CREDIT SPREAD PREDICTION USING

Graham L. Giller Data Science Research, Equity Division

April 2017 What is Compressed Sensing?

Parameter estimation technology developed by Tao, Candès, Donoho.

When estimating coefficients of sparse linear systems, minimizing 퐿1 norms dominates traditional “” analysis in accuracy.

Only requires 표 푘푆 data points for perfect reconstruction of a noiseless 푆-sparse system of 푁 ≫ 푆 data points. 푘 ≈ 4, 12, ln 푁 … (depending on proof) Nyquist-Shannon sampling theorem says 2푁 ≫ 푘푆

Results are extendable to noisy systems.

Applications: MRI Images, Radar Images, Factor Models! Why does it Work? (This due to Tao, who is very much smarter than me!)

2 2 Minimize 푥1 + 푥2 Minimize 푥1 + 푥2

Such that 푎푥1 + 푏푥2 = 푐 (푎, 푏, 푐 are random) Such that 푎푥1 + 푏푥2 = 푐 푥 푥 And 1 is sparse (lies on axis). And 1 is sparse. 푥2 푥2

Probability of sparse solution: Probability of sparse solution: 4 ∞ i.e. almost never. 4 4 i.e. almost always. Interpolating Polynomials Joseph Louis Lagrange, 1795

푁 10 푖−1 푝푁 푥 = 휃푖푥 푖=1 8

6

4

2

0

푁 -2 For any set of 푁 pairs 푦푖, 푥푖 푖=1 I can create a polynomial of order 푁 − 1, 푝푁 푥 -4 such that 푦푖 ≡ 푝푁 푥푖 ∀ 푖 i.e. The polynomial reproduces the observed data with zero error! -6 0 1 2 3 4 5 6 7 Sparse Interpolating Polynomials

10

5

0

-5

-10 If 푦푖 = 푝푆 푥푖 ∀ 푖 and 휽 0 = 푆 ≪ 푁 we have a sparse interpolating polynomial. -15 From an estimation point-of-view, this is a linear system since the polynomial is linear in the 2 푁−1 푇 -20 parameters vector: 푝푁 푥 = 1 푥 푥 ⋯ 푥 휽

Thus this problem is amenable to compressed sensing. -25 0 1 2 3 4 5 6 7 Minimizers and

It is well known that the solution to It is a little less well known that the solution to 푁 푁 2 휇 = arg min 푥푖 − 휇 휇 = arg min 푥푖 − 휇 휇 휇 푖=1 푖=1

Is the arithmetic mean Is the 푁 휇 = median 푥푖 1 푖 휇 = 푥 푁 푖 푖=1 Minimizers and Weighted Medians

It’s a pretty obscure piece of information that The weighted median of a set of ranked pairs the solution to 푁 푁 푓 = arg min 푥푖 − 훽푖푓 푥푖, 푤푖 푖=1 where 푥푖 ≤ 푥푗 ∀ 푖 < 푗 푓 푖=1 + and 푤푖 ∈ ℝ ∀ 푖

is the “weighted median” 푥 is the arithmetic mean of the values 푥 and 푥 푖 + − 푓 = wgt med , 훽푖 푖 훽푖 which are chosen such that

푗 푥− = 푥푗 where 푗 = arg max 푖=1 푤푖 푗 1 such that 푗 푤 ≤ 푁 푤 푖=1 푗 2 푖=1 푗

and

푁 푥+ = 푥푗 where 푗 = arg min 푖=푗 푤푖 푗 1 such that 푁 푤 ≤ 푁 푤 푖=푗 푗 2 푖=1 푗 Structural Model for Credit Spreads

Assume piecewise linear evolution of latent company specific credit spread process.

Credit spread rates are constant for random intervals and have random values.

Actual bond trades are observed at random times and relative spreads (premia).

A given bond has a quasi-constant idiosyncratic premium. Method for Solution

Compressed Sensing?  This model is a piecewise linear factor model.  Within intervals it is a 1-sparse polynomial in time.  So compressed sensing should work…

1. Cross-Sectional Sessionization  Find a set of intervals such that at least 푚 = 푘푆 related bonds have traded at least 2 times within each interval.  Estimate of common spread rate for that interval as the weighted median of the individual bond spread rates.  The weights are the time intervals between the first trade and the last trade for each bond.  I used 푘 = 12 so 푚 = 12

퐶 푡+ − 퐶 푡− 푗 푖푗 푗 푖푗 + − 푔 푇푖 = wgt med푗 + − , 푡푖푗 − 푡푖푗 푡푖푗 − 푡푖푗

2. Factor Intercept Extraction

+ + 푆 푇푖 = median 퐶푗 푡푖푗 − 푔 푇푖 × 푡푖푗 − 푇푖 푗

3. Longitudenal Sessionization  Find a set of intervals for each bond such that the bond has traded at least 푘푆 = 12 times within the interval.  These are different intervals to those used in step 1.

훼 푗 = median 퐶푗 푡푗푘 − 푔 푇푖 × 푡푗푘 − 푇푖 − 푆 푇푖 푘

So let’s do it… Common Credit Spread Factor for 22 IBM Bonds Source data is yield spread to benchmark bond, with trade data from TRACE Spread Reconstructions for 3 Specific IBM Bonds Ex Ante Pricing Errors for 459200HG9 Process Dealer Buys, Dealer Sells and Dealer-Dealer trades separately

퐸푇푖+1퐶푗 푡 = 훼 푗 + 푆 푇푖 + 푔 푇푖 × 푡 − 푇푖 for 푡 > 푇푖+1 Pricing errors for the other IBM bonds? Overall: mean error = 0.333 bp, r.m.s. error = 15.2 bp, algorithm tuning done: none Need to take better account of deal size…