0
( I V
Formula Simulation Absolute Diff. Relative Diff. S.E.
T
(A) (B) (C)=(A)-(B) (D) (E)
20 0.14875 0.14856 0.00019 0.00127 0.00068
25 0.20600 0.20454 0.00146 0.00708 0.00092
30 0.26883 0.26813 0.00070 0.00260 0.00119
35 0.33666 0.33507 0.00159 0.00472 0.00148
40 0.40908 0.40779 0.00129 0.00316 0.00179
Notes: Weletσ= 0.01,λ= 0.1, and keep all the other basic parameter estimates (except T) unchanged.
Column (A) lists the values derived by the formula in Proposition 3. Column (B) lists the values derived by simulation based on 50,000 paths. Column (C) describes the absolute difference between (A) and (B), and column (D) describes the relative difference, which is equal to the value in (C) divided by the value in (A). Column (E) reports the standard error of the simulation estimates.
6. Conclusions
We have derived the closed-form formulae to price the interest rate guarantees embedded in DC pension plans. Different from previous studies, we consider the relative guarantees linked to the -year spot rates. The guarantee design mixes several features commonly observed in exotic options, such as exchange options, path-dependent options and forward start options. Given that the volatility term in the HJM model is constant, the formulae for two types of guarantees, maturity guarantee and multi-period guarantee, are respectively described in Propositions 1 and 2. In addition, we obtain the pricing formula for the type-I guarantee in Proposition 3 under
the assumption that the volatility term in the HJM model is exponentially decaying.
Numerical results and sensitivity analyses are also presented. We compare the values derived by the formulae with those derived by simulation, and justify that our explicit formulae are accurate and efficient.
This paper could be further extended. First, guarantee providers may care about how to hedge the risks from providing the relative guarantees linked to the -year spot rates. It is essential to explore suitable hedging strategies. Second, we assume that the volatility term in the HJM model is constant or exponentially decaying, which may not be the case in the real world. It is worth analyzing how to value interest rate guarantees under different term structure models. Third, we overlook the case that the participant changes the investment portfolio during the accumulation phase. An interesting extension is to incorporate investment strategies into this study and investigate how these strategies affect the terminal values of both guarantees.
References
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Appendix
easily shown by Ito’s Lemma.It is easy to derive that
) 3
need to calculate
expressed as follows:
T
Therefore,
In addition,
Substitute these expressions into equation (A.1) and rearrange, we have
t
. Furthermore,
t
Then we obtain the equation. □
(C) Prove that
0S
(D) Details of Reducing Discretization Error
We follow the concept in Glasserman (2004) to reduce discretization error when
simulating investment portfolio price by equation (3) and discount factor,
0Truduexp . From equation (7), we know that given f
( t , u )
f, the stochastic differential equation for short rates ist
Glasserman (2004), we have
2 2 2 2
where
is time interval and is equal to 1/52 year in our simulation setting.Equations (D.1)-(D.3) are used to simulate short rates, asset prices and discount factor under the assumption that f
( t , u )
f.If f(t,u)e(ut) 0, we follow the next expressions to simulate short rates, portfolio prices and discount factor:
(0, ) 2 (20, ) 2 2 2(E) Prove that
1
Proof:
Impacts of Investment Strategies and Pa r t i c i pant s ’ Behaviors on Guarantee Costs and Retirement Benefits in
Defined Contribution Pension Plans
Abstract
This paper investigates how investment strategies and behaviors affect guarantee costs and retirement benefits in defined contribution plans that provide interest rate guarantees. Several investment strategies are considered, including both a buy-and-hold strategy without portfolio modifications and those that allow frequent modifications of the portfolio during the accumulation phase. According to the numerical results offered by simulation techniques, a participant chasing the highest expected retirement benefits should adopt the buy-and-hold strategy and distribute all contributions in the portfolio with high expected rate of return and high volatility in our guarantee designs. After incorporating frequent modification behaviors, averages and risk measures of guarantee costs may be higher than when the participant always holds the portfolio with high expected rate of return and volatility. Therefore, when the pension plan provides rate of return guarantees, the plan provider cannot ignore the impact of the participant’s frequent modification behaviors.
