Liability Driven Dynamic Hedging Strategies for CashBalance Pension Plans

HARDY, MARY, SAUNDERS, DAVID, AND ZHU XIAOBAI

1Department of Statistics and Actuarial Science, University of Waterloo

abstract here

1 Introduction

Cash Balance (CB) pension plans are the fastest growing pension plan by far in the US, accord-

ing to Kravitz (2015). The number of CB plans has increased from around than 3% to 28% of

all Defined Benefit (DB) plans over the past decade. The number of CB plans in the US was

9648 in 2014 and the number of participants reached 12.3 million.

The CB design originated in the 1980s, first established by the Bank of America. The moti-

vation was to provide sponsors with more predictable, less volatile costs within a DB plan. CB

plans were supposed to preserve lower risk properties of Defined Contribution (DC) plans for

pension sponsors, with relatively predictable benefits for participants. CB plans remained rare

for nearly a decade and started gaining popularity in the mid-1990s. Paulson (2004) explained

that, from 1995 to 2001, companies seeking ways to avoid the reversion tax applied to the ex-

cess pension assets for DB Plans frequently switched to CB plans. Coronado and Copeland

(2003) on the other hand, drew a different conclusion that the conversions from DB to CB were

1

primarily driven by the labor market conditions. Prior to 2006, Cash Balance pension faced fre-

quent legal challenges concerning the age discrimination. However, the legal uncertainty issue

has been resolved by the 2006 Pension Protection Act, see Lunder and Staman (2008). In 2010,

the IRS proposed further regulations that offered companies more options for crediting rates

which came into effect on November 16, 2015 1.

CB plans are classified and regulated as DB plans but share some features of DC plans.

Participants have individual accounts showing their up-to-date accrued benefits; however, those

accounts are notional. The assets are not allocated to individuals and the amounts paid into

the accounts need not be equal to the notional contributions. Moreover, members have limited

or no control over the investment decisions, and their benefits are not directly affected by the

performance of the investment portfolio. The crediting rates applied to the notional account

are either a fixed rate, or tied to an index approved by the IRS. Popular choices for the credit-

ing rates are fixed rates of return, which account for 42.2% of all CB plans, 30-Year Treasury

Rates with no floor, which account for 29.9% and 30-Year Treasury Rate with a Floor, which

accounts for 23% (Based on 553 Kravitz Cash Balance clients, see Kravitz (2015)). Another

option gaining popularity recently is the actual rate of return of the investments (4.4% of plans).

This option shifts the investment risk back to employees but with some protection such as the

preservation of capital. Overall, using 30-Year Treasury Rate as the crediting rate is the most

common choice, which accounts for more than half of the CB plans.

Traditional actuarial valuation approach to CB plans have been studied by Murphy (2011),

who concluded that the traditional methods may understate the actual liability, and generate a

loss on early termination of the plan. Brown et al. (2001) adopted a market value approach.1Internal Revenue Bulletin, T.D.9743, 2015-48 I.R.B.679, “Transitional Amendments to Satisfy the Market

Rate of Return Rules for Hybrid Retirement Plans”

2

They used a certainty model and Vasicek model to forecast future yields and evaluate the costs

of CB liabilities. Hardy et al. (2014) followed a similar approach, but with more sophisticated

arbitrage-free interest rate models. Moreover, Hardy et al. (2014) derived explicit valuation

formulae, and considered the sensitivity of valuation with respect to changes in the economic

conditions, the interest rate model and the parameters. In Hardy et al. (2014), crediting rates are

assumed to be paid continuously. Part of this work extends the results to the discrete case. In

practice, crediting rate are usually applied annually or semi-annually.

Unlike the traditional actuarial approach, Hardy et al. (2014) viewed CB benefits as finan-

cial liabilities. Under a complete and arbitrage-free market, the price of transferring the pension

liability to a third party would be same as the liability value if retained. Although this tech-

nique involves subjective selection of an interest rate model and parameters, Hardy et al. (2014)

show that their results are relatively insensitive to the choice of the interest rate model and the

parameters, even using larger time horizons and long-term crediting rates. Market consistent

valuation techniques have been studied for other pension and annuity liabilities. Some exam-

ples can be found in Marshall (2011) and Chen and Hardy (2009). Despite the fact that the US

public and private sector pension plans have not yet adopted market consistent valuation, much

of the literature has discussed the benefit of using market values to assess the funding status of

pension plans. Novy-Marx and Rauh (2009) and Biggs (2010) recalculate the funding ratios of

US public pensions and conclude that traditional actuarial techniques significantly overstate the

funding ratio. Moreover, Biggs and Smetters (2013) outlined common misconceptions about

using a long-term market rate as the discount rate for pension liabilities. In this paper, we re-

inforce the advantage of market consistent valuation from a risk management perspective, and

demonstrate that it provides a better measure of risks, and provides a natural way of hedging

the maturity benefit.

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Hedging the liability is an equally important topic as pricing and it is one of the crucial cri-

teria in the selection of models. In the broad area of interest rate derivatives, a frequently asked

question is how many stochastic drivers are required to construct an effective hedging portfolio.

