The Shattered Crystal Ball
The Immunization of the Retirement Liability against Interest and Inflation Uncertainties
By Leonard Wissner (Len)
Ah if only we had a crystal ball that would tell us the future. We live in uncertain times. In tackling the retirement dilemma we are presented with many uncertainties. How long will I live? How much will I need each year for the remainder of my life? What rate of interest will I be able to invest my savings at in the future? Will inflation ravage my retirement income? My prior article, Immunizing the Retirement Annuity against Interest Rate Uncertainty, dealt with interest rate uncertainty. In this article, I will demonstrate that a bond immunization strategy augmented with inflation swap contracts presents an effective means of immunizing a retirement annuity against the duel uncertainties of interest and inflation rate fluctuations. I will conclude the article with a simulation of an immunization strategy using historical interest and inflation swap data observed over the past 20 years.
I would like to say at the outset I would have preferred to use an inflation futures contract to hedge my inflation exposure. A futures contract would be guaranteed by a regulated futures exchange. The contract would be priced continuously making price discovery easily available to the public at a minimal cost. Historical price data would be easy and generally inexpensive to obtain. Variation margin and collateral requirements would be uniform and clearly defined.
Unfortunately, an inflation futures contract was not successful when it was launched by the Coffee Cocoa Sugar Exchange in 1984 and the Chicago Mercantile Exchange in the early years of the current century. I am hopeful an inflation futures contracts will again be reintroduced in the near future. True there are advantages in the over-the-counter Inflation Swap Market. Swaps can be individually tailored to the needs of investors. The tenor or maturity of a swap can range from one to 30 years. Inflation hedges can be set up more easily than dealing with a handful of futures contracts whose actively traded settlement dates might span only a couple of years on the futures exchange. There do exist sources of data in the over-the-counter swap market. There is, MarkitSERV, which is the predominant trade matching and processing platform for interest rate derivatives. A subscriber to Bloomberg can obtain current and historical data on inflation swaps but such a platform could prove expensive to the average investor.
I worry about the current situation we find ourselves in the opaque Inflation Swap Market. Will the dealer community have the resources to honor the long-term commitments which can extend to thirty years? Will we experience the collusion of rates we saw with interest rate swaps tied to the Libor interest rate? Indeed, when we have a situation where the fox is in charge of counting the chickens in the coop, we are setting ourselves up for a recipe for disaster. We all remember the financial debacle of 2008–2009. Back then we relied on the dealer community to police the credit default swap market and determine the value and collateral requirements of credit swaps each day. Where would we have been if the Fed did not arrange for the bailout of American Insurance Group (AIG) a major originator of credit default swap at that time?
The problem of immunizing a Fixed Annuity against the fluctuation of Interest rates was tackled by FM Redington in his seminal 1952 paper Principles of Life Office Valuations. In this paper, Redington introduced an immunization strategy utilizing fixed income instruments to fund a fixed annuity schedule without resorting to a cash flow matching design. I was always intrigued whether immunization could be applied to tackle the dual uncertainties of both inflation and interest rates. The latter two uncertainties, interest and salary growth, are critical assumptions in determining the solvency of a pension plan. Could the concepts of immunization be generalized to apply to the pension plan investment problem? In the Treasury market today, we do have Treasury Inflation-Protected Securities (Tips). However, a matching strategy applied to an annuity will require TIPS of many maturities which could prove to be very inflexible. Certain TIPS might be priced richly in the marketplace at a given point in time. Immunization strategies would be preferable as they could capture temporary inefficiencies throughout the spectrum of maturities and credits. A dynamic immunization strategy presents an opportunity for the bond portfolio manager to enhance the yield of a bond portfolio above the yields presented in the TIPS market priced at a static point in time.
