## Real Bond Immunity in the 21st Century

Immunizing A Deferred Annuity Against Interest and Inflation Uncertainties

This is the third article on employing an immunization strategy to fund a deferred annuity. The first article, The** Immunization of the Retirement Annuity against Interest Rate Uncertainty** compared a classical Redington immunization design to an immunization design utilizing the principal component analysis of changes in the Treasury yield curve. The second article, **The Shattered Crystal Ball**, dealt with the immunization of a deferred annuity obligation against the dual uncertainties of both inflation and interest rate fluctuations. Both the prior articles dealt with the problem of employing a bond immunization strategy to fund a single payment deferred annuity. The current article deals with employing an immunization strategy to hedge the inflation and interest uncertainties one will encounter in the funding of a** scheduled contribution deferred annuity.** The problem at hand can be summarized as follows:

Let us say you are an individual in January 2000 who earns $100,000 per year and you intend to retire in January 2010. You have committed to contributing a lump sum and contributions of 10% of your inflation-adjusted salary for 10 years and wish to receive $80,000 per year for 10 years adjusted to the inflation experience. Hence in the year 2010, you will receive approximately $102,650 given a 2.52% inflation experienced in the period 2000–2010. One quickly realizes there are **two major uncertainties **to this problem. What will be the** investment return** and the **inflation** one will experience over the time interval 2000–2021? This is a trivial problem to the individual with the hindsight of history, but to our hypothetical individual in the year 2000, this is a considerable problem. One can perhaps forecast inflation and interest for one year but a 20-year projection is fraught with error.

The current article will simulate the performance of an immunization strategy over the time period 2000–2021 to fund our hypothetical annuity independent of** interest** and** inflation** uncertainty. We will employ treasury **strip securities** and **inflation swaps **available over this time period to meet this objective.

# Redington Immunization

An Immunization strategy was first utilized by FM Redington in his seminal 1952 paper Principles of Life Office Valuations. The object of his analysis was to fund a **fixed** schedule of future obligations of a life office without having to resort to a cash flow matching strategy. The other advantage of an immunity design was to **achieve second derivative profits as interest rates fluctuate.** In his analysis, however, there was an implicit assumption of a single interest rate and a parallel shift in the yield curve as rates fluctuate. Using differential calculus, he developed 3 equations to immunize the investments of a life office against the future fluctuation in interest rates without resorting to a cash flow matching strategy. My first article showed that the immunization design adapted to the **principal component analysis of changes in the yield curve** greatly improved the performance of the strategy when simulated on live treasury yield curve data observed in the current century.

**Immunization Strategy**

Although Redington dealt with a fixed payment annuity the analysis could be extended to payments that must be adjusted to inflation conditions. The advent of inflation swaps allows us to use an immunization strategy to solve this latter problem if the portfolio manager engages in an inflation swap which will fix the liability payments at a predetermined inflation rate. For example, the 10-year inflation swap rate initiated at the beginning of the year 2000 was approximately 3.75%. Hence the buyer of inflation would target payment of approximately $115,651 in the year 2010 to protect the payment of $80,000 in the year 2000 dollars. Inflation swaps trade at tenors of 1–30 years and my second article demonstrated an immunization design simulated for a single payment deferred annuity set up in the year 2007.

The current design allows for **an initial pool of assets** and **contributions **of 10% of salary which is assumed to grow at the inflation rate over time. This design is more akin to the problem of the defined benefit pension fund which has a pool of assets and contributions are received based on a **salary growth assumption. **The model used in the second study is adapted to handle the 10-year annual contributions of $10,000 as negative liability payments. These payments are also fixed by engaging in an inflation swap at the onset of the program on December 26, 1999. At that time, the 1-year inflation swap rate is approximately 3.60%. Hence a nominal -$10,360 is targeted to be contributed(disbursed) for the year January 2001. For our hypothetical individual earning $100,000 per year in January 2000, we have determined an initial pool of assets of $556,561 and contributions of 10% of salary growing at the annual inflation for ten years would fund a 10-year annual annuity of $80,000(the year 2000 dollars) in the years 2010–2021 locking in a potential **real return** in excess of 3.08%.

