Master’s Thesis:
On Dollar Cost Averaging and Value Averaging
for small investors
Antwan van der Burg s1962426
Abstract
Dollar Cost Averaging (DCA) and Value Averaging (VA) are commonly applied in financial prod-ucts. Behavioural economics explains the attractiveness of both strategies to small investors, while neo-classical finance shows their inefficiency. In this paper we present a fair comparison of VA and DCA by taking into account criticism on the internal rate of return and other performance measures. We provide analytical results on DCA and VA and compare their performance based on historical data and two simulation models. In the stimulations we consider quarterly investments with a investment horizons of five and twenty years. Our research shows that VA outperforms DCA as VA is most likely to provide the desired final wealth to the small investor at the lowest costs. Furthermore, we examine the benefits of using a money account in the VA strategy. The money accounts induces additional costs and no real benefits for the five year investment horizon. For the twenty year horizon, it largely increases the likelihood of obtaining the desired final wealth.
Master’s thesis MSc Econometrics, Operations Research & Actuarial Studies Specialization track Actuarial Studies
Acknowledgements
We would like to thank the University of Groningen for allowing us to use Peregrine, their super-computer. Without Peregrine it would not have been possible to complete the process of simulation in time.
1
Introduction
Households and small investors often face strategic decisions on how to save money for big expenses. Examples include savings for retirement, the purchase of a car, repayment of mortgages, and home renovations. The question how these small parties can make an informed strategic decision to ef-ficiently save for these big expenses is still relatively unanswered. Given today’s low interest rate, saving is expensive and investing is more attractive. Long term investment in the stock market normally creates a higher return than savings. However, participation in the stock market requires facing aversion to regret, harmful lack of self control, and recognizing and mitigating cognitive errors with respect to the framing of risky choices. These issues are extensively discussed in the framework of behavioural economics, starting with Kahneman and Tversky (1979).
We will analyse three investment strategies that take the insights of behavioural economics into ac-count. These strategies, Dollar Cost Averaging (DCA), Value Averaging (VA) and Value averaging with a Money account (VM) are designed to allow households and small investors to participate in the stock market. From a standard neo-classical finance perspective they are deemed irrational, since one can obtain the same pay off distribution at a lower price or a better one for the same price. Constantinides (1979) shows this for DCA and Hayley (2010) for VA. Nevertheless, banks offer a wide variety of products based on these three strategies, which is explained by behavioural economics. This paper aims to assist the individual investor to make a well informed choice between DCA, VA and VM. We present useful analytical results for a continuous time model and offer both historical evaluations and model simulations.
market overheating. Consequently the likelihood of a decrease in share price in the next period is increased. In that scenario, not selling shares increases the exposure and risk. Furthermore, by selling shares one tries to follow an investment wisdom: “buy low, sell high.”
Summarizing, DCA requires fixed investments and provides a random final wealth, while VA and VM aim for a fixed final wealth and requires random investments. We will focus on describing the randomness of this strategies, generating insights such that the small investor can choose and fine tune the strategies.
2
Literature review: perspectives of neo-classical and behavioural
finance on DCA, VA & VM
In this section we will discuss the critics who deem DCA, VA and VM inefficient and irrational as discussed in the neo-classical literature. However, they are not able to explain why these strategies are still popular in practice. We conclude with a discussion of the perspective from behavioural finance.
As discussed before, DCA ensures that an investor buys his shares at a price below the average share price. However, Weston (1949) criticizes that it does not ensure any gains. In neo-classical finance, investors are assumed to be rational, among other things, they value investments based on the pay off distribution, and the share price process is assumed to be symmetric around a trend. That is, an arbitrary increase above the trend is just as likely as a decrease below the trend of the same proportion. Constantinides (1979) describes the suboptimality of DCA. DCA does not take into account any new information in later periods of time. Constantinides concludes that there are better strategies that creates returns just as good as DCA and sometimes better. Rozeff (1994) shows by simulation that DCA is inefficient compared to lump sum investing. Bierman et al. (2004) find that delaying investments as DCA prescribes, is costly and DCA does not reduce risk or increase expected return in a normal situation. However, they explicitly do not account for behavioural considerations of the investor. These critics base themselves on neo-classical finance which is not completely applicable to small investors.
Hayley (2010) uses the model and calculations presented by Dybvig (1988a, 1988b). He derives a closed form for the inefficiency of VA in the continuous case. However, the inefficiency is zero for an expected market return or growth rate equal to zero, and for small time periods the inefficiency is small as well. Furthermore, if the growth rate is close to the expected market return as Edleson (2011) suggests, the inefficiencies are small for individual investors.