Fan et al. (2001) have studied up to four factor models in the swaption market and conclude

that low-dimensional models are capable of accurately pricing swaptions but are insufficient for

hedging purposes. Gupta and Subrahmanyam (2005) and Driessen et al. (2000) draw a similar

conclusion for the cap/floor market. Their work is closer to ours, as they used delta hedging

strategies, whereas Fan et al. (2001) focus on bucket hedging. In the case of CB plans, Hardy

et al. (2014) show that the difference in pricing using one and two factor Hull-White (HW)

models is insignificant. However, the hedging of the CB liability is still unexplored. Brown

et al. (2001) studied the duration of the CB plans, which leads to a model-independent duration

hedging strategy. On the other hand, Harvey (2012) discussed the difficulties in calculating

the duration of a CB plan, and outlined several practical investment strategies including credit

default swaps, Treasury futures and swaptions, but without supporting theoretical or empirical

analysis. Motivated by the results from the swaption and cap/floor market, and to match our

pricing paradigm, we consider delta hedging in one-factor and two-factor Hull White models.

The remainder of the paper is structured as follows. Section 2 introduces the assumptions,

notation and models we adopt in this paper, and also the valuation formulae. Section 3 presents

the construction of hedging portfolio. In Section 4 we analyze the effectiveness of dynamic

hedging strategies under both simulations and real data. Section 5 concludes.

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2 Model and Assumptions

In this section we outline the formulae and assumptions used for valuating the cash balance

pension liability. The results are from Hardy et al. (2014) and Zhu (2015).

2.1 CB Benefit

This paper focuses on solely the hedging of interest rate risk. We ignore mortality and other

demographic considerations. Treating the individual CB account balance at the retirement as

a contingent payment, the market value of the liability should be the risk neutral expectation

of the discounted value of the account value at the termination. Our valuation approach is

accrual based, and so are all the hedging strategies we construct. Under the accrual principle,

the accrued liability is based only on past contributions. Future contributions into the plan can

be each regarded as a separate derivative which will be evaluated and hedged in the same way

as the existing account value. We denote by Ft the notional account value at time t, and let ic(t)

and rc(t) be the discrete and continuous crediting rates at time t, respectively. The starting date

of the plan is t = 0, and we value the liability at arbitrary dates between 0 and T. The lump-sum

payment at the retirement date is

FT = F0

Tn−1∏t=0

(1 + ic(t))

for discrete crediting, where n is the crediting frequency per year, and

FT = F0e∫ T0 rc(t)dt

for continuous crediting.

The frequency of interest crediting varies between plans. The most common choice is annual

crediting. We present the calculations under both continuous crediting and discrete crediting,

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which, in most cases, display a similar numerical result.

The selection of the crediting rate also varies in practice. According to Kravitz (2015),

among 553 of their clients, 51.2% use a fixed rate between 2% and 5.5%, and 44.1% use the

30-year Treasury Rate with or without a floor (floor ranging from 3% to 5%). Other studies

such as Hill et al. (2010) report similar findings. Since the liability using a fixed rate can be

perfectly replicated using zero coupon bonds, this paper will only consider Treasury rates.

2.2 The Yield Curve

We use standard notation for the yield curves. Let r(t) denote the continuously compounded

short rate of interest at time t, p(t, T ) denote the price of a zero coupon bond at time t with face

value of $1, maturing at T, and rk(t) denote the k-year spot rate at time t. The relations between

spot rates, bond prices and short rates are as follows:

r(t) = limk→0

rk(t)

p(t, T ) = e−(T−t)rT−t(t)

p(t, T ) = EQt [e−

∫ Tt r(s)ds]

where Q denotes to the risk neutral measure.

The Valuation Formula

Using option pricing theory, the value at time t of the payoff FT at time T is

Vt = EQt

[FT e

−∫ Tt r(s)ds

]where both FT and r(t) can be random. In the CB liability context, the valuation formula is

Vt = FtEQt

[e∫ Tt rc(s)−r(s)ds

]6

for continuous crediting, and with a similar expression for discrete crediting.

In this article, we use joint models for rc(t) and r(t) to value Vt, as they are naturally

dependent. For the rest of the paper, we denote V (t, T ) as the ‘valuation factor’. This is the

market value at time t, for a CB plan with $1 in the participant’s account balance at time t,

where the benefit matures at time T ≥ t.

2.3 Short Rate Models

We use both the One-Factor and Two-Factor versions of the Hull-White model. Both of these

maintain analytical tractability and allow a perfect match between the model and market starting

yield curve. One disadvantage of these models is that the short rate has a Gaussian distribution,

with the possibility of being negative, but Hardy et al. (2014) showed that this has a rather

insignificant impact on the valuation factor. For the One-Factor Hull-White model, the instan-

taneous short rate under the risk-neutral measure Q has the following SDE:

dr(t) = (θ(t)− ar(t)) + σdW (t), r(0) = r0

where a > 0, σ > 0 are constant, and θ(t) is a deterministic function of t chosen to match

the market term structure at the starting date, W (t) is a standard Brownian motion under the

measure Q, and r0 is the observed short rate.

