Immunizing the Retirement Annuity
Against Interest Rate Uncertainty
by Leonard Wissner(Len)
Few things in life are invariant to the forces of nature. In physics, the speed of light is invariant to the measurement system adopted by an observer. In funding a fixed annuity, an immunized bond portfolio strategy is invariant to the future fluctuation in interest rates. This is quite remarkable since a retirement annuity could span a time horizon in excess of twenty years. There is a great deal of uncertainty in the course of interest rates over such a long-time horizon. There would be much room for error if one relied on forecasting techniques to accomplish such an important task. In this article, I would like to suggest three approaches to this investment problem.
- Cash Flow Matching
- Redington Immunization — periodically equating the duration of the bond portfolio to the duration of the annuity obligation
- Redington Immunization modified with Principal Component Analysis of the treasury yield curve
After defining an immunization strategy, I will present simulations using actual yield curve data observed over the years 1986–2020*. On the basis of these simulations, I have concluded that a classical immunization strategy, augmented with principal component analysis of changes in the Treasury yield curve, shows great promise in immunizing an annuity obligation against adverse changes in the interest rate level.
The Simple Solution- Cash Flow Matching
A particularly opportune time to fund an annuity is when interest and inflation rates are at historically high levels. Such was the case on December 30, 1985, when the Treasury Strip curve was as shown below. Ah yes, to our new students of the bond market, there was a time in recent history when long risk-free treasury securities were yielding in excess of 10%.
*https://www.federalreserve.gov/econresdata/researchdata/feds200628_5.html
Finance and Economics Discussion SeriesThe U.S. Treasury Yield Curve: 1961 to the Present
Refet S. Gurkaynak, Brian Sack, and Jonathan H. Wright
2006–28
Let us say you desire on December 30, 1985, to receive a 20-year annuity of $1 Million a year to commence in 5 years, January 4, 1991 — January 4, 2010.
You could purchase a $6,025,985 portfolio consisting of 20 $1 million principal amount treasury strip or zero-coupon bonds which mature on the exact dates when each of the $1 million annuity payments is due. This strategy would lock in a cash flow yield of 9.65% independent of the course of interest rates over the next 25 years. It is interesting to note if performance is measured quarterly looking only at the quarterly price performance of the strip portfolio a matched portfolio strategy would appear extremely risky. Most performance measurement systems fail to look at both the asset and liability sides of the equation. Financial analysts are generally trained in the school of the Capital Asset Pricing Model where the treasury bill would be considered the risk-free rate of interest. This presents a difficult dilemma for money managers who employ Bond immunization strategies. In a period of falling rates the portfolio of strips, in our example, would exhibit extraordinary good returns in a customary performance measurement system, making investment officers look like geniuses. However, due diligence should be paid to the fact that the opportunity cost of funding the annuity is greater in a low-interest rate setting. In a period of rising rates, traditional performance measurement systems will show poor returns for the portfolio manager, ignoring the ease of funding an annuity in a higher interest rate setting.
In most investment situations there is little coordination between the actuarial and investment functions in either a life office or a pension plan. The great American actuary, Irwin Vanderhoof in his paper published in The Transactions of Society of Actuaries 1972, The Interest Rate Assumption and the Maturity Structure of Assets of a Life Insurance Company remarked that the only coordination between the actuarial and investment departments of an insurance company takes place at the annual golf outing each year. We are deluged with many economists and financial gurus but I have yet to see an actuary interviewed on national T.V discussing the challenges in the public and private pension systems.
The latter discussion is particularly relevant when looking at public Pension plans in the present day. In periods of falling rates and raging equity markets, there is great danger in celebrating the robust performance of the asset portfolio without paying due diligence to the opportunity cost of funding future obligations in a low-interest rate setting. Those plan sponsors who reduced contributions to the pension plan during the robust stock market years at the turn of the 21st century, when asset returns were robust paid a dear price during the market collapse of 2008–2009.
Redington Immunization
In a matched portfolio design the ability to enhance the yield of the immunized structure utilizing noncallable high-grade corporate or municipal bonds becomes extremely difficult. Trading opportunities to capture transient inefficiencies in the bond market can not be exploited. It is difficult to enhance yield as market interest rates fluctuate during the term of the annuity payouts.
To accomplish the effect of cash flow matching without resorting to its rigid structure F.M. Redington, a British Actuary, published in 1952 a seminal paper In the Journal of the Institute of Actuaries, Review of the Principles of Life-Office Valuations. The topic of immunization was also discussed by the American actuary, Irwin Vanderhoof in his paper cited above. In the appendix to the Redington paper 3 important equations were formulated to immunize a bond portfolio against adverse changes in interest rates. With Bond portfolio immunization, a portfolio manager can accomplish the funding of a specified annuity schedule without resorting to cash flow matching with the bonus opportunity of enhancing yield beyond a matched portfolio. These immunization equations are summarized below.