As a consequence of these critics, DCA, VA and VM are not presented in most modern finance books. Even though these strategies are not presented in these books, DCA is still popular under small investors as Statman (1995) explains. He bases his argumentation on the prospect theory of Kahneman and Tversky (1979), aversion to regret, cognitive errors and self-control. In addition to Statman (1995), Milevsky and Posner (2003) show that for investors who know the final value and assets with enough volatility, the return on DCA outperforms that of the underlying asset. They use Brownian bridges in their analysis, where a Brownian bridge is a random walk with fixed final value. This approach can be justified as for individual investors the stock market is claimed to exceed current price levels over longer periods of time. Hence, the investor implicitly assumes a sort of fixed final value.
money account, as a consequence the investment has an exposure of about thirty to forty percent to shares. They find that in this setting VA statistically outperforms DCA in terms of final wealth and smoothes total risk more efficiently.
3
Methodology
In this section, we present the research approach we use. First, we will make some assumptions and after that the mathematical framework of the strategies is provided. Next, we make explicit which measures we use to compare DCA, VA and VM. Then, we discuss the historical evaluation based on empirical data and we back test with simulation models. We will use two different simulations to increase the robustness of the back test. The models are the Continuous Stochastic Volatility model with Jumps in the share price (CSVJ) and the Discrete Stochastic Volatility model with Jumps in the share price (DSVJ).
Assumptions
We assume that the big expense is known in real terms. That is, the investor requires L at time T . Then, t = 1, ..., T denotes time periods with normalized unit length. Furthermore, we consider two investment options, shares and a bank account that pays a rate of rt at time t. The shares give an
expected total return over time t, E[ St
St−1] = e
µ. Furthermore, we assume that the investor uses µ
as continuous discount rate.
Constant increments in the value path are not appropriate for longer time periods. It makes disinvestment in the later time periods more likely. Edleson (2011) explains that due to inflation and portfolio growth, the relative importance of increment decreases. For any portfolio size the return distribution in terms of relative return is assumed equal. Therefore, the return on a portfolio is more likely to exceed an absolute level of return if the portfolio increases in size. The likelihood that the return on the portfolio exceeds the increments in the value path increases as time develops, t → T . Potentially this could lead to disinvestment in the final stages of VA and VM. If in the last time steps the increments are relatively small to the total portfolio size, it is likely that the strategy dictates to sell. Especially VM has this potential as profits are stored for possible later purchases reducing the likelihood of investing even more compared with VA. DCA faces the same pitfall; for a constant number of monetary units invested, the actual power of purchase is not equivalent over time, but decreases instead. We follow the approach of Edleson (2011), who uses increasing investments / increments, which we discuss in the next subsection.
3.1 Strategies
For the implementation the investor picks a growth rate g. The literature recommends to pick g close to µ for efficient investment. Let Is
t define the periodic investment at time t, where s = {d, v, vm}
denotes the strategy, d for DCA, v for VA, and vm for VM. First, for DCA:
Itd= Cdegt, (1)
where Cd is such that the sum of total investments equals the target wealth,
L =
T
X
t=1
Hence, Cd= L(e
g−1)
eg(egT−1). Let Wts denote the accumulated wealth at time t. This gives for DCA:
Wtd= Wt−1d St St−1 + Itd, Wtd= St t X j=1 Itd Sj . (2)
Next, for VA and VM, let Vt∗ denote the value path at time t. Define Vt∗ = CteRt, where C is such
that VT∗= L and R is the growth rate of the value path. So, for R = 0 we have a linear value path. The actual investments do not only depend on the value path but also on the share returns. If the growth rate R is smaller than µ, the returns on the share price will reduce the actual growth in the expected investments to a level below R and vice versa. Edleson (2011) suggests to take the average of both, that is, R = g+µ2 . We then have for VA:
Itv = Vt∗− Vt−1∗ St St−1
, Wtv = Vt∗.
VM also requires defining the money account at time t, Mt. We have for VM:
Itvm= Vt∗− Vt−1∗ St St−1 − (1 + rt)Mt−1 + , Mt= Vt∗− Vt−1∗ St St−1 − (1 + rt)Mt−1 − , Wtvm= Vt∗+ Mt,
where [·]+and [·]−denote the positive and negative part, respectively. Moreover, the money account is empty at time t = 0, so M0 = 0.