There are two common parameterizations for the Two-Factor Hull-White model. The one

we adopt here is commonly referred to as G2++ (Brigo and Mercurio (2001)). The dynamics

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of the instantaneous short rate under Q are

r(t) = x(t) + y(t) + ϕ(t), r(0) = r0

dx(t) = −a1x(t)dt+ σ1dW1(t), x(0) = 0

dy(t) = −a2y(t)dt+ σ2dW2(t), y(0) = 0

dW1(t)dW2(t) = ρdt

where r0 is the initial observed short rate, a1, a2,σ1,σ2 are all positive constants, and−1 ≤ ρ ≤ 1.

ϕ(t) is a deterministic function of t for matching the implied yield curve with the observed one.

(W1(t),W2(t)) is a two-dimensional Brownian motion under the measure Q with correlation ρ.

As we described before, the most common crediting rate is the par yield on the 30-year

Treasury bond, with no margin. However, there is no explicit solution to the valuation factor

and simulation is required for the approximation. Hardy et al. (2014), and Zhu (2015) used the

k-year spot rate as the crediting rate, leading to analytical solutions for the One- and Two-Factor

Hull-White models, under both the discrete and continuous crediting. This approximation is

valuable as it offers some insight into the valuation and hedging results. The valuation formulae

for both continuous and discrete crediting are provided in the Appendix; interested readers can

refer to Hardy et al. (2014) and Zhu (2015) for more detailed derivation.

3 Constructing the Dynamic Hedging Portfolio

In recent years, driven by the new regulatory landscape, pension sponsors have increased their

attention to liability driven investment (LDI) strategies. To minimize risk, traditional methods

of immunization are often adopted. The Cash Balance Pension liability differs from the tradi-

tional DB pension liability in that the retirement benefit is strongly related to the treasury rate,

which is stochastic, and no marketable asset exists with similar returns. This complexity makes

perfect immunization impossible, but one may find a partial solution using mathematical fi-

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nance techniques. Dynamic hedging provides at least a theoretically appealing way to replicate

the maturity payoff. It closely ties the value of the asset portfolio with the value of the liabili-

ties, and it is very simple to construct. Compared with the duration hedging, which is popular

in practice and has the same goal of immunizing interest rate risk, dynamic hedging does not

assume parallel shifts in the term structure, and performs better when payments are random.

In this article, we consider delta and delta-gamma hedging strategies under the One-Factor

Hull-White model, and delta hedging under the Two-Factor model. The hedging instruments

we use are the bank account, with return equal to the short rate, and treasury STRIPs of vary-

ing length. In practice, hedging with only treasury STRIPs is impractical, due to the size of

the STRIPs market. In December 2014, the outstanding principal was only about 213 billion.

However, the dynamic hedging techniques may be applied with other fixed income securities,

such as treasury bonds. Moreover, we re-calibrate the parameters using Swaptions on the date

of valuation, and re-balance the portfolio. Although re-calibration of parameters at portfolio re-

balancing dates is inconsistent with our valuation paradigm which implicitly assumes constant

parameters, it is more realistic. Also, since the drift term of the Hull-White model is implied by

the observed term structure, it is natural to update the parameters to reflect yield curve changes.

3.1 Greeks

The delta and gamma for a particular derivative are defined as the first and second order deriva-

tives respect to the underlying process. Here we provide the delta and gamma for the CB

liability (subscript V) and for the zero coupon bond price (subscript B) under the One-Factor

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Hull-White model:

∆B(t) =∂P (t, T )

∂r(t)= −B(a, T − t)P (t, T )

∆V (t) =∂V (t, T )

∂r(t)= −γ(a, k)B(a, T − t)V (t, T )

ΓB(t) =∂2P (t, T )

∂r(t)2= B(a, T − t)2P (t, T )

ΓV (t) =∂2P (t, T )

∂r(t)2= γ(a, k)2B(a, T − t)2V (t, T )

B(a, T ) =1− e−aT

a

γ(a, k) = 1− B(a, k)

k

Similarly under the Two-Factor Hull-White model, the deltas for the CB liability and for the

zero coupon bond are the first partial derivatives respect to each stochastic driver (x and y).

∆xB(t) = −B(a1, T − t)P (t, T )

∆yB(t) = −B(a2, T − t)P (t, T )

∆xV (t) = −γ(a1, k)B(a1, T − t)V (t, T )

∆yV (t) = −γ(a2, k)B(a2, T − t)V (t, T )

where B(a, T ) and γ(a, k) are defined the same as in the One-Factor model.

3.2 Position of hedging instruments

The hedging instruments we use for delta hedging under the One-Factor Hull-White model are

the bank account, with return equal to the short rate, and a zero-coupon bond with maturity

T1. For gamma hedging under the One-Factor Hull-White model and delta hedging under the

Two-Factor Hull-White model, we add another zero-coupon bond with maturity T2 > T1. The

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position in each instrument can be obtained by solving linear equation(s)2. For the One-Factor

Hull-White model, the position in the T1-year zero coupon bond is

ΛHW =γ(a, k)B(a, T − t)V (t, T )

B(a, T1 − t)P (t, T1)

The position in the T1-year zero coupon bond (ΛHW1 ) and the position in the T2-year zero coupon

bond (ΛHW2 ) of the delta-gamma hedging strategy under the One-Factor Hull-White model are

ΛHW1 =

γ(a, k)B(a, T − t)(γ(a, k)B(a, T − t)−B(a, T2 − t))V (t, T )

B(a, T1 − t)(B(a, T1 − t)−B(a, T2 − t))P (t, T1)

