The market conditions
The causes of inflation are not unique and can originate from government spending, an increase in costs in the private sector or from a shortage of commodities. Whatever the initial source, the persistence of inflation is most likely a monetary phenomenon that, in modern financial economies, is heavily dependent upon the level of money supply orchestrated by central authorities.

Since the early stages of the financial crisis, both the US Federal Reserve (the Fed) and the European Central Bank have engaged in substantial intervention to insulate the financial system from an Armageddon scenario, leading to an unprecedented expansion of money supply. Between 2007 and 2012 alone, the Fed has grown its balance sheet from less than $1 trillion to almost$3 trillion.

As a result of this protracted intervention of ‘printing money’, gold has soared in price as it is broadly seen as the ultimate resort of value against inflation, being the only ‘currency’ that cannot be printed. However, inflation is a phenomenon that plays in the background and is not always transparently understood. Once returns are expressed in ‘real’ terms by means of the Fisher equation, the gold picture might change as $1.00 in 1980 is worth$2.81 in 2012.

Despite monetary easing, inflation has not spiked while economies stalled in protracted recessionary environments. Today’s investors are facing increasing uncertainty, as the threat of inflation is never far away. This risk must not be disregarded in tactical asset allocations. By including stochastic estimates of inflation-adjusted returns on simulations over time adds value to the search for optimal portfolio allocation.

The key drivers
Since the introduction of government bonds linked to inflation indexes, like Treasury Inflation-Protected Securities (Tips) in the US, portfolio managers can benefit from more streamlined ways of buying inflation protection and enriching the catalogue of strategies alongside gold investments, short currency bets or commodity/energy exposures. However, rebalancing global wealth portfolios by accounting for inflation views requires non-trivial calibration techniques of correlated dynamics of risk factors over time.

Empirical evidence suggests that long-term portfolio performance is sensitive to the following:

• Inflation trend – Asset performance relates to the direction of inflation rate changes, not to absolute rates. Environments of spiking inflation favour gold, silver, energy and inflation-linked bonds, while disinflation favours treasuries and fixed-income securities.
• Deflationary environments – Portfolio optimisation struggles to lock in profitable asset allocations when pure capital preservation is the key goal (guarantee products, highly rated treasuries).
• Fisher equation on interest rates – The real yield of inflation-linked bonds can become negative draining on portfolio potentials, making Tips investments unappealing to long-term investors.

The stochastic methods
Here, the dynamics of the term structure of interest rates are modelled using the Hull-White two-factor model (1994), which evolves the instantaneous short rate under the risk-neutral measure, according to the following stochastic differential equation:

dr(t) = ( θ(t) + u(t) – ar(t) )dt + σ1dZ1(t), r(0) = ro

where the stochastic mean-reversion level satisfies

du(t) = -bu(t)dt + σ2dZ2(t), u(0) = 0

with

(Z1, Z2) a two-dimensional Brownian motion,

dZ1(t)dZ2(t) = ρdt

r0, a, b, σ1, σ2 positive constants,

-1 ≤ ρ ≤ 1

The function θ is deterministic and fits the initial term structure of interest rates. The model parameters are calibrated on market prices of swaptions. The same framework is used to simulate the inflation expectation curve over time.

The relation between nominal and real environments is described by the Fisher equation as when the difference between nominal and real rates equals the expected inflation. Mathematically, this is equivalent to trading fixed-income nominal amounts as one currency and real amounts as a foreign currency, where the exchange rate is the consumer price index (CPI). The future exchange rate between the two ‘economies’ is defined by the interest rate parity: the nominal curve is ‘domestic’ and the real curve is foreign.

The simulated value of financial instruments can be displayed in both nominal and real terms.

The optimal portfolio
Consider the process of a US investor searching for an optimal asset allocation with a key constraint on inflation-adjusted capital preservation. The following investment opportunities are available:

• US Treasury bond, 10 years, fixed rate
• US Tips bond, 10 years, linked to CPI
• US exchange-traded fund MSCI energy index

The investment exercise was drawn in 2007, at the first stages of the financial crisis, with the Fed funding rate at 525 basis points, real rates in the positive region and Tips yielding positive expected returns.

The objective function of the investor is to attain capital protection and 40% nominal return over an eight-year investment horizon. Total returns of the potentially inflation-performing assets are simulated using mark-to-future techniques. Investment portfolio performance can be expressed in both currencies.

By looking at the simulation of individual bond investments, the impact of the inflation adjustment on total returns is clearly visible.

The optimisation exercise – maximum probability of beating the target return with capital protection – is performed in the ‘nominal environment’ and in the ‘real environment’ starting with 2007 market conditions. The effective backtesting – attained total performance after five years from initial investment – reveals the added value of the approach, while gold and Tips are adding risk alongside return potential in ‘nominal environments – resulting in less appealing opportunities in the optimal asset allocation – they become performance drivers in both states of the world when asset allocation is optimised in real terms.

Conclusion
Embedding stochastic inflation in products and portfolios return simulations has become an important element in actuarial work and in wealth management advisory. Hedging inflation and profiting from changes in inflationary environments require meaningful metrics that add value. This demands that investment decision-makers simulate and stress test portfolios against different long-term assumptions to drive investment profitability in real terms.

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