Modelling with negative interest rates

Question

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 "liquidity trap" one and the plasma currency (software money taking place of the regular cash). Here there are some points (which might contradict themselves) about which I am asking for some actual formulae/algorithms, if you happen to know any:

The value of higher presently outgoing cash flows is less important than the value of future incoming cash flows. This sounds to be the case for intertemporal rate marginal substitution (eg: pensions), a model to model a margin and not the whole rate. From computational point of view, it can be seen as marking to the future market rather than marking to the presently observed market prices; with calculations to be done not in the base currency but into a different unit, not even necessarily a currency, and fit the number of such units at future times.

The discounting curves are generally inferred from zero coupon bonds, especially for short rates, where such zero coupon bonds exist. Given the recent governmental interest rate bonds, this implies that the discounting curve has to be negative, at least for short rates. 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. But the negative interest rates modelling with a inversed(/mirrored in the region of negativity) Japanese style yield curve might not be optimal, given that it is based on the historically known inverse curves. The historical Japanese curves 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 (lower the magnitude of negative rates) is difficult to be done, due to reserves constrains. In addition, the credit quality of the issuer might not be constant at the terms considered when comparing short-term to long-term interest rates.

Because the increase in magnitude of negative interest rate can be justified by an increase in the perceived credit risk, the momentum profitability will drive the future investments, leading at the observed market rate. For the high risk, momentum profitability decreases with size, so the A1 available volume might be capped by the specific credit rating of the portfolio's owner (entity).Moreover, the amount which can be borrowed by an entity is limited. 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.

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.

Hoping that you could give your opinion and come with an actual mathematical documentation or ideas on how to model the influence of future negative rates on risk at the portfolio level, I am very much looking forward for your answer.

Answer 1

There is the white paper "New volatility conventions in negative interest environment - Current developments and necessary adjustments of IT systems in trading, risk management and accounting" by M. Beinker and H. Planck (2013), which discusses recent developments and also gives an introduction to the displaced diffusion model, which can handle negative interest rates.

For more detail on the displace diffusion model, see "Volatility and Correlation: The Perfect Hedger and the Fox" by R. Rebonato which contains more detailed information in Chapter 16.

Answer 2

The NPV of a zero coupon bond Z, in the currency in which it has been emitted:

Z(t)= Notional*[Z(0)* exp (yiled (Z;t)*t)]

and

yield(Z;t)=Z(0)*t+ epsilon

Where

----- t is the time since the bond has been emitted

---- epsilon is the the normal distribution N(0;1)

----- Z(0)=m(0) ; it's the unit selling price when the Zero coupon bond went on the market

---

Or, more generally, for a treasury bond émission Z=g, emitted at t=t0=23.02.2005 by the government G=government(UK) with a maturity Tau=10 years (23.02.2015) its price g at emission is:

g(N;t0)=M*f(t0)=999'000'000 GBP(2005)=999'000'000 GBP <*(!)****>

N=total notional of the treasury bonds of the government G(the total amount of money the government G borrowed at t0, and not the notional of 1 piece of emitted bond)=1'000'000'000=1Ml

The government G(23.02.2015) returns (no government defaults in own currency) at the maturity $\Tau$ of the bond emission an amount

$N * f($$\tau$$)=N*GBP(23.02.2015)=10^9 $GBP

N=cardinal of cash_amount/face_value at time $\tau$

M=purchasing_cash/face value at time $t_0$=999'000'000

f(t)=face value at moment t =utility, function of purchasing power parity as government can choose to print more money at $\tau$ For a non-government bond or for a G bond, but not in own currency, f() represents the credit risk. <*(!!)****> $N*f(\Tau)=M*f(t_0)=M*1$ f(t0)=1 because it is the reference/scaling value

Purchasig power(t)=N*currency(t) = f(t)

g(f(TAU))=f(t0)

where

f(t)=instant face value at moment t = c(t)=instant unit of currency of country with government G at t =c(t;G) at t

c=country function <*(!!)****>

$0<=f(t)=f(t;c)=P(t)^2$

P^2(t)=purchasing power parity(t) of the country with government G at time t

g(P^2(2015))=P^2(2005)

g(t;tau(t))=ExpectedValue[ g(t;t) * exp( (y_g(t;tau(t))t) ) ] *!!!**

With:

yield y_g(t;tau(t))=g(t;t)tau(t) + b_g (!!!!!)**

timeToMaturity from moment t < 2015 tau=tau(t)=2015-t

y_g(t;T(t))=y_g(t) (4)

!!:The instant yield of the treasury bond at moment t = instant risk-free rate at t

y(t)=r(t)=y(t) <*(!!)****>

y(t;tau(t))=r(t)

y(t;2015-t)=r(t)

r(t)=a_g(2005)*(2015-t) + b_g

b_g = constant specific to bond g = epsilon, extracted from a Normal(0,1)=N(0,1)

a_g(2005) = alpha= a_g(2015) drift specific to bond g (on planet Earth, as the drift on planet risk-free is zero)