The Immunization Strategy applied to a Deferred Annuity
Let us say, in 2007 we wish to fund a deferred annuity consisting of $1,000,000 per year after inflation, to commence in 2010 for 10 years till 2019. Rather than buying a laddered portfolio of TIPS maturing each year from 2010–2019 we wish to employ an immunization strategy that will fund this obligation independent of future inflation and interest rate fluctuation. Given the strip curve and inflation swap rates observed in January 2007, we have determined that $ 8,671,980 would be sufficient to fund this obligation. As chart 3 below indicates a real immunity rate of close to 2% could be realized in January 2007 based on the Treasury strip curve and inflation swap rates in January 2007. The data for a simulation was difficult to obtain. The inflation swap curve was approximated from the Cleveland Fed Database of monthly inflation expectations 1982-present. The discussion of the inflation expectations model used by the Cleveland Fed can be found in the excellent working paper Inflation Expectations, Real Rates, and Risk Premia: Evidence from Inflation Swaps by Joseph G. Haubrich, George Pennacchi, and Peter Ritchken. This paper suggests that inflation swap rates were priced at a premium of 1/3/-2/3% above the inflation expectation of market participants derived from their model.
We will employ the Redington immunization strategy adapted to the principal component analysis of the yield curve discussed in my prior article Immunizing the Retirement Annuity. You will recall that the principal components of the yield curve were estimated from historical yield curve shifts observed from 1986–2007. In the simulations discussed later, I rebalance our bond portfolio each week from 2007–2019 so that the following immunization constraints are observed. Portfolios are selected weekly using the scipy linear programming package with the objective function of choosing the portfolio of maximum yield satisfying the constraints (1–3) listed below.
from scipy.optimize import linprog
Immunization Adapted to the Principal Component Analysis of the Treasury Yield Curve 1986–2007
Our challenge is somewhat different than the challenge Redingtom faced in immunizing a stream of fixed payments in the future. Here the payments must be adjusted to the inflation rates experienced over the course of the next 13 years. A simple explanation of the mechanism of a zero-coupon inflation swap would be helpful.
The buyer of the swap agrees to pay a fixed inflation rate over the time period(tenor) of the swap. Since this rate is fixed and established at the beginning of the contract it establishes the cash flow target amount for our immunization liability. In our problem, the 3-year swap rate on January 2007 is approximately 2.68% so that the target amount for the $1 million payment due in January 2010 would be $1,082,573 ($1million x(1.0268)³). Similarly, the target payments for each payment of our retirement annuity 2011–2019 can be determined. We, therefore, begin our immunization strategy with a portfolio of $8,671,980 and follow an immunization strategy on the fixed liability payments contracted at the rates of the swap curve on January 2007.
Let us consider two alternative scenarios to see how the inflation hedge will operate. In a rising inflation rate-setting, our annuity payment due will be higher than the swap rate agreed to in January 2007. The profit made on the settlement of the swap on the maturity date of the swap will offset this higher annuity payment due. Conversely, in a falling inflation rate setting, the annuity payment due will be less than the swap rate agreed to in January 2007. The loss incurred on the swap on settlement day will be offset by the lower annuity payment due. I will go into this mechanism in more detail in the discussion of the simulation of the portfolio strategy using historical data from 2007–2019.
During the life(tenor) of the swap, the dealer who sold the swap to the buyer will be constantly monitoring the value of the swap based on current market conditions. In a falling inflation setting the buyer of the swap, who contracted to pay a higher rate than the current market, will be forced to post additional collateral. I treat this amount as one would treat variation margin in a futures contract and record this amount throughout the simulations that follow. It should be noted if inflation rates rise, I record the variation margin profits that would result from holding a swap contract at a lower rate than current market conditions. I have estimated the variation margin with a simple algorithm for pricing existing swaps based on current market conditions which can be explained as follows.
The buyer of a swap has two choices available. He could sell his existing contract and roll into a new contract priced in the current market for the remaining tenor of the contract. Alternatively, he could hold his existing contract to maturity. The future value of the price received or /paid from his sale added to the maturity value of a new contract initiated at the current market (swap roll) should be roughly equal to holding the existing contract to maturity. The reader might find the enclosed python notebook helpful in pricing an inflation swap in the current market environment. The python program also gives a check on this pricing mechanism for each swap contract in our simulation maturing from 2010–2019. The routine calculates the difference between the future value of the price received for selling an existing swap added to the value of a swap roll in the current market minus the value of holding the original swap to maturity.
