May 21, 2025

A word on "cross-currency basis"

A few weeks back, ESTER/SOFR cross-currency basis briefly crossed zero before retreating back into negative territory. This event flashed in news, but details were omitted. I will try to fill-in the gaps, using the latest market data: 

Have you ever wondered ?

1) What does cross-currency basis mean? 
2) Why it exists? 
3) Why it has been traditionally negative?

When I did my CFA more than 10 years ago, this basis was not mentioned anywhere in the curriculum. Instead, all three levels were grinding covered interest rate parity (IRP). According to this parity, the basis shouldn't exist:
FxFwd / FxSpot = 1 + (r1-r2) × n/360
Later, quants gave me the following explanation: 
🗣️ "Cross-currency discount curve (such as EUR) is used to discount EUR leg of an xccy swap, collateralized using USD rate. We fit this curve in order to correctly reprice both legs of xccy swap". 
Next, traders on desk told me: 
🗣️ "The basis is caused by supply and demand" 
I found none of these explanations helpful because they don't carry any economic meaning, nor do they reconcile with what I learned about IRP earlier. 

So, here is the main reason why this basis exists and IRP is seemingly broken: 

Funding preferences on credit markets


The covered interest rate parity (IRP) condition holds in simplified, risk-free settings (e.g., SOFR vs. ESTER). However, in real-world markets, most cross-currency transactions occur on the credit side beyond the reach of any parity/arbitrage rules. 

 If you are German AAA-rated corporation seeking 5-year funding as of 20/5/2025, you have two options: 

1) Borrow in EUR at the risk-free rate of 2.14% plus a 46 bp credit spread (all-in 2.60%). 

2) Borrow in USD at the risk-free rate of 4.06% plus a 29 bp credit spread (all-in 4.35%) + hedge USD exposure using 5Y FxSwap 

Without credit spreads, the risk-free rate differential (4.06% - 2.14% = 1.92%) would be offset by FX fwd premium (remember IRP), leaving the issuer indifferent between markets.

However, the USD AAA-rated market has historically offered a lower credit premium (here, 17 bp less than EUR), making it more attractive for international borrowers. This demand dislocated interest rate parity, requiring EUR/USD cross-currency market participants to give-up 8 bp on their ESTER rate to execute the trade. 

The attached graph shows that cross-currency basis on derivatives market has been traditionally correlated with the differential of credit risk premium in respective countries. Yet because credit markets are incomplete, this relationship holds only in a weaker form - no strict arbitrage forces these rates to converge perfectly. 

But how about the arbitrage ? 


Executing the IRP arbitrage effectively requires significant leverage. 

XCY swaps are essentially the only instruments which can be used to effectively arbitrage the long-end of the yield curve. However here is the catch: they are all collateralised in USD regardless of which counterparty is posting collateral. 

In fact, the asymmetric demand for USD collateral is therefore another reason why cross-currency basis exists. 

That closes the circle.

Jun 24, 2020

Factor Investing and Fama-French model

This notebook illustrates factor investing and five-factor Fama-French model.


Risk Factor

Certain characteristic of economy (Inflation/GDP) or stock market itself (S&P 500)

Factor Model

Factor model uses movements in risk factors to explains portfolio returns

Questions which factor investing answers

  • Why different asset have systematically lower or higher average returns?
  • How to manage the asset portfolio with the underlying risks in mind?
  • How to benefit of our ability to bear specific types of risks to generate returns?

Fama-French Model


Assumes linear relationship between empirical factors and stock returns:

  • Market Factor (MER)
  • Size Factor (SMB)
  • Value Factor (HML)
  • Profitability Factor (RMW)
  • Investment Factor (CMA)

Factors are constructed daily from definitions, as illustrated previously

  • They are global for the entire stock market

Factor sensitivities are calibrated using regression

  • They represent “reward for taking a specific risk”, which is different for every stock
  • Risk/Reward relationship is expected to hold over time
  • Objective: maximize the model’s predictive power R2

Market Excess Return (MER)

  • Market excess return (over RF rate) alone explains around 80% of asset movements
  • Daily returns are ~normally distributed
  • Relationship between returns of the overall market and returns of selected portfolio

Size (SMB) factor

  • Small-cap companies typically bear additional risk premium - was it always the case?
  • Python can help you to see that this factor has a different prevalence in different economic regimes

Value (HML) factor

  • Value companies trade at higher yields to compensate for lack of growth potential
  • Python can help you to see that this factor has different explanatory power in different market situations and on different portfolios (very interesting)

Profitability and investment factors

  • Profitability factor (RMW) to attribute superior returns of companies with robust operating profit margins and strong competitive position among peers

  • Investment factor (CMA) to segment companies based on their capital expenditures

  • Analysts opinion: High capex structurally associated with growth companies, which puts usefulness of this factor in question


Evaluating 5-factor model

  • Analyst opinion: High correlations between risk factors puts usefulness of 5-factor model into question.
  • R2 10-20% for RMW, CMA
  • 5 factor improvement only by 0.2%