ΛHW2 =

γ(a, k)B(a, T − t)(γ(a, k)B(a, T − t)−B(a, T1 − t))V (t, T )

B(a, T2 − t)(B(a, T2 − t)−B(a, T1 − t))P (t, T2)

The position in the T1-year zero coupon bond (ΛG21 ) and the position in the T2-year zero coupon

bond (ΛG22 ) of the delta hedging strategy under the Two-Factor Hull-White model are

ΛG21 =

(γ(a1, k)B(a1, T − t)B(a2, T2 − t)− γ(a2, k)B(a2, T − t)B(a1, T2 − t))V (t, T )

(B(a1, T1 − t)B(a2, T2 − t)−B(a2, T1 − t)B(a1, T2 − t))P (t, T1)

ΛG22 =

(γ(a1, k)B(a1, T − t)B(a2, T1 − t)− γ(a2, k)B(a2, T − t)B(a1, T1 − t))V (t, T )

(B(a1, T2 − t)B(a2, T1 − t)−B(a1, T1 − t)B(a2, T2 − t))P (t, T2)

Last but not least, the amount invested in the savings account S(t) equals the difference between

the liability and the total value of the position in zero coupon bonds.

For simplicity, we select the first zero coupon bond with the same maturity as the horizon of

the liability, and select the longest maturity zero coupon bond available as the second (30-year

for the U.S. Treasury Strip market).

2For example, for the delta hedging under One-Factor Hull-White model, we solve for the position in T1-yearzero coupon bond (ΛHW

1 (t)) as:

∆V (t) = ∆B(t, T1)ΛHW1 (t)

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3.3 Estimating the Parameters

We will assess the effectiveness of the hedging strategies by applying the 5-year CB liabilities

maturing in the period 2005-2015. In both Hardy et al. (2014) and Zhu (2015), the same param-

eters are used. For the One-Factor model, we have α = 0.02, σ = 0.006, and for the Two-Factor

Hull-White model, we have

a1 = 0.055, a2 = 0.108, σ1 = 0.032, σ2 = 0.044, ρ = −0.9999

While those parameter values imply a long-term unconditional standard deviation for the short

rate that is close to experience over the past 30-40 years, they should not be used to examine the

effectiveness of the hedging strategies using the historical data. Intuitively, the parameters used

at each valuation date should depend on the past data only. Therefore, in order to accurately

analyze the performance of the dynamic hedging strategy through the period 2000-2015, we

need to estimate the parameters at each valuation date (or re-balancing date).

The most common choices of market derivatives used for calibration of interest rate models

are caps and swaptions. We choose swaptions as they contain more information on the corre-

lations between forward rates (see Brigo and Mercurio (2001)). The choices for the number of

swaptions used in the calibration strongly affects the parameter values, for example, Gurrieri

et al. (2009) compare the calibration results using three sets of swaptions, for the One-Factor

Hull-White with time-varying mean reversion and volatility. We use at-the-money swaptions

with expiration dates between one year and the maturity of the CB liability. The tenor of the

swaptions are chosen to be 30 years, with quarterly payments. We minimize our objective func-

tion as the sum of the relative squared difference between the theoretical swaption price and the

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market price.

obj =n∑i=1

(HW implied swaption pricei −market swaption pricei

market swaption pricei

)2

For the Two-Factor Hull-White model, to distinguish between the parameters of the two

stochastic drivers, we set constraints as

0 < a2 < a1 < 1, 0 < σ1, σ2 < 0.5, −1 < ρ < 1

The swaption data are quoted as a black-volatility matrix and obtained from Bloomberg.

Figure 1: Calibrated Parameters for One-Factor HW

Figures 1 and 2 present the calibrated parameters. At first glance, the estimated parameters

for the Two-Factor Hull-White model do not match with the estimated parameters for the One-

Factor Hull-White model. Indeed, the variance of future short rates under these two sets of

13

parameters differs significantly. The reason is that the swaptions we used for calibration are

focused on the long rates over the next five years. In fact, the two sets of parameters imply

a close match for the variance of spot rates with long maturity in the next one to five years.

Also, notice that all calibrated parameters are very unstable, especially for the Two-Factor Hull-

White model (where parameters are often close to the constraints). This phenomenon has been

observed elsewhere in the literature (for example, Enev (2011) and Gurrieri et al. (2009)).

Figure 2: Calibrated Parameters for Two-Factor HW (volatility and correlation only)

4 Hedging Performance

Under certain assumptions, a dynamic hedging strategy with no transaction costs can perfectly

replicate the liability. However, this is impossible in practice. The hedge error comes from

three sources: the discretization of hedge periods, the mismatch between the model and the real

movements of interest rates, and parameter uncertainty. There are also additional costs, such

as transaction fees, which can be significant, especially for frequent re-balancing. Dynamic

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hedging involves continuously re-balancing the portfolio, but in practice, only periodical re-

balancing is possible. Here we consider weekly, monthly and annual re-balancing, where the

monthly case serves as the benchmark. We also present result for simulations, which quantifies

the effect of the discrete hedging error. The parameter uncertainty effect is hard to evaluate.