https://colab.research.google.com/drive/122DF-qtOeOl8oFB26dlGQ_pBZGgjU4np?usp=sharing
The chart below measures the actual difference in this pricing mechanism at maturity measured weekly over the life of the ten inflation swap contracts purchased in 2007. The differences are quite small suggesting a maximum difference at maturity of $5000 per million notional on the 2010 maturity date contract and at most $20,000 on the contract with a maturity date of 2019. Hence this pricing algorithm has an accuracy sufficient for the purpose of conservatively determining variation margin on the swap portfolio. It should be noted if inflation rates rise, I record the variation margin profits that would result from holding a swap contract at a lower rate than current market conditions.
The bond portfolio manager must be cognizant of two major risks in operating an immunization strategy to fund an inflation-adjusted annuity. There is the risk that the immunization strategy will fall short of the present value of the remaining annuity due at any given point in time (the immunization strategy risk). There is also the risk that there could be calls for greater collateral requirements if inflation swap rates fall below those contracted at the commencement of the strategy. This latter risk is interesting since the manager will be penalized today and wait to be rewarded with a lower disbursement amount at the maturity date of the swap. A Contingency Reserve Fund quantifies the immunization risk and the valuation of the inflation swap portfolio quantifies the variation margin risk. These risks are measured weekly throughout the simulation period from 2007–2019(Chart 5).
Simulation Results
A weekly simulation of the immunization strategy from 2007–2019 was designed to fund an annuity consisting of $1,000,000 per year after inflation, to commence in 2010 for 10 years till 2019. The immunization contingency reserve and the value of the inflation swap portfolio were recorded each week to monitor the immunization strategy risk and the risk that the inflation swap portfolio will require additional collateral or variation margin. Let us observe the week of January 4, 2010, when the first annuity payment comes due.
Simulation Results January 4, 2010($000)
At inception in January 2007, a portfolio of $8,671,980 was determined to be sufficient given the swap curve and strip curve at that time. The CPI was 202.2, and the swap rate for a 3-year tenor inflation swap maturing in 2010 was approximately 2.68%. The actual annual inflation experience(cpi 216.2) was approximately 2.26% per annum in the 2007–2010 time period which explains the approximately $13,300 loss on the first Swap Settlement at maturity in 2010. The remaining 9 contact swap portfolio (2011–2019) is about $439,000 underwater since inflation swap rates have declined on average approximately .36% since 2007(Chart 2). Hence additional collateral would be needed to be posted to fund this deficit. However, the Cumulative immunization surplus to date of $405,000 largely covers this possible contingency.
The actual Annuity payment of $1,069,323 reflects the actual inflation experience to fund the growth of a $1 million payment due in January 2010. Note this annuity disbursement is roughly $13,250 lower than the $1,082,573 targeted disbursement based on the rate of the swap contract, initiated on January 2007, whose maturity date is January 2010. Hence the lower annuity disbursement will offset the loss on the settlement of the swap contract, demonstrating the inflation hedge. The reader will note the CPI is reported monthly by The Bureau of Labor Statistics and yet the simulation is performed weekly. Weekly CPI data is derived using the Scipy linear interpolation method fitted to the monthly CPI data.
from scipy.interpolate import interp1d
The immunized bond portfolio of $10,687,930 will be sufficient to fund the annuity payment($1,069,323) in January 2010, the swap settlement payment, and the $9,604,865 needed to fund the remaining 9 annuity payments due (constraint 1), leaving a slight surplus of $464 which is added to the cumulative surplus to date.
The linear program optimization was successful in selecting a $10,687,930 portfolio of 4 securities in the prior week to January 4, 2010. Immunization obviates the need of using a large laddered portfolio of Strips matched to the maturity structure of the annuity payments due. The maturity structure is somewhat barbell in nature but not as extreme as the barbell structure that would be suggested in a classical Redington immunization strategy where only 2 securities would be selected, a short and very long bond.