The immunization strategy:
S(i)-Surplus of the value of bond portfolio less the present value of the annuity payments evaluated at the current interest rate level i
A(i)-value of Bond portfolio at current interest rate i
L(i)- the value of Annuity at current interest rate i
The immunization strategy requires that the Bond Portfolio is Periodically rebalanced such that
The derivation of the latter 3 conditions follows a classical minimization problem in differential calculus where 2 heroic assumptions are made.
- The yield curve is flat hence there is one interest for all bonds independent of maturity
2. The yield curve shifts in a parallel manner from one time period to the next, such that the changes in yields across all maturities are the same.
The latter assumptions almost never occur in practice as seen in the history of interest rates from 1986 -2020 shown in the graph below. It is interesting to note if one assumes the latter two assumptions regarding the behavior of interest rates, it is possible to increase the realized yield in funding the annuity due as second derivative profits are made as interest rates fluctuate during the time horizon of the annuity. Hence the immunization strategy is more flexible than a matched structure and one obtains the opportunity for second derivative profits as interest rates fluctuate, both extremely favorable outcomes.
My interest lies in answering the question of how effective is the Redington model in immunizing a retirement annuity using actual yield curve data* from December 30, 1985- January 4, 2020*? Despite the simplicity of the Redington model, and its unrealistic assumptions, did it perform better than the matched portfolio we constructed in the latter section? A simulation would be helpful in answering this question.
During the bullish and bearish bond market settings during the simulation period from 1986–2010, the immunized bond portfolio is rebalanced weekly to satisfy the 3 Redington constraints. I used the Scikit linear programming package to select each week the portfolio satisfying the 3 Reddington constraints with the objective function of maximizing the average yield of the bond portfolio in the given yield curve setting:`
obj=list(-1*yield_df_zero_weekly.loc[loc].values)#objective function
lhs_eq = [list(prices),list(durations)] # Redington Constraints1&2 rhs_eq = [port_val,dur_val] # Redingtton Constraints 1&2
lhs_ineq=[list(-1*convexity)]#Redington Constraint 3
rhs_ineq=[-1*con_val]#Redington Constraint 3
opt = linprog(c=obj, A_ub=lhs_ineq, b_ub=rhs_ineq,
A_eq=lhs_eq, b_eq=rhs_eq,method=’revised simplex’,options={‘tol’: 2.0e-9})After pricing the optimal portfolio(opt.x) chosen by the linear program, the following week, I would record the surplus or deficit in the market value of the bond portfolio over the market value of a strip portfolio matched to the remaining payments in the retirement annuity. I would then place the surplus each week in a contingency reserve fund earning interest at the overnight money market rate. The cumulative surplus in the contingency reserve fund is a measure of the excess/deficit performance to date of the Redington immunization strategy over the matched portfolio constructed on December 30, 1985, in the last section.
Throughout the simulation period under study, the linear program was able to satisfy the Redington constraints and an optimal solution was obtained. In general, the optimal portfolio was a barbell structure consisting of a short bond and the longest bond available that would satisfy the duration constraint equating the duration of the bond portfolio to the duration of the remaining payments in the annuity. The barbell portfolio structure makes intuitive sense since in general yield curves are increasing with the higher yields in the long maturities. Hence longer bonds will satisfy the objective function of maximizing the average yield of the portfolio. The convexity constraint is achieved when the spread of the maturities of the bond portfolio is greater than the spread of the maturities of the laddered annuity, reinforcing the choice of a barbell bond portfolio structure.
Unfortunately, the immunization strategy does not appear to be independent of the bond market cycle. During the bull bond market cycles from 1986–1991, the immunized portfolios generally outperformed a portfolio of strips matched to the remaining annuity payments, with an increase in cumulative surplus to almost $1.6 million by 1991 as shown in the graph above. This bullish interest rate cycle was followed by a bear market in 1993–1994 and cumulative surplus levels dropped to about $400,000. It should be noted that I almost lost my job managing an immunized bond portfolio during this bear market cycle as the plan sponsor used customary performance measurement techniques. Fortunately, the sponsor persevered with the strategy enjoying robust returns in the ensuing bull markets. The general widening of yield spreads across maturities in the period of 2000–2004(shown in the history of rates 1986–2020) is accompanied by a decrease in cumulative surplus to approximately $500,000. This type of fluctuation in cumulative surplus is hardly what one would expect from a true immunization strategy.
The reason for the behavior observed in the period 2000–2004 could stem from the fact that as the annuity payment schedule shortens there is an overweighting of the portfolio in the long sector to meet both the duration and convexity constraints of the classical immunization strategy. The strategy has difficulty funding an annuity schedule of short to intermediate payments with a barbell structure where long bonds will be relatively penalized when spreads across maturities widen and the short annuity payments have the advantage of riding down the yield curve. Overall, we end the simulation with a cumulative surplus of approximately $900,000. Following the immunization strategy resulted in a cash flow yield of 10.31%, a pickup of 0.66%, above the 9.65% return of the matched portfolio constructed in the previous section, hence fulfilling the important objective of enhancing the yield above a matched portfolio.