Here we compare the performance using (i) constant parameters, (ii) calibrated parameters for

valuation only at the beginning of the CB plan, and (iii) adjusting the parameters throughout the

hedge period. We follow Marshall (2011) by choosing 10 basis points as the transaction cost

for treasury strips, which is slightly larger than the median of the bid-ask spread in the recent

STRIPs market.

4.1 Hedging Error

We denote H(t) as the value of the hedge portfolio at time t, given the participant remains in

the plan. By definition we have H(t) = V (t). Let H(t−) be the value of the hedge portfolio at

time t immediately before re-balancing, and we let E(t) denote the hedge error at time t before

re-balancing.

E(t) = V (t)−H(t−)

The funding ratio is often cited as the basis for the financial health of a pension plan. Assuming

that at each re-balancing date, we invest/borrow the amount equal to the hedge error into bank

account, the maturity hedge error is defined as

T∑t=1

E(t)e∫ Tt r(s)ds

and we subtract this value from the CB account balance to get the maturity funding level of the

plan.

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A transaction cost is incurred whenever the hedge portfolio is rebalanced. Let κ be the

transaction cost rate, which is set to be 10 basis points. We assume no transaction costs for

the bank account. The cumulative transaction cost (C) of T1-year zero coupon bond for delta

hedging, using the One-Factor model is

C =T∑t=1

κ∣∣ΛHW (t)− ΛHW (t− 1)

∣∣P (t, T1)

where Λ(t) is the number of the T1-year zero coupon bond in the portfolio at time t. Similar

equations hold for delta-gamma hedging, and delta hedging using the Two-Factor model.

4.2 Simulation

The distribution of hedge errors under simulation can be obtained by generating a large number

of sample paths for the future crediting rate under the real-world probability measure, then

calculating the maturity hedge error for each sample path. Notice that, under simulation, we do

not have any other external information to adjust the estimate of the parameter values, thus we

assume the same parameter values throughout the liability horizon. Moreover, the parameters

used to value the liability are under the risk-neutral measure. To simulate the real-world path,

we need to shift the drift according to the market price of risk. See details in Bjork (2009). For

simplicity, we follow the work of PricewaterhouseCoopers (2014), which assumed a constant

market price of risk λ, estimated through historical zero rates of maturities between fifty and

twelve years from 2/01/2001 to 2/01/2013. The dynamics of the short rate under real world

measure is

dr(t) = (θ(t)− ar(t) + λσ)dt+ σdW P (t)

The estimates of λ lie between 0.46 and 0.55. We assume 0.5 would be a reasonable approxi-

mation for λ, and since we use 0.006 for volatility term, the drift adjustment would be 0.003.

16

Figures 3 and 4 display the distribution of maturity hedge errors under simulation using

One-Factor Hull-White model, assuming the initial fund value is $1000. The starting date of

the plan (or the initial term structure) is chosen to be 2009-02-27, but we find that the maturity

hedge error is not very sensitive to the initial term structure under the simulation. Immediately,

we observe that increasing the frequency of the re-balancing will reduce the overall hedge loss.

Also, for delta hedging, the effect of switching from annual re-balancing to monthly is much

greater than from monthly to weekly (in both the magnitude and the shape of the distribution).

Notice, even for annual re-balancing, the maturity hedge error for a 5-year plan is less than

0.2%, which is insignificant. In terms of terminal funding level, a monthly hedge under simula-

tion perfectly replicates the maturity benefit with error less than 0.01%. Cumulative transaction

costs exhibit similar patterns as shown in Zhu (2015), where transaction costs increase with re-

balancing frequency and the number of hedging instruments, but all less than $0.2 for a $1000

CB plan. Combining the transaction costs with hedge error, re-balancing monthly and weekly

result in a very close hedge.

17

Figure 3: Simulation Results for Delta Hedging (One Factor HW), plan started at 2009-02-27,initial account value F0 = 1000

18

Figure 4: Simulation Results for Gamma Hedging (One Factor HW), plan started at 2009-02-27,initial account value F0 = 1000

4.3 Hedging Performance using Historical Data

The outstanding performance of the delta hedging strategy under simulation is to be expected,

as it does not capture the model error or parameter uncertainty. The One-Factor Hull-White

model implicitly assumes a perfect correlation between long rates and short rates which con-

tradicts the empirical findings. Although Hardy et al. (2014) demonstrated that the valuation

19

factor is not very sensitive to the model we use, the delta of the valuation factor (and thus the

hedging portfolio) is a different story. In this section, we examine the performance of dynamic

hedging strategies using historical data.

To be as realistic as possible, here we consider a Cash Balance plan with annual crediting

rate that is settled in advance. Due to the limit of data size (with swaptions data only for 15

years), we considered a 5-year plan. Judging on the results from the previous subsection, here

we display the result for monthly re-balancing. To illustrate the parameter effect, we display

the results in the order of (i) constant parameters, (ii) calibration of parameters only at the plan

starting date, and (iii) re-calibration of parameters along the hedging path. Since the valuation

factors evaluated using different parameters result in different capital requirements at the set-

tlement date, the hedge error alone does not provide a full picture of the effectiveness of the

strategies. Figure 5 presents the valuation factor under different parameters. This demonstrates

that the valuation factors are not heavily dependent on the parameters, since the largest differ-

ence is within 0.6% (during the crisis). The robustness of valuation factors to the parameters

allows us to focus solely on the hedge errors.