3.2 Measurements
In finance and investing, one frequently uses the IRR to compare investment performances as do Edleson (2011) and Marshall (2000, 2006) in their analysis of VA and VM. However, Hayley (2010) criticises the appropriateness for VA. VA reduces the weight given to low or negative returns and vice versa. Hayley defines this as the automated bias of value averaging, but the question is whether the individual investor feels the same? All in all, the IRR gives him insight in the return he makes on his investment. A strategy that increases the IRR implies larger final wealth and/or lower costs to obtain the final wealth. However, there is some truth in the argument of Hayley. As the investments in VA and VM are random and volatile, it might be that the investor needs to hold additional cash to be able to keep his strategy feasible. If we do not consider the implications in our analysis, the IRR might be biased upwards. We will exploit other properties of DCA, VA and VM and consider the need and implications of a buffer.
When we discuss summary statistics we will report on the minimum, 10% quantile, average, 90% quantile, maximum and sample error. Then, we will discuss the following measurements of the risk associated to the investment strategies.
• For downside risk, the likelihood that L is not obtained at time T , P (Wi
T < L) and the
• For upside potential, the summary statistics of final wealth, Wi
T and E[WTi|WTi > L].
• The summary statistics of the total discounted costs of the strategy,
T
X
t=1
Itie−µt.
• The summary statistics of the IRR, i, is the argument that solves:
T
X
t=1
Itie−it= WTie−iT.
• The summary statistics of costs on the individual time period level, Ii t.
• The summary statistics of the costs in k consecutive periods,
j+k−1
X
t=j
Iti, where j = 1, ..., T −k +1
denotes the starting period.
In addition to these strategy specific measurements we will also provide a comparison by counting the number of times a strategy gave the highest IIR and the lowest total discounted costs.
When these measurements are obtained we will look at the following aspects. First, what if we specify a downside risk, for example P (WTd < L) < 0.05. How much should we increase the DCA investments to realise this? Then, we will limit the periodic investments and the cumulative sum of consecutive investments of VA and VM. If VA or VM requires the investor to invest more than the limits allow he will only invest what the limits allow for. We will evaluate two different sets of limits for two investors. Investor A perceives L as smaller compared to his income as Investor B does. A is able to invest at most 25 percent of L in any given period given that in rolling year he does not invest more than half of L. B has a lower income and is only able to invest at most 15 percent of L per period and 30 percent of L per rolling year. The limits imply risk of underfunding of L for both strategies, on the other hand they ensure a strategy that is feasible for the investor. We will report on the implications of the probability of underfunding, expected shortfall and the average and sample error of the IRR.
Lastly, the impact of longer time horizons will be explored. We will increase the time horizon from five to twenty years (T = 20). By multiplying the time horizon by four and maintaining L, the investment limits become rather loose. To overcome this, we divide the limits by four. We use constant limits for simplicity, a more advanced technique would let the limits grow over time. However, the implications are limited as the highest costs will occur in the final periods of the time horizon. Hence, the difference between limits increasing over time with the same final values as the fixed limits will not influence the results. We recommend to use limits that increase over time if one wants to investigate different time horizons more exactly.
3.3 Historical evaluation
For historical evaluation we require empirical data. We use quarterly observations on the S&P 500 and the quarterly forward rates1. We combine both data sets to obtain quarterly observations from January 1960 till April 2016. Taking quarters consisting of 63 trading days we obtain 224 data points. In Table 1 the summary statistics of daily, quarterly and yearly log returns on the share
price are presented. Furthermore, the T-bill rate averages 4.72% per annum. Figure 1 shows the historical share prices from 1960 till 2016.
Table 1: Summary on the S&P log returns Period average sample error skewness kurtosis
day 0.0003 0.0101 -0.9991 29.8694 quarter 0.0163 0.0782 -0.7218 5.6030
year 0.0661 0.1405 -0.5669 2.8921
Figure 1: Empirical distribution of S5.
3.4 Simulation Models
Jumps in the share price process (DSVJ). They are generalizations of the famous Black-Scholes (1973) model, which is found accurate for option pricing by Black and Scholes (1972). However, the model suffers a bias, found by Rubinstein (1985). He identifies two biases, first the continuity assumption and second the non-stochastic volatility. Heston’s (1993) model assumes a stochastic volatility and Broadie and Kaya (2006) combine this with a jump diffusion process. The latter model addresses both the volatility by including a stochastic volatility process and the jumps create discontinuity.
For simulation of the CSVJ and DSVJ model we will follow Broadie and Kaya (2006). Therefore, we use their specification of the CSVJ and DSVJ model. First, we give some definitions then we introduce the parameters and after that the CSVJ model followed by the DSVJ model.
