The European Investment Bank (EIB) is the European Union's investment bank and is owned by the EU Member States. It is one of the largest supranational lenders in the world. The EIB is a not-for-profit organisation which funds projects that achieve the policy aims of the European Union through loans, guarantees and technical assistance.The EIB focuses on the areas of climate, environment, small and medium sized enterprises (SMEs), development, cohesion and infrastructure. It has played a large role in providing finance during crises including the 2008 financial crash and the COVID-19 pandemic. Since its inception in 1958 the EIB has invested over one trillion euros. It primarily funds projects that "cannot be entirely financed by the various means available in the individual Member States".The EIB is one of the biggest financiers of green finance in the world. In 2007, the EIB became the first institution in the world to issue green bonds. In 2019 it committed to stop funding fossil fuel projects by the end of 2021. The EIB plans to invest 1 trillion euros in climate-related projects by 2030 including a just transition. The EIB is not funded through the budget of the EU. Instead, it raises money through the international capital markets by issuing bonds. The EIB is rated triple-A, the most credit-worthy rating on the bond market, by "the Big Three" credit rating agencies: Moody's, Standard and Poor's, and Fitch. Each member state pays capital into the EIB's reserves which is broadly in line with their share of EU gross domestic product.The EIB was founded by the Treaty of Rome, which came into force on 1 January 1958. It was the first of the world's regional development banks and is sometimes referred to as the largest multilateral development bank (MDB). The EIB was established to facilitate equitable development in the EU through lending to regions that are less developed and to support the EU's internal market. The EIB is active in 140 countries throughout the world. It makes around 10% of its investments outside the EU to support the European Union's development aid and cooperation policies. The EIB plans to invest 1 trillion euros in climate-related projects by 2030 including a just transition.
This paper proposes a methodology to estimate the aggregate financing needs of firms that are bankable yet discouraged from applying for a loan. Our data come mainly from the 2018-2020 Enterprise Survey and cover 35 emerging and developing economies. Drawing on the literature on corporate bankruptcy prediction, we develop a model with elastic net regularization to predict the outcome of loan applications. Our approach suggests that 38% of discouraged firms would have had their loan application approved, signaling inefficient self-rationing. Using this information, we estimate an aggregate credit gap of 5.4% of GDP, with significant variation across countries. Small and medium-sized enterprises account for more than two-thirds of the total, reflecting both their contribution to economic activity and the fact that they are more likely to be credit-constrained.
The origin of the red colouration in Jupiter's Great Red Spot (GRS) is a long-standing question in planetary science. While several candidate chromophores have been proposed, no clear conclusions have been reached regarding its nature, evolution, or relationship to atmospheric dynamics. In this work, we perform microphysical simulations of the reddish haze over the GRS and quantify the production rates and timescales required to sustain it. Matching the previously reported chromophore column mass and effective radius in the GRS requires column-integrated injection fluxes in the range 1 x 10(-12)-7 x 10(-12) kg m(-2) s(-1), under low upwelling velocities in the upper troposphere (v(trop) less than or similar to 1.5 x 10(-4) m s(-1)) and particle charges of at least 20 electrons/mu m. Such rates exceed the mass flux that standard photochemical models of Jupiter currently supply via NH3-C2H2 photochemistry at 0.1-0.2 bar, the most popular chromophore pathway in recent literature. We find a lower limit of 7 years on the haze formation time. We also assess commonly used size and vertical distribution parameterisations for the chromophore haze, finding that eddy diffusion prevents the long-term confinement of a thin layer and that the extinction is dominated by particles that can be represented by a single log-normal size distribution.
This paper derives the exact transition density and cumulative distribution function of a linear combination of two independent Cox–Ingersoll–Ross (CIR) processes. By combining the Poisson–Gamma mixture representation of the non-central chi–square law with the Kummer-type convolution of Gamma densities, it derives a closed-form analytical expression involving confluent hypergeometric functions. This result extends the classical single-factor CIR transition law to a multifactor framework, providing the first explicit analytical characterization of the sum of two independent CIR diffusions. The proposed density admits stable numerical evaluation and facilitates exact likelihood computation, enabling rigorous parameter estimation in multifactor affine term–structure, stochastic volatility, and credit risk models. Numerical experiments confirm that the analytical density and CDF closely match Monte Carlo simulations across various parameter regimes, demonstrating high accuracy and computational efficiency.
We present a regime-split Black–Scholes implied volatility solver in which every initial seed is a fully closed-form analytical expression, derived from the asymptotic structure of the Black–Scholes price in its natural domain. At the money, series reversion of an exact Gaussian identity yields a fourth-order seed with error 𝒪(s^8). In the moderate out-of-the-money region, successive Gaussian CDF approximations of increasing order produce explicit initial seed formulas whose accuracy is proved numerically, with no iteration or numerical inversion at the seed stage. In the deep out-of-the-money region, a Gaussian tail cancellation identity – the Mills ratio – reveals the asymptotic structure of the Black–Scholes price and motivates a ratio-corrected seed that achieves near-machine-precision initialisation for large moneyness. All regime boundaries are derived analytically from CDF truncation tolerances and numerical solver theoretical error bounds, with no empirically tuned constants. A universal fourth-order Householder polisher then drives all regimes to machine precision, with mean update iterations strictly below two on both standard and granular benchmark grids – meeting and surpassing the two-iteration target established by the highest-accuracy reference implementation in the literature (Jäckel, 2015). The resulting C implementation achieves a 1.73–1.85× throughput gain over the state-of-the-art benchmark (Jäckel, 2015) under identical hardware and compiler conditions, with maximum absolute error 𝒪(10^-14), stable across grid configurations. A Python/Numba implementation confirms portability. All source code is publicly available.
Young, innovative small and medium-sized enterprises (SMEs) often face financing constraints due to information asymmetries, limiting their access to traditional bank credit. Venture capital (VC) can alleviate these constraints, yet its supply remains insufficient in Europe, motivating public intervention. How publicly backed VC compares with fully private VC in terms of firm-level outcomes, however, remains an open empirical question. We address this issue by evaluating the European Investment Fund’s (EIF) intermediated investment model, through which public resources are channelled via privately managed VC funds. Unlike previous studies comparing VC-backed firms with firms that never received VC, we compare EIF-backed firms with a matched sample of firms financed by fully private VC investors. This research design allows us to isolate the incremental effect of the public component of the investment while substantially reducing selection bias associated with access to equity finance. Using a quasi-experimental framework combining exact matching, propensity score matching, and staggered Difference-in-Differences estimation, we find that firms receiving EIF-backed VC perform on par with comparable firms financed by fully private VC. Our findings suggest that publicly backed VC, when delivered through private financial intermediaries, can effectively address equity market failures without compromising firms’ post-investment growth and innovation performance.