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Figure 5: Valuation Factor evaluated using different parameters

To illustrate the value of using hedging strategies, Figure 6 demonstrates the performance

where all assets are invested in stocks (with return linked to the S&P 500) or all invested in 30-

year zero-coupon bonds or a 60/40 balanced portfolio . Despite the fact that there is potentially

a large gain at the maturity date for both strategies, the key finding is that the strategies can

also lead to huge losses. At first glance, investing in the bond seems an ideal investment option

since it only has a few small losses in the past. However, this is due to the fact that over the

past few decades, the term structure had a overall downward trend, arrived to the historical

lows. An upward trend in interest rates would leads to large losses on CB plans using the

long bond investment strategy. More importantly, the volatility of the hedge error is huge,

and a simple mixed portfolio of stocks and bonds can not achieve a satisfactory level for risk

diversification. We note that the purpose of hedging is not to guarantee gains, but to minimize

the losses as much as possible. Figure 7 include the performance of dynamic hedging strategies

using constant parameters, it clearly shows that risks will be greatly reduced when a dynamic

21

hedging strategy is implemented.

Figure 6: Maturity Hedge Loss for 5-year CB plan mature at 1994-2015, initial account valueF0 = 1000, and V0 calculated using One-Factor Hull-White model.

Figure 7: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Compare different asset allocation, using constant parameters.

22

Figure 8: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Delta-Hedging with One-Factor Hull-White Model

Figure 9: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Gamma-Hedging with One-Factor Hull-White Model

23

Figure 10: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Delta-Hedging with Two-Factor Hull-White Model

Figure 11: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Compare different hedging strategies, constant parameters used

24

Figure 12: Maturity Hedge Loss for 5-year CB plan mature at 2005-2015, initial account valueF0 = 1000, Compare different hedging strategies, parameters re-calibrate monthly

Figure 8 shows the hedging results for the delta strategy, using the One-Factor Hull White

model. Figure 9 shows the gamma strategy hedging result, and Figure 10 shows the delta strat-

egy hedging result using the Two-Factor Hull White model.

First notice that for the One-Factor Hull White model, using calibrated parameters does gen-

erally decrease the volatility of the hedge errors compared with using the constant parameters.

Re-calibration of parameters at each re-balancing date gives the best performance. It is less

significant for the delta-hedging, but for delta-gamma hedging, re-calibration of parameters ef-

fectively controls most terminal hedge errors within −5% to 5% of the initial notional amount.

Things get reversed for the Two-Factor Hull White model. Using calibrated parameters intro-

duces extra volatility to the maturity hedge errors. Increasing the number of re-calibration dates

during the hedge period does not have significant improvements. However, it is interesting to

25

note that the Two-Factor Hull-White model does not perform worse than the gamma hedging

in the One-Factor model, with less volatility hedge errors. As shown in the previous section,

estimated parameters fluctuated wildly during the crisis. Clearly, parameter uncertainty is a key

factor. All three cases demonstrate that even though the valuation factors are not sensitive to the

parameters, the performance of the hedge clearly is.

Figure 11 compares the performance of different hedging strategies when parameters are as-

sumed to be constant, Figure 12 does the same comparison but the parameters are re-calibrated

monthly. Three important notes observed from these two graphs. Firstly, the delta-hedging

under one-factor can not achieve the best performance regardless the choice of the parameters.

Secondly, although the parameter uncertain greatly reduce the effectiveness of delta-hedging

under two-factor model, the volatility of its maturity hedge error still remain smaller than the

delta-hedging under one-factor model. Lastly, the best hedging performance under both cases

can keep most of the maturity hedge loss between -5% and 5% of the initial account value.

Figures 13, 14 and 15 display the cumulative transaction costs under different hedging

strategies with different sets of parameters. As expected, changing parameters will incur more

transactions costs. From the graphs, we observe that the transaction or the position of each

instrument is sensitive to the changes in the parameters. In addition to the conclusion in Zhu

(2015) that the hedging portfolio constructed by the Two-Factor model is more sensitive to the

changes in interest rate, here we see the opposite for the changes in the parameters. It is also

clear that transaction costs are relatively small in general compared to the hedge loss. Using

less frequent calibrated parameters would result in a difference in the cumulative transaction

cost of less than $2 for most cases.

26

Figure 13: Cumulative Transaction Cost for 5-year CB plan mature at 2005-2015, initial accountvalue F0 = 1000, Delta-Hedging with the One-Factor HW Model

Figure 14: Cumulative Transaction Cost for 5-year CB plan mature at 2005-2015, initial accountvalue F0 = 1000, Gamma-Hedging with the One-Factor HW Model

27

Figure 15: Cumulative Transaction Cost for 5-year CB plan mature at 2005-2015, initial accountvalue F0 = 1000, Delta-Hedging with the Two-Factor HW Model

5 Conclusion

In this article, we have studied dynamic hedging strategies for Cash Balance Pension plans.

Dynamic hedging strategies should be considered when the pension sponsor’s investment ob-

jective is to match the terminal liability. Dynamic hedging is consistent with financial valuation

techniques, it is relatively simple to construct, and offers significant protection for interest rate

risk.