Definition 1 Zt(j) are independent Brownian motions.
Definition 2 Nt is a Poisson process with intensity parameter λ.
Definition 3 log(ξs) ∼ N (µs, σs2), a lognormally distributed random variables.
Definition 4 ∆Zt(j) ∼ N (0, ∆t), normal variables with mean 0 and standard deviation √∆t.
Definition 5 N∆t is a Poisson random variable with intensity parameter λ∆t.
Let Stand Vtdefine the share price and variance at time t respectively. Then, µ defines the drift and
√
Vt the volatility of the share price process. ρ the correlation between the two processes and ξsthe
jumps in the share price as in Definition 3. Let dZt(1) and dZt(2) Brownian motions as in Definition 1 and dNt a Poisson process as in Definition 2. Furthermore, ξs, dZt(j) and dNt are independent.
Let κ denote the speed of mean reversion, θ the long-run mean and σ the volatility of the variance process which is also known as a square-root process. Then, the CSVJ model is defined as:
dSt= µStdt + p VtSt h ρdZt(1)+p1 − ρ2dZ(2) t i + (ξs− 1)SsdNt, (3) dVt= κ(θ − Vt)dt + σ p VtdZt(2). (4)
Notice that Ss is missing in the paper of Broadie and Kaya (2006). Define Ss− as the share price
just before the jump and Ss+ as the share price directly after the jump. Then by definition,
Ss+ = Ss−+ dSs. Broadie and Kaya (2006) claim for a jump at time s, Ss+ = Ss−ξs. One can
verify that this is only true when dSs= (ξs− 1)Ss−dNs= (ξs− 1)Ss−. Furthermore, Jessen (2010)
defines the share price process as in Equation 3.
Broadie and Kaya (2006) use the Euler discretization to obtain the DSVJ model. For a time line ti = i/n, where i = 0, 1, ..., n, the DSVJ model is defined as:
Sti = Sti−1+ µSti−1∆t + q Vti−1Sti−1 h ρ∆Zt(1) i + p 1 − ρ2∆Z(2) ti i + N∆t X ti−1≤s≤ti (ξs− 1)Ss, (5) Vti = Vti−1+ κ(θ − Vti−1)∆t + q Vti−1σ∆Z (1) ti , (6)
where ∆t = ti− ti−1= 1/n, and ξs, ∆Zt(j) and N∆t are independent and as in Definition 3, 4 and
4
Analytical comparison
In this section we analyse the strategies in the Scholes framework. In Definition 6 the Black-Scholes (1973) model is presented, where without loss of generality we assume S0 = 1. Furthermore,
in the subsequent parts we analyse DCA, VA and VM, where we assume without loss of generality L = 1.
Definition 6 Black-Scholes (1973) model defines share price St according to the process:
dSt= µSt+ σStdZt,
Then Itˆo’s lemma, Theorem 1, gives us:
St= S0exp{(µ −
1 2σ
2)t + σZ t}.
Theorem 1 (Itˆo’s lemma) let dXt= µtdt + σtdZt and f (t, x) ∈ C2, then:
df = ∂f ∂t + µt ∂f ∂x + σt2 2 ∂2f ∂x2 dt + σt ∂f ∂xdZt.
4.1 Dollar cost averaging
The investments per period, Itd = Cdegt are known, see Equation 1. From Equation 2, recall that
the final pay off depends on the share price process: WTd= ST
T
X
0≤s≤T
Isd/Ss.
Vorst (1996) shows that the impact of different sampling frequencies of the share price is minimal on the final payout of an Asian option, where an Asian option pays at maturity the average share price over a certain time period minus the strike. Milvesky and Posner (2003) use the result of Vorst (1996) to write the pay off distribution of DCA as the pay off distribution of an Asian option with zero strike. Replace each investment by n investments and divide the size of the investment through n such that the total investment volume is equal:
WTd= ST T n X i=1 1 nIi/nd Si/n .
By letting n go to infinity and replacing St by its definition, we find:
where the last equality follows from replacing τ by T − t. Then, ˆZτ = ZT− ZT −τ is a time reversed
Brownian motion that equals 0 for τ = 0. ˆSτ then the time reversal property of a Brownian motion
gives that ˆSt has the same properties as St, ˆSτ ∼ St. We have:
WTd= 1 Te gT Z T 0 e−gτSˆτdτ.