Principal Component Analysis of Shifts in theTreasury Yield Curve
Can we improve the performance of the Redington Immunization strategy with the application of Principal Component Analysis to the weekly changes of the Treasury strip curve? This seems like a fertile area of inquiry given the significant correlation of changes in weekly Treasury strip yields observed from 1986–2007.
Using the linear algebra package in NumPy I was able to compute the principal components of the changes in the weekly yields of Treasury strips from 1986–2007. A significant 98% of the total variation in the change in weekly strip yields is explained with only the first 3 principal components. A significant 88% of the variation is explained with just 1 principal component
from numpy import linalg as LA V_matrix=LA.eig(del_yield_df_zero_weekly_test.cov())
np.sum(vars[0:3])/np.sum(vars)
0.9819324872467556
np.sum(vars[0:1])/np.sum(vars)
0.8814896142380747The loadings of the first three most significant eigenvectors of the covariance matrix are a measure of the change in yield of each maturity of the treasury strip curve given a unit shift in the respective principal component. Based on the historical experience of 1985–2006 greater shifts in yield will occur in the intermediate maturities of 5–10 years, approximately 20 basis points, as compared to about 17 basis points change in yield in the long maturities of 25 -30 years given a unit shift in the first principal component. This first principal component has all shifts in yield going in the same direction given a unit shift in its component. The second principal component will capture rotations in the yield curve where short rates are going down as the intermediate to long end of the curve shifts upward given a unit shift in its component. The third component is interesting as it appears to capture an inversion in the yield curve with intermediate maturities shifting upward relative to the long end shifting down.
I modified the three Redington constraints so that we are looking at partial derivatives with respect to the first 3 principal components:
The Modified immunization strategy requires that the Bond Portfolio is Periodically rebalanced such that:
S-Surplus of the value of bond portfolio less the present value of the annuity payments evaluated at current strip yield curve rates
A-value of Bond portfolio at current strip yield curve rates
L- the value of Annuity at current strip yield curve rates
Pi-change in the ith principal component i=1,2,3
Simulation 2007–2019
A simulation was then performed, over the time period 2007–2019 comparing classical Redington immunization and immunization modified with the principal component analysis of the Treasury strip curve performed from 1986–2006. In this manner, the simulation comparison is fair since the principal component analysis is trained on data not seen in the test simulation period. In the simulation to follow we are funding on January 1, 2007, a 10-year annuity of $1 million/year to commence on January 1,2009-January 2019. This deferred annuity could be funded with a matched portfolio of treasury strips for $7,141,496.
The results of this simulation are dramatic. Over the period from January 3,2007-January 3,2019 the classical immunization strategy had difficulty outperforming a matched portfolio as evidenced by a buildup of a cumulative surplus of close to a $460,000 deficit by the end of 2018, and a deficit as high as $720,000 by the year 2012. The cash-flow yield of 3.93% of the Redington strategy was far lower than the yield of 4.71% for a matched portfolio design. The standard deviation in surplus in the classical Redington Strategy was $26,000 per week. By contrast, the modified immunization strategy utilizing principal component analysis, recorded a cumulative surplus of $709,000, and a cash flow yield of 5.98%, outperforming a matched portfolio(4.71%) and classical immunization(3.93%). Note the steady buildup of surplus independent of the interest rate cycle, which is the behavior one would expect from a robust immunization strategy. The standard deviation in the fluctuation of surplus of only $12,400 per week represents a dramatic improvement to classical immunization.
The other interesting feature of this simulation experiment is the portfolio recommendations of the respective strategies. In classical immunization, the portfolios chosen are largely a barbell structure with a very short maturity mixed with the bond of the longest maturity possible given the shortening duration of the remaining payments in the annuity. The optimal portfolios were chosen, when principal components are taken into consideration, consisted of four securities. These optimal portfolios mixed short bonds and intermediate maturity bonds which are intuitively better at capturing the gyrations of the yield curve as the annuity payments come due in the latter years of the simulation period.
We are fortunate to be living in a world today where long-duration assets, treasury strips, are freely traded and priced daily in the marketplace. These government-guaranteed instruments are ideal to meet the long-duration liabilities in many applications in the life insurance industry and the pension plan. Such long-duration assets were nonexistent in the days of Redington and Vanderhoof. Today we have derivative and swap markets which greatly facilitate the modification of asset portfolio durations to meet liability objectives. However there is no free lunch here as a derivative interest rate instrument might incur counterparty risk and there is sometimes the difficulty of price discovery, risks which do not occur using market traded instruments valued on a daily basis.
The application discussed in this article dealt with a single premium deferred annuity. An interesting modification of this design is a multiple premium deferred annuity, where the future premiums must be hedged against falling interest rates with Treasury bond futures which are actively traded at the Chicago Board Of Trade. I had the honor of presenting this design before the Canadian Institute Of Actuaries in 1984 and applied it to the defeasance of the retired life obligations of a major pension plan in the U.S. I can not conclude this discussion without addressing the elephant in the room, inflation uncertainty. When we tackle the major uncertainties of both inflation and interest rates we will finally possess the true immunization that Redington saught. This is a problem that has haunted me my entire life and I plan to present a solution in a later article.