We have restricted the crediting rate to the 30-year spot rate. This assumption differs from

current practice which uses par rates, but the main driver of uncertainty is the interest rate risk.

Also, for simplicity, we include only zero coupon bonds in our asset portfolio, but the principles

can be easily extended to other interest rate derivatives. Our analysis is based on both simula-

tions and historical data.

28

Clearly, performance under real world data is worse than the simulation result. This is a

measure of model and parameter risk. However, dynamic strategies do provide a satisfactory

hedge against interest rate risk in our real world analysis. We show that the maturity hedge error

could be well controlled within ±10% of the initial funding value (funding ratio greater than

90%). However, in contrast to the analysis of valuation factors, we have showed that the final

performance of the hedging strategy greatly depends on the parameters adopted.

The result from dynamic hedging strategies are not perfect. In future work, we will explore

other types of hedging strategies, for instance, semi-static hedging. See Marshall (2011) and

Liu (2010) for discussion of semi-static hedging for different guaranteed options on variable

annuity.

References and Notes

Biggs, A. 2010. An options pricing method for calculating the market price of public sector

pension liabilities. American Enterprise Institute Working Paper 164.

Biggs, A. and K. Smetters 2013. Understanding the argument for market valuation of public

pension liabilities. American Enterprise Institute.

Bjork, T. 2009. Arbitrage Theory in Continuous Time. Oxford University Press.

Brigo, D. and F. Mercurio 2001. Interest Rate Models: Theory and Practice - with Smile,

Inflation and Credit. Springer Verlag.

Brown, D., P. Dybvig, and W. Marshall 2001. The cost and duration of cash balance plans.

Financial Analysts Journal.

29

Chen, K. and M. R. Hardy 2009. The DB underpin hybrid pension plan: fair valuation and

funding. North American Actuarial Journal, 13(4):407–424.

Coronado, J. L. and P. C. Copeland 2003. Cash balance pension plan conversions and the new

economy. Federal Reserve Board Finance and Economics Discussion Series. Working Paper

No. 2003-63.

Driessen, J., P. Klaassen, and B. Meleberg 2000. The performance of multi-factor term structure

models for pricing and hedging caps and swaptions. Working Paper.

Enev, E. 2011. Gaussian affine term structure models within the duffie-kan framework and their

use for forecasting. Master’s thesis, University of Amsterdam.

Fan, R., A. Gupta, and P. Ritchken 2001. On pricing and hedging in the swaption market: How

many factors really? Working Paper.

Gupta, A. and M. Subrahmanyam 2005. Pricing and hedging interest rate options: Evidence

from cap-floor markets. Journal of Banking and Finance, 29:701–733.

Gurrieri, S., M. Nakabayashi, and T. Wong 2009. Calibration methods of hull-white model.

SSRN Electronic Journal. ISSN 1556-5068.

Hardy, M. R., D. Saunders, and X. Zhu 2014. Market-consistent valuation and funding of cash

balance pensions. North American Actuarial Journal, 18(2):294–314.

Harvey, J. 2012. Strategies for hedging interest rate risk in a cash balance plan.

Hill, T., G. Pang, and M. Warshawsky 2010. Hybrid pension plans: A comprehensive look at

their history, economics and features. Towers Watson Perspectives Report. New York.

Kravitz 2015. 2015 national cash balance research report.

30

Liu, Y. 2010. Pricing and Hedging the Guaranteed Minimum Withdrawal Benefits in Variable

Annuities. PhD thesis, University of Waterloo.

Lunder, E. and J. Staman 2008. Cash balance pension plans: Selected legal issues. United

States Congressional Research Service, RS22214.

Marshall, C. 2011. Financial Risk Management of Guaranteed Minimum Income Benefits Em-

bedded in Variable Annuities. PhD thesis, University of Waterloo.

Murphy, R. 2011. The Cash Balance Funding Method. Society of Actuaries.

Novy-Marx, R. and D. Rauh, Joshua 2009. The liability and risks of state-sponsored pension

plans. Journal of Economic Perspectives, 23(4):191–210.

Paulson, B. 2004. Replacing defined benefit pensions: An analysis of the trend toward defined

contribution and cash balance plans.

PricewaterhouseCoopers 2014. Embedded options in pension plans - valuation of guarantees in

cash balance plans.

Zhu, X. 2015. Performance of dynamic hedging strategies for cash balance pension plans.

Master’s thesis, University of Waterloo.

A Continuous Crediting Based on k-year spot rate: One-Factor Hull-White Model

This section and the following three sections provide formulae for valuation factors, detailed

proof can be found in Hardy et al. (2014) and Zhu (2015). For CB plans with the k-year spot

rate as the continuously crediting rate, the valuation factor using the One-Factor Hull-White

31

model is

V (t, T ) = EQt

[e∫ T0 rk(s)ds−

∫ Tt r(s)ds+m(k)T

]= e

∫ t0 rk(s)dse−

∫ Tt

A(s,s+k)k

dsPγ(t, T )em(k)T

where

Pγ(t, T ) = exp {Aγ(t, T )−Bγ(a, T − t)r(t)}

Aγ(t, T ) = γ log

{PM(0, T )

PM(0, t)

}+

{σ2γ

4a{2

a(γ − 1)(T − t) + (e−2at − γ)B2(a, T − t)