Notice that WTd is not lognormally distributed, since the sum of or integral over lognormal distri-butions is not lognormal. It follows that the pay off distribution of DCA is equal to a constant, egT multiplied by the arithmetic average of the stock price process, where g can be interpreted as an artificial dividend rate. The latter also represents the stochastic pay off of an arithmetic Asian option with zero strike. The arbitrage free price, ˆp, of this option is equal to the discounted expectation under the risk-free discount rate. Hence,
ˆ p = 1 Te (g−µ)TE Z T 0 e−gτSˆτ dτ = 1 Te (g−µ)T Z T 0 e−gτEh ˆSτ i dτ = 1 Te (g−µ)T Z T 0 e(µ−g)τdτ = 1 Te (g−µ)Te(µ−g)T − 1 µ − g = 1 T 1 − e(g−µ)T µ − g
where the second equality holds as the integral is an infinite sum of random variables and the expectation is a linear operator. The third equality follows from Eh ˆSτ
i
= eµτ. One can observe that for g < µ, ˆp < 1. This implies that one can obtain the pay off distribution of an Asian option that is just as good as the pay off distribution of DCA for a smaller price. Hence, instead of a DCA contract investing L over time, we can buy an Asian option on the underlying with 0 strike for a price smaller than L that gives us the same pay off distribution at maturity. And vice versa for g > µ. We summarize this in Result 1.
Result 1 The DCA strategy has the same pay off distribution as an zero strike Asian option. When the growth rate of the investments, g is not equal to µ their prices differ. The rational individual has a preference between the two depending on g and µ, if g < µ he prefers the Asian option and vice versa.
For the probability of underfunding, P (WTd< L), and L = 1 we have:
P (WTd< 1) = P 1 Te gT Z T 0 e−gτSˆτdτ < 1 ,
which is the probability that an Asian option with strike S0 is out of the money at maturity, time
T . The internal rate of return on DCA can be found by solving:
Unfortunately, there is no closed form solution available, in the Section 6 results of the simulation give more insight in the IRR.
4.2 Value averaging
The final wealth is known and equal to L, therefore P (Wv < L) = 0. On the other hand, the individual investments at time t are random and can be written as:
Itv = Vt∗− Vt−1∗ St St−1
, E[Itv] = Vt∗− Vt−1∗ eµ
= CeR(t−1) teR− (t − 1)eµ .
Recall R = g+µ2 , for g > −µ the first term is positive and increasing in t, the second term decreases at a higher rate in t. Hence, for small g and large t the expected investment can become negative. An investor can pick g large enough such that the expectation of the investments is positive and increasing over the complete investment period. This has its implications on the variance. We first calculate the second moment and then the variance follows:
E[(Itv)2] = Vt∗2+ Vt−1∗2e2µ+σ2 − 2Vt∗Vt−1∗ eµ, V ar(Itv) = Vt−1∗2 e2µ+σ2 − e2µ
= C2(t − 1)2e2R(t−1)
e2µ+σ2 − e2µ.
One can observe that the variance increases with t for g > −µ. Next, we consider the effects on the variance and the expectation of the sum of two consecutive payments:
Itv+ It+1v = Vt+1∗ − Vt−1∗ St+1 St−1 − Itv St+1 St − 1 = Vt+1∗ − Vt−1∗ St+1 St−1 − Vt∗ St+1 St − 1 + Vt−1∗ St St−1 St+1 St − 1 = Vt+1∗ − Vt∗ St+1 St − 1 − Vt−1∗ St St−1 . (7)
Hence, at time t + 1 VA requires Vt+1∗ to be invested. Subtract the starting capital of two periods back, Vt−1∗ , and the return we made on it up to time t. Secondly, the proceeds on the investment at time t are subtracted to find Equation 7. This gives us the expectation:
E[Itv+ It+1v ] = Vt+1∗ + Vt∗ 1 − E St+1 St − Vt−1∗ E St St−1 = Vt+1∗ + Vt∗(1 − eµ) − Vt−1∗ eµ = E[Itv] + E[It−1v ],
and the variance:
Hence, in the Black-Scholes framework risk is not reduced if we consider the sum of consecutive investments. Lastly, considering the IRR, i solves the following equality:
T X t=1 Itve−it= WTve−iT, T X t=1 Vt∗− Vt−1∗ St St−1 e−it= VT∗e−iT, T −1 X t=1 Vt∗e−it= T X t=1 Vt−1∗ St St−1 e−it, T −1 X t=1 Vt∗e−it= e−i T −1 X t=1 Vt∗ St St−1 e−it,
where the last step follows from substituting t = t + 1 and V0 = 0. So we are looking for i that
solves: T −1 X t=1 Vt∗e−it e−i St St−1 − 1 = 0.