1. Introduction
Defined contribution (DC) pension plans are becoming more and more popular in many countries. With a DC plan, retirement benefits depend on the accumulated account value at retirement. The participant usually has the right to select investment portfolios but also must bear the downside risk of poor investment performance. To avoid the inadequacy of pension benefits, minimum rate of return guarantees often appear in DC plans. If the actual accumulated account value is less than the guaranteed account value, the participant withdraws the guaranteed amount, and in some scenarios, the related guarantee costs might be substantial. We investigate how the participant’s investment strategies and behaviors affect these guarantee costs and retirement benefits in DC plans that contain interest rate guarantees.
Mitchell and Utkus (2004) link behavioral finance research to pension plan design and analyze how participants make decisions to save and invest. Dowd and Blake (2006) note that insurance policies with long horizons increase the pressure associated with taking policyholder behavior into account. These authors imply that the pension plan provider should consider how participants behave and incorporate possible behaviors into its risk management. Participants might adopt a variety of strategies to allocate their contributions during the accumulation phase or modify portfolios, such that the
portfolios at retirement date are not necessary the same as those at inception. To estimate guarantee costs accurately, it therefore is essential to consider a participant’s allocation and modification strategies, especially when the plan has long horizons.
The failure to do so can lead to misestimated guarantee costs and thus great harm for the guarantee provider.
Many papers discuss how to determine the fair value of a contract embedded with various guarantees, including Brennan and Schwartz (1976), Sherris (1995), Ekern and Persson (1996), Boyle and Hardy (1997), Persson and Aase (1997), Miltersen and Persson (1999), Pennacchi (1999), Hansen and Miltersen (2002), and Lindset (2004).
These articles derive fair values without incorporating any investment strategy, whereas other articles focus on investment strategies in a DC pension plan. For example, Blake et al. (2001) discuss how value-at-risk estimates, which measure the risk that participants face in DC plans, vary with allocation strategies. Vigna and Haberman (2001) study which investment strategies minimize the cost incurred by the deviation of the actual pension fund level from the target level. In addition, Boulier et al. (2001) explore optimal portfolio composition in a plan that contains a guarantee, which relates to the level of the interest rate at retirement, and the evolution of short rates is described by Vasicek (1977) model. Menoncin (2002) considers background
risk and inflation risk to derive the optimal portfolio allocation. Battocchio and Menoncin (2004) deal with stochastic interest rates, salary risk, and inflation risk in an optimal portfolio allocation problem. In the latter three articles, the objective is to maximize the expected value of the terminal utility function with specific constraints.
In this research, we attempt to fill an existing gap in the literature by taking investment strategies into account and thereby estimate the costs of guarantees provided in DC plans. We consider seven strategies, each of which describes different investment behavior, and investigate how these strategies affect guarantee costs and retirement benefits. In the simplest strategy, known as the “buy-and-hold strategy,”
the participant neither changes portfolio allocation methods nor sells his or her portfolios during the accumulation phase.1 We consider several allocation approaches, and the proportion of contributions allocated to each portfolio is always invariant.
Benartzi and Thaler (2001) document that participants tend to allocate contributions evenly among the assets available in the plan and call it the “1/n heuristic.”2 We therefore take this allocation approach into account in the buy-and-hold strategy.
1 Zweig (1998) asks several economists how they put their money on the TIAA-CREF, the largest DC plan in the world. William F. Sharpe, the professor known for the Sharpe index, says that: “I am strictly a buy-and-hold investor. In fact, I have never sold an equity mutual fund.”
2 For example, if there are 10 available assets in the pension plan, the participant spreads his or her contributions evenly across all 10 assets. Huberman and Jian (2006) and Agnew (2006) call this strategy the “framing (effect) 1/n heuristic.”Huberman and Jian (2006) show that many participants follow the “conditional 1/n rule,”in which case they do not use all assets available in the plan but instead distribute contributions evenly among the assets they use, typically no more than three or four.