+1

a(2− 2γ)B(a, T − t)}+ γfM(0, t)B(a, T − t)

}Bγ(a, T − t) = γB(a, T − t)

B(a, k) =1− e−ak

a

A(t, t+ k) = logp(0, t+ k)

p(0, t)+ f(0, t)B(a, k)− σ2

4aB(a, k)2(1− e−2at)

γ = 1− B(a, k)

k

B Discrete Crediting Based on k-year spot rate: One-FactorHull-White Model

A CB plan with the k-year spot rate as the crediting rate and crediting frequency n times per

year, has the valuation factor using the One-Factor Hull-White model

V (t, T ) = e∑tn

s=1

rk( sn )

n eA(t,T )−B(a,T−t)r(t)eEH(t,T,n)+ 12V H(t,T,n) (1)

32

where

EH(t, T, n) =Tn∑

i=tn+1

−A(in, in

+ k)

nk

+B(a, k)

nk

Tn∑i=tn+1

[r(t)e−a(

in−t) + ϕ

(i

n

)− ϕ(t)e−a(

in−t)

−σ2

a2

(1− 1

2e−a(T−

in) − e−a(

in−t) +

1

2e−a(T+

in−2t))]

V H(t, T, n) =

(σB(a, k)

nk

)2 Tn∑j=tn+1

(e−

an(j−1) − e−aT

ean − 1

)2e2a

jn − e2a j−1

n

2a

ϕ(t) = fM(0, t) +σ2

2a2(1− e−at)2

fM(0, t) is the observed instantaneous forward rate at time t, and A(t, T ), B(a, k) are defined

in the previous section.

C Continuous Crediting Based on k-year spot rate: Two-Factor Hull-White Model

For a CB plan with the k-year spot rate as the continuous crediting rate, the valuation factor

using the Two-Factor Hull-White model is

V (t, T ) = exp

{∫ t

0

rk(s)ds

}exp {m(k)T} exp

{−∫ T

t

AG(s, s+ k)

kds

}exp (A∗(t, T )− γ(a1, k)B(a1, T − t)x(t)− γ(a2, k)B(a2, T − t)y(t))

33

where

AG(t, T ) = logp(0, T )

p(0, t)+

1

2(ν(T − t) + ν(t)− ν(T ))

ν(k) =σ21

a21(k − 2B(a1, k) +B(2a1, k)) +

σ22

a22(k − 2B(a2, k) +B(2a2, k))

+2ρσ1σ2a2a2

(k −B(a1, k)−B(2a2, k) +B(a1 + a2, k))

B(a, k) =1

a(1− e−ak)

A∗(t, T ) = logp(0, T )

p(0, t)+

1

2(ν∗(T − t) + ν(t)− ν(T ))

ν∗(k) =γ(a1, k)2σ2

1

a21(k − 2B(a1, k) +B(2a1, k)) +

γ2(a2, k)σ22

a22(k − 2B(a2, k) +B(2a2, k))

+2ργ(a1, k)γ(a2, k)σ1σ2

a1a2(k −B(a1, k)−B(a2, k) +B(a1 + a2, k))

γ(aj, k) = 1− B(aj, k)

k, j = 1, 2

D Discrete Crediting Based on k-year spot rate: Two-FactorHull-White Model

The CB plan with the k-year spot rate as the crediting rate and crediting frequency n times per

year, has the valuation factor using the Two-Factor Hull-White model

V (t, T ) = e∑tn

s=1

rk( sn )

n eAG(t,T )−B(a1,T−t)x(t)−B(a2,T−t)y(t)eEG(t,T,n)+ 1

2V G(t,T,n)

34

where

EG(t, T, n) =Tn∑

i=tn+1

−AG(in, in

+ k)

nk+B(a1, k)

nk

[x(t)e−a1(

in−t) −MT

x (t,i

n)

]+B(a2, k)

nk

[y(t)e−a2(

in−t) −MT

y (t,i

n)

]V G(t, T, n) =

(σ1B(a1, k)

nk

)2 Tn∑j=tn+1

(e−

a1n(j−1) − e−a1T

ea1n − 1

)2e2a1

jn − e2a1 j−1

n

2a1

+

(σ2B(a2, k)

nk

)2 Tn∑j=tn+1

(e−

a2n(j−1) − e−a2T

ea2n − 1

)2e2a2

jn − e2a2 j−1

n

2a2

2ρ

(σ1B(a1, k)

nk

)(σ2B(a2, k)

nk

)Tn∑

j=tn+1

(e−

a1n(j−1) − e−a1T

ea1n − 1

)(e−

a2n(j−1) − e−a2T

ea2n − 1

)e(a1+a2)

jn − e(a1+a2) j−1

n

a1 + a2

MTx (s, t) =

(σ21

a1+ ρ

σ1σ2a2

)B(a1, t− s)−

σ21

a1e−a1(T−t)B(2a1, t− s)−

ρσ1σ2a2

e−a2(T−t)B(a1 + a2, t− s)

MTy (s, t) =

(σ22

a2+ ρ

σ1σ2a1

)B(a2, t− s)−

σ22

a2e−a2(T−t)B(2a2, t− s)−

ρσ1σ2a1

e−a1(T−t)B(a1 + a2, t− s)

35