For the IRR of a T period VA strategy we are required to find the relevant root of a T − 1 order polynomial since VT∗ cancels out. The relevant root gives an real IRR between -1 and 1. Result 2 gives us a summary of our findings.
Result 2 The analysis of VA shows that for g > −µ if t → T , the expectation of investment decreases while the variance increases. For g large enough, i.e., close to or exceeding µ, the ex-pectation of investment increases as t → T . However, the variance also increases for increasing g. Furthermore, there is no time diversification effect when considering multi period risk of investment.
4.3 Value averaging with money account
VM requires us to deal with positive and negative parts which makes analysis though. We start with the periodic investment, Itvm. Define the event A as the event of investment, Vt∗−Vt−1∗ St
St−1−Mt−1>
0, and Ac as the event of a sell signal where the proceeds are invested in the money account. Then we can write: E[Itvm] = E " Vt∗− Vt−1∗ St St−1 − Mt−1 +# = E Vt∗− Vt−1∗ St St−1 − Mt−1 A P (A) + 0 = P (A) Vt∗− Vt−1∗ E St St−1 A − E [Mt−1] .
E[Mt−1] and P (A) depend on all previous Ms, s = 0, ..., t − 2. The dependency leads to
computa-tional difficulties due to the positive and negative parts. For an upper bound on E[Itvm] we assume that Mt−2= 0, a strong assumption for t → T . This yields us:
Next, evaluating each of the terms individually starting with: P (A|Mt−2= 0) = P Vt∗− Vt−1∗ St St−1 − Mt−1> 0 Mt−2= 0 = P Vt∗− Vt−1∗ St St−1 − Vt−1∗ − Vt−2∗ St−1 St−2 − > 0 ! = P Vt∗− Vt−1∗ St St−1 − Vt−1∗ − Vt−2∗ St−1 St−2 − > 0 Bt−1 ! P (Bt−1) + P Vt∗− Vt−1∗ St St−1 − Vt−1∗ − Vt−2∗ St−1 St−2 − > 0 Bt−1c ! P (Bct−1),
where Bt is the event, V
∗ t Vt−1∗ > St St−1, and B c
t its complement. We then have:
P (A|Mt−2= 0) = P Vt∗− Vt−1∗ St St−1 > 0 P (Bt−1) + P Vt∗− Vt−1∗ St St−1 + Vt−1∗ − Vt−2∗ St−1 St−2 > 0 P (Bt−1c ) = P (Bt) P (Bt−1) + P Vt∗+ Vt−1∗ − Vt−1∗ St St−1 − Vt−2∗ St−1 St−2 > 0 P (Bt−1c ) = P (Bt) P (Bt−1) + P Itv+ It−1v > 0 P (Bt−1c ).
Hence, we need to evaluate the sum of two lognormals which can be solved numerically. Furthermore, P (Bt) can be written as the inverse of the standard normal distribution function as follows:
P (Bt) = P Vt∗ Vt−1∗ > St St−1 = P Vt∗ Vt−1∗ > exp{(µ − 1 2σ 2) + σ(Z t− Zt−1)} = P Zt− Zt−1< 1 σ log V ∗ t Vt−1∗ − µ + 1 2σ 2 = N−1 1 σ log V ∗ t Vt−1∗ − µ + 1 2σ 2 ,
where N−1 denotes the inverse of the standard normal distribution function. Next, the expectation of St/St−1 under the event A and Mt−2= 0:
We now observe a sum of lognormal variables in the condition of the expectation, which can only be solved by numerical computation. For the second expectation we have:
E [Mt−1|Mt−2= 0] = E " Vt−1∗ − Vt−2∗ St−1 St−2 −# = E Vt−1∗ − Vt−2∗ St−1 St−2 P (Bt−1c ) + 0 = P (Bt−1c ) Vt−1∗ − Vt−2∗ eµ ,
which can be solved as discussed before. The assumption of Mt−3 = 0 is less restricting and will
give a tighter upper bound on E[Ivm
t ]. However, it requires to work with a sum of three lognormal
distributed variables and numeric evaluation of theses sums is not possible. Result 3 summarizes our results.