For the entire simulation period 2007–2019, the immunization portfolio design performs quite well with a steady buildup in cumulative surplus of $890,000 after the last annuity payment is disbursed in 2019. Note how the cumulative surplus is generally much larger than the swap portfolio adverse variation margin or collateral requirements shown in Chart 5 below. Swap rates generally fell during the period 2007–2019(Chart 2). This would explain the generally negative value of the swap portfolio during the simulation period.
The strategy is tested during periods of extreme stress in the financial markets. The sudden bankruptcy of Lehman Brothers and bailout of American International Group (AIG) in the latter part of 2008, resulted in a dramatic decline in inflation swap rates (Chart 2). Collateral, in excess of $880,000 would have to be posted in January 2009 to cover the negative market value of the swap portfolio. Fortunately, there was a $350,000 cumulative immunization surplus at this time. For most of the history of the simulation Cumulative Surplus was more than the negative market value of the swap portfolio supplementing the requirements of additional collateral.
All would agree that testing the strategy under such severe conditions as the panic of 2008–2009 is a testament to the robustness of the portfolio design. There is an important lesson to be learned here. Holding an inflation swap portfolio in a period when market inflation expectation suddenly collapses will present an extremely challenging time for the portfolio manager. True the distant annuity obligations will be smaller than the contracted rates of the inflation swap contracts but the pain of an adverse margin call will be occurring immediately. One must have faith in the model during these times of crisis which I am confident will again occur periodically in the future!
The time period of the simulation witnessed an active program of quantitative easing by the Fed in the aftermath of the recession in 2008- 2009. This had dual effects, lowering the yields on Treasury Strips (Chart1) and reducing market inflation expectations thereby lowering inflation swap rates (Chart 2). The cash flow real yield earned on our immunization program was approximately 2.90% over the simulation period when collateral requirements for the swap portfolio are taken into consideration. This represents a pickup of about 1/2% in real return above Treasury Inflation-Protected Securities(TIPS) rates in January 2007.
In hindsight, initiating an immunization program in January 2007 was a fortuitous decision despite the market turbulence witnessed in the ensuing years. It can be seen from Chart 3 real immunity returns between 1–2% would be difficult to achieve in the past 5 years. The situation might very well improve in the future, as the Fed in late 2021 has indicated the end of their Quantitative Easing program. This might hopefully make the yields on fixed income attractive again. If the rise in rates occurs in the backdrop of an easing of inflation expectation, we could expect real returns promised on Immunity programs to again approach 1-2%.
I conclude the discussion of Immunization I presented in my last 2 articles with a note of caution. Would I recommend this strategy to all investors? The returns promised by an immunized structure might be in the range of 1–2% above inflation. In a pension plan, such narrow spreads between investment return and a proxy for salary growth would prove to be extremely expensive. This explains the phenomenon witnessed in the past 20 years of largely shunning fixed income strategies in favor of the investment in equities and alternative asset classes promising higher real returns. In many ways, the Quantitative Easing policies, adopted by Central Banks in the western world, have exacerbated these trends which have led to historically high valuations on equity securities, and alternative asset classes. The jury is still out whether these promised higher real returns are delivered, and the fate of our pension system might very well rely on this promise.
However, certain entities such as insurance companies and banks I regard as fiduciaries with the role of conservators of capital in our society. These institutions should certainly consider immunization strategies to hedge their distant liability obligations. They should be regulated and penalized when they deviate substantially from their role as conservators. When we have conservators behaving as speculators the results have proven disastrous as we have seen in the S&L crises in the 1980s and the mortgage derivative, housing crises of 2008–2009. Older individuals in the twilight years of their life might consider immunization strategies to synthetically fund their living expenses with Treasury securities rather than assume the risk of the default of an annuity provider. Younger investors can and should take risks as sometimes the great rewards in life come with the assumption of risk. There is a necessary place for both conservators and speculators in the capital markets.
Much has been said and written on ways to profit, by forecasting future market trends, and the long-term rewards obtained through the assumption of risk with stocks and alternative investment vehicles. My goal was to present 2 articles discussing bond immunization strategies. These strategies do not rely on forecasting techniques as they attempt to minimize the risk of funding distant liability obligations in the face of a hazardous and uncertain future.