 5percentile) 
8.30% 
21.56% 
If we were to turn off the interest rate effect, which provides some amount of selfhedge, the PL impact above would be even more pronounced:
BBB spread : 1% 
BBB spread: +1% 

CVaR + (best 5percentile) 
+34.50% 
+7.08% 
Expected PL 
+12.49% 
9.97% 
CVaR  (worst 5percentile) 
5.25% 
24.61% 
3.3 Oil shocks
Here we propagate a macro shock that is exogenous to the risk factors used to price the bonds. We use shocks on the 6 month WTI crude future, ranging from 10% to +10%. First the impact on the Tesla bond:
Internally, our calculation engine propagates an oil shock as follows:
Oil  SP500  VIX  BB spread  5yr Treasury 
10.0%  1.3%  19.8%  +12bps  7.5 bps 
An oil shock is a flight to quality shock and the credit spread exposure on Tesla's bond will dominate the behavior and result on a loss of 44bps when oil drops 10%. Conversely, 3M's bond is more sensitive to rates and a drop in oil will result in a positive PL of 48 bps:
Putting both bonds together, we pretty much immunize the expected PL to oil shocks.
4. Summary:
We demonstrated in this short report how Everysk can reliably propagate macro shocks within a multiasset portfolio containing corporate bonds.
We shocked rates and credit spreads and provided an intuition on how our model differs from a localized approach that stress tests bonds via their pricing formulas. We contend that our methodology is more relevant to buy side investors interested on big picture effects.
Additionally our methodology generalizes to shocks on any exogenous index, even when the shock does not directly affect the bond via its pricing formula. We used oil futures as an example above.
In a follow up report, we will extend the concepts above to bonds with embedded options.