Result 3 For analytic analysis at time t one requires the assumption that Mt−2 = 0. This allows
for numerical exploration of the sum of two lognormal variables. If one would choose to set only Mt−3to zero one would obtain the sum of three lognormally distributed variables which gives numeric
5
Simulation
In this section we discuss the simulation from both the CSVJ and the DSVJ model. We use the procedures as presented by Broadie and Kaya (2006). The independence of the jump diffusion process allows to separate the simulation in two parts, the simulation of the Heston (1993) model and the jumps, a compounded Poisson process. Broadie and Kaya are only interested in the final value of the share price. However, as the performance of DCA, VA and VM are path dependent, quarterly observations are required. That results in the following time line, t = 1, ..., mT where T the target date in years from now and m is the number of moments per year, let u < t indicate time as well. We consider quarterly observations, therefore m = 4. We discuss the simulation of the CSVJ model first and then the DSVJ model.
During the simulations it is possible to create negative variances, whenever this occurs we abort the simulation and start a new one. Secondly, price paths in which the share price at any point has more than 10 folded or decimated are deemed very unlikely. We want to avoid that those simulation influence our analysis and discard these simulations as well.
5.1 Continuous stochastic volatility model with jumps in the share price
Broadie and Kaya (2006) give the following exact simulation algorithm:
Exact Simulation Algorithm for the SVJ Model:
1. Split the jump process from the SVJ model to obtain Heston’s model and simulate from the consequent Heston model.
a. Generate a sample from the distribution of Vt given Vu.
b. Generate a sample from the distribution ofRt
uVsds given Vt and Vu. c. Recover Rt u √ VsdZ (1) s .
d. Generate a sample from the distribution of St given
Rt uVsds and Rt u √ VsdZs(1).
2. Simulate the jumps and update the stock prices.
a. Generate N , a Poison random variable with mean (λT ) to determine the number of jumps in the time horizon.
b. Generate Jj ∼ dU [0, mT ]e for j = 1, ..., N , where d.e denotes the ceil operator and U
the uniform random variable. Then Jj denotes the period in which jump j occurs.
c. Generate the independent jump sizes εJj, for j = 1, ..., N as in Definition 3.
d. Find the adjusted stock price, ¯St= StQI(J,t)j=1 εJj, where I(J, t) is equal to the number
of Jj < t.
5.1.1 Step 1
Removing the jump process from the CSVJ model yields the following set of equations: dSt= (r − λ¯µ)Stdt + p VtSt h ρdZt(1)+p1 − ρ2dZ(2) t i , (8) dVt= κ(θ − Vt)dt + σ p VtdZt(2). (9)
Then St and Vt conditional on Su and Vu can be written as:
St= Suexp (r − λ¯µ)(t − u) − 1 2 Z t u Vsds + ρ Z t u p VsdZs(1)+ p 1 − ρ2 Z t u p VsdZs(2) , Vt= Vu+ κθ(t − u) − κ Z t u Vsds + σ Z t u p VSdZs(1), (10)
by applying Itˆo’s lemma, Theorem 1, to d log(St) and d log(Vt) as in equation 8 and 9, respectively.
Step 1a
According to Theorem 2 the distribution of Vt given Vu is equal to the product of a multiplicative
constant and the noncentral chisquare distribution, see Definition 7. According to Theorem 3 and Definition 7, for d > 1 one can sample from χ2nc[d, λ∗] by sampling from a standard normal random variable and a chisquare distribution. Hence,
Vt|Vu∼ σ2(1 − e−κ(t−u)) 4κ (Z + √ λ∗)2+ χ2[d − 1].
Furthermore, Theorem 4 allows us to sample if 0 < d ≤ 1. Sample n from a Poisson distribution with mean 12λ∗, then χ2nc[d, λ∗] ∼ χ[d + 2N ]|N = n for 0 < d ≤ 1.
Step 1b
In order to sample from Rt
uVsds|Vt, Vu, observe that we deal with the evaluation of a bridge from
Vu to Vt. We apply a Laplace transform to obtain its characteristic function and for convenience
we use Vu= V (u) notation in this step. Next, we rewrite the volatility process in Equation 10 as a
square root process with volatility equal to 2. We have: V (t) = V (0) + Z t 0 κ(θ − V (s))ds + Z t 0 σpV (s)dZs(2).
Focussing on the last term and dropping the irrelevant index of the Brownian motion: Z t 0 σpV (s)dZs= 2 Z t 0 σ 2 p V (s)dZs ∼ = 2 Z t 0 p V (s)dZσ 2s (in law) = 2 Z t 0 s V 4 σ2 σ2 4 s dZσ 2s.