Another consideration we address is that participants may modify their investment portfolios irregularly. Fund management companies and administrators periodically disclose the performance of all portfolios available in the plan through analyses of to-date performance that employ a variety of indexes, such as the realized rate of return or the Sharpe index. We assume that some participants always track these performance indexes and prefer the portfolio with the best past performance.3 If the past performance of the current portfolio is not the best among the available portfolios, the participant may make modification decisions and invest all savings in the portfolio with the best performance. The portfolio selected at inception therefore is not always the same during the accumulation phase, and the time to modify the portfolio cannot be predicted ex ante.
Furthermore, Barber and Odean (2003) show that investors prefer to realize gains rather than losses in their tax-deferred accounts, consistent with the “disposition effect,”as first labeled by Shefrin and Statman (1985).4 Dhar and Zhu (2006) also
3 Mitchell and Utkus (2004) summarize that the “representativeness heuristic”and “availability heuristic”can account for reliance on past performance. The representativeness heuristic means that when making decisions, people tend to be affected by some descriptions or patterns, even if those descriptions are uninformative or the patterns are random; the availability heuristic implies that people are prone to rely on the most easily or cheaply available information. Readers interested in these two heuristics could refer to Tversky and Kahneman (1974). Moreover, Voronkova and Bohl (2005) demonstrate that Polish pension fund investors (institutional investors) tend to purchase stocks with excellent past performance. Pension funds under their analysis are classified into defined contribution plans. During the period they investigate, these plans provide minimum guaranteed rates of return, which are announced quarterly.
4 The central concept of the disposition effect is that the investor tends to sell winners (portfolios with
show that the disposition effect exists among the investors in individual retirement accounts (IRA) and Keogh accounts. Therefore, we consider strategies related to the disposition effect herein.
This research deals with pension plans that include interest rate guarantees. We analyze two types, type-I (maturity) and type-II (multi-period) guarantees, as we introduced in the first essay. The dynamics of interest rates are described by the CIR (1985) model. It is difficult to obtain explicit formulae for fair values of these two guarantees after incorporating modification behaviors. We therefore adopt a Monte Carlo simulation approach in our analysis. By simulating modification behaviors, guaranteed rates, and actual rates of return at all record dates in the real world, we can derive the statistics of guarantee costs and retirement benefits at retirement date.
Our numerical results offer several significant findings. First, guarantee costs and retirement benefits are both affected by investment strategies. Low retirement benefits ensue when participants put all their contributions in a portfolio with low expected
rate of return and low volatility. This conservative allocation approach costs more for
value moves above a reference point (e.g., purchase price), the investor becomes more risk averse and sells the winner as soon as possible to either feel pride or avoid regret if the value declines. If the portfolio depreciates and the value falls below the reference point, the investor becomes risk seeking and maintains the loser so that he or she does not have to concede a wrong investment decision or miss the buoyancy in the future. Many authors have demonstrated the existence of the disposition effect in financial markets, including Shefrin and Statman (1985), Odean (1998), Grinblatt and Keloharju (2001), Genesove and Mayer (2001), Locke and Mann (2005), and Shu et al. (2005).
the type-I guarantee provider but less for the type-II guarantee provider in our basic parameter settings. Second, the buy-and-hold allocation strategy with higher retirement benefits may cause lower type-I guarantee costs but higher type-II guarantee costs. Therefore, the type-II guarantee provider has an incentive to impose restrictions on allocation methods to reduce its risks, but such restrictions deter the participant from reaping the highest possible retirement benefits. Third, frequent modifications of the portfolio may not do good to the growth of pension benefits, regardless of whether transaction costs exist. This finding is consistent with Barber and Odean’s(2000) conclusion that frequent trading hurts the investment performance.
Frequent modifications also may increase guarantee costs in some guarantee designs.
If the guarantee provider neglects the effect of frequent modifications, its guarantee costs likely will be much higher than its ex ante estimates.
The remainder of this article is organized as follows. Section 2 describes the characteristics of guarantees, the financial models, and the strategies under analysis.
Section 3 provides the numerical results and presents how guarantee costs and retirement benefits vary with strategies. The implications emerging from the numerical results are also provided. Section 4 concludes the paper.