# Simulation 2000–2021

Simulation is an effective technique to assess the risk of an investment program utilizing actual historical data. Utilizing the 3 Immunization constraints and an objective function of maximizing portfolio yield we employ **linear programming** to select a portfolio weekly from January 2000-January 2022. The surplus is recorded each week (constraint 1) and represents a measure of the degree to which the Assets exceed the present value of the remaining Liabilities. Hence a buildup in **cumulative surplus** would indicate the strategy is performing well as it is funding the annuity disbursement schedule leaving an excess of assets in the fund. The second measure of risk would be the performance of the **inflation swap portfolio.** In a falling inflation rate setting, the inflation swap portfolio would decline, necessitating the need for **additional collateral**. The amount that additional collateral exceeds cumulative surplus I define as **drawdown**. This represents the additional funds needed to perform the immunization strategy over the simulation period.

For example, in January 2010 the CPI index was 215.949as shown in Table 1 below. When the program was initiated on December 26, 1999, the CPI was 168.3. An $80,000 distribution in 2000 dollars was equal to $102,649 in January 2010. The 10-year inflation swap rate in January 2000 was 3.75%. When this tenor swap expires its value is approximately $(13,101) since the actual inflation experience is approximately 2.52% per annum. Note that the actual disbursement is approximately $13,000 less than the targeted disbursement of $115,650 for 2010 shown in Chart 1 above. This demonstrates the inflation hedge where a lower disbursement on the annuity is offset by a loss on the swap portfolio when the swap matures. The 9 remaining inflation swaps would be valued at approximately $221,000 underwater. Since the cumulative surplus to date is only $18,228 you could be asked to post approximately $203,000 in additional collateral(**drawdown**). The surplus for this week is approximately $109,000 underwater, which is the value of the bond portfolio less the value of the 9 remaining annuity disbursements less the swap maturity and distribution outlay.

Chart 2 below indicates that there has been a dramatic decline in the yield of treasury strips rates since January 2000. The 21 st century has already witnessed three climatic events in the first twenty years. There was the bursting of the technology bubble and the 9/11 attack of 2000–2001, the housing market implosion of 2008–2009, and the covid crises in 2018–2021. Each of these periods has resulted in a dramatic decline in strip yields. Note in the midst of this decline in long yields the spread between long and short strip rates widened dramatically (chart 2). This type of environment would present a great deal of difficulty for a classical Redington strategy emphasizing long and short-duration bonds. However, the Redington strategy adapted to the principal component analysis of changes in the yield of Treasury strips, measured in the years **1980–2000**, can tackle this difficulty much more effectively.

To counter the recessionary impact of the 2008–2009 recession cycle the Fed undertook 3 programs of quantitative easing. This had the effect of lowering strip yields relative to inflation sending real expected returns down dramatically. Immunization programs could be expected to earn a real return close to 2- 3% early in the century but would be hard-pressed to earn a positive real return in the periods from 2018–2022 as shown in Chart 4 below.

Despite these challenges, the buildup of cumulative surplus (Chart 5) is impressive. The cumulative surplus climbed to approximately $1.2 million at the end of the simulation in January 2022. Indeed, the simulations indicate an impressive real return on this immunity program in excess of 5.32% taking the $300,000 reserve fund needed to cover possible collateral demands on the swap portfolio into account. There were challenges, however, particularly in the period after the housing implosion of 2008–2009. During this time period, cumulative surplus dropped dramatically as bond yields declined particularly in the shorter maturities of the yield curve (Chart2). At the same time as we entered recession inflation swap rates declined which would have necessitated additional collateral for the inflation swap portfolio. The maximum drawdown of approximately $300,000 occurred in February 2011 when the cumulative surplus was -$60,000 and the value of the inflation swap portfolio was about $232,000 below water. It should be noted that at this time the Fed was in the second phase of its quantitative easing program, actively buying long-term treasury bonds which had the effect of lowering expected real return. A maximum drawdown of about $300,000 would have been needed which could have been returned by the end of the year.

Can we expect to see real returns in excess of 3% that we saw early in the century? In late 2021 the Fed indicated an end to their quantitative easing program and a desire to increase rates as inflation has emerged. Would inflation have emerged in any case as a result of the quantitative easing programs prior to the 2018 covid crises? This will be a subject of debate for many years to come. As the Fed combats inflation in 2022 we could expect to see equity returns falter and bond yields climb. This would suggest a possible return of real yields to the historical experience observed in the twentieth century. Real returns in excess of 3% could prove possible, especially taking second derivative profits derived from an immunization strategy. I see a bright future for bonds immunization strategies, again, in the future.