For a project, I am interested to model the impact of recently negative interest bonds on the portfolio. The literature on modelling negative interest rates is limited, and the only theory I could find was the "liquiduity trap" one and the plasma currency (software money having to take over regular cash). Here there are some points (they might be contradictory) about which I am asking for some actual formulae/algorithms, if it happens you to know any, or to have your opinion on which method below seems economically and finacially justified (or maybe all of them):

The value of higher presently outgoing cash flows is less important than the value of lower future incoming cash flows. This sounds to be the case for intertemporal rate marginal substitution (eg: pensions), used to model a margin and not the whole rate. But, from the computational point of view, it can be seen as marking to the future market rather than marking to the presently observed market prices, in order to do (spreads/model) calibration.

Since the real cash presence in the transactions is the main issue, in the present netting system one should consider additional netting: the one of all the electronic /credit cards/checks based (software) cash flows, with formulae to be therefore different for the software and non-software settlement. With this one more netting in place, from the portfolio's owner point of view, the non-software flows could be considered observed values. The negatives (outgoing principals) will be discounted at the effective funding rate (the by-principal-weighted average of all the incoming/positive non-software flows) and the positives (deposited) should be discounted at the average opportunity of investing rate (the weighted-by-principal return rate of all the non-software future flows). The treasury's rates (where they exist) might be taken as a proxy for effective funding and opportunity of investing rates when calculating present value of non-software cash flows.

The discounting curves are generally inferred from zero coupon bonds, especially for short rates, where such zero coupon bonds exist. Given the recent governmental negative interest rate bonds, this implies that the discounting curve has to be negative, at least for short rates. But the negative interest rates modelling with a Japanese style inverted yield curve might not be optimal, given that it is based on the historically known inverse curves. They were due to short rates becoming larger than the longs ones and short positive rates stopped to increase by the government stepping in and lowering them. Such a move cannot be done and also, the considered credit quality might not be the same at the terms considered when comparing short-term to long-term interest rates. The presently observed negative interest rates seem to increase in magnitude with the term. It might be a possibility for their modelling to follow the one for positive rates, without taking care of the sign, modelling just the magnitude, via imaginary rates, that become negative the moment they are observed with a properly defined metrics.

Because the observed increase in magnitude of negative interest rates can be justified by an increase in the perceived credit risk, the momentum profitability will drive the future investments, generating the observed market rate, via statistical average (of, by consequance, future high risk cash flows). For the high risk, momentum profitability decreases with size, so the non-software volume might be capped by the specific credit rating of the portfolio's owner (entity). Observed is that the amount which can be borrowed by an entity is limited (even at LIBOR rate, one can borrow in a "reasonablel market size", which is carefully monitored). Depositing now might be treated as collateral to borrowing in the future, therefore involved in calculations with an associated haircut. This might lead to a maximum depositing amount, with the magnitude of negative interest rates increment depending on the left size of the deposit-able amount. Portfolio-specific negative interest rates need to be modeled taking into account not only the time, but also the frequency and the magnitude of the non-software flows.

The magnitude of the negative interested rate is correlated with the increase in fixed assets prices and with cross-currency basis spreads. Their volatility / correlation coefficients could be historically fitted. If credit risk is to be considered completed integrated in the actually observed market prices, the change in the credit rate will trigger the change in the interest rate/market value, and no additional credit risk needs to be calculated, having an integrated market and credit risk. Two options might be considered here: the negative interest rates evolution continues to be smooth, with rating states neglected in practice and only reflected in continuous spreads, or has jumps, reflecting the discrete credit rating system.

An autoregressive model to get the future evolution a non-jumpy evolution of the interest rates seems a good opition , but not taking into account the possible future variations in credit rate states decreases its forecasting power. Using a split in the R actual rate, into a rate P which is always positive (floored at 0) and a second rate N, taking negative values, including zero does not make sense as well. One could use (see the approach of RiskMetrics after the negative interest in Switzerland appeared) an empirical approach: R=P+N to model a set of the instruments, gives the option to use either R or P for an other part of instruments and uses strictly P to model the remaining part of instruments; but this is inconsistent. Otherwise, one could use a two-factor model, based on the same spit of R into P and N; but this is based on the assumption that the market price for short interest rate risk is zero which is not true, taking into account that the negative interest rates in France / USA are not supported by a triple A rating, and that Germany might loose its triple A as well.

Hoping that you could give your opinion and come with an actual mathematical formula/algorithm on how to model the influence of future negative rates on risk at the portfolio level, or links to such a documentation (not mandatory describing the points above), and that you might give your opinion on which of the points above make sense to you (there shoule be no problem in modelling them myself) I do look forward to receiving your answer.