The scaling property of a Brownian motion allows the second equality. Let u = σ2s/4, then du = (σ2/4)ds and the square root process can be written as:
Let Q(u) = V (4u/σ2), we have Q (t) = Q(0) + 4 σ2 Z t 0 κ [θ − Q (u)] du + 2 Z t 0 p Q (u)dZu.
Then for η = 4κθ/σ2 and ζ = −2κ/σ2, we obtain
Q(t) = Q(0) + Z t 0 [2ζQ(u) + η]du + 2 Z t 0 p Q(u)dZu. (11)
According to Definition 8, the infinitesimal generator equals 2xD2+ (2ζx + η)D, where D = dxd. We can apply Theorem 6 to the Laplace transform if we first rewrite Equation 11 such that ζ = 0. That is, remove the stochastic drift, 2ζQ(u)du, from Q(t). The Laplace transform gives us:
E exp −a Z t u V (s)ds V (u), V (t) = E exp −a Z t u Q σ 2 4 s ds Q σ 2 4 u , Q σ 2 4 t = E " exp ( −4a σ2 Z σ2t/4 σ2u/4 Q (s) ds ) Q σ 2 4 u , Q σ 2 4 t # = ˜ E h exp n −4a σ2 − ζ2 2 Rσ2t/4 σ2u/4Q (s) ds o Q σ2 4 u , Q σ2 4 t i ˜ E h exp n −ζ22 Rσ2t/4 σ2u/4Q (s) ds o Q σ2 4 u , Q σ2 4 t i
where the last equality follows according to the work of Pitman and Yor (1982) by applying Gir-sanov’s theorem, Theorem 5, with Us = exp
n
−ζ22 Rσ2t/4
σ2u/4Q (s) ds
o
. Next, we apply Theorem 6 to both the numerator with b = 2h(a)/σ2, and denominator with b = ζ, where h(a) = √2aσ2+ 4k.
Then rewriting in terms of V (·) yields: E exp −a Z t u V (s)ds V (u), V (t) =h(a)e −1 2 (h(a)−κ)(t−u)(1 − e−κ(t−u)) κ(1 − e−h(a)(t−u)) · exp ( V (u) + V (t) σ2 " κ(1 + e−κ(t−u)) 1 − e−κ(t−u) − h(a)(1 + e−h(a)(t−u)) 1 − e−h(a)(t−u) #) · In/2−1 4h(a) √
V (u)V (t)e−h(a)(t−u)/2 σ2(1−e−h(a)(t−u))
In/2−1
4κ√V (u)V (t)e−κ(t−u)/2
σ2(1−e−κ(t−u))
. (12)
We now have the characteristic function of the random variable V (u, t) ∼
Z t
u
Vsds|Vu, Vt.
That is, for a∗ = ia in Equation 12 we have:
Φ(a) = E exp −ia Z t u V (s)ds V (u), V (t) . (13)
Theorem 7 gives us a numerical approximation: F (x) ≈ hx π + 2 π ¯ N X j=1 sin hjx j Re[Φ(hj)],
where Re[.] defines the real part operator and h and ¯N are such that both e∗d(h) is and e∗T( ¯N ) are small. For 1 − F (u∗) < e∗d(h), h ≥ π/u∗. Broadie and Kaya (2006) advice u∗ equal to the mean plus five standard deviations, which both follow from the characteristic equation in Equation 13. Furthermore, we truncate the summation at j = ¯N iff 2|Φ(h ¯N )|/ ¯N π < e∗T( ¯N ) = 0.1. According to Abate and Whitt (1992) and Broadie and Kaya (2006) the consequent grid size, h, and truncation point, ¯N , are rather conservative.
Lastly, we apply the inverse transform method to obtain a sample from R0tVsds|Vu, Vt. That is,
we generate a uniform random variable U and then solve P (V (u, t) ≤ x) = U . We apply Hayley’s method as in Theorem 8. For the initial guess we use the normal inverse of U , with the same mean and variance as we use to obtain µ∗.
Step 1c
We recoverRut√VsdZs(1) from Step 1a and 1b. We have:
Z t u p VSdZs(1) = 1 σ Vt− Vu− κθ(t − u) + κ Z t u Vsds , since we assume the path of Vtgiven.
Step 1d
Next, we generate St. One can verify in Equation 4 that Vt and Zt(2) are independent. Therefore,
Rt
u
√
VsdZs(2) given the path of Vt is normal with mean 0 and variance
Rt
uVsds. Then:
log(St) = m(u, t) + σ(u, t)N (0, 1),
m(u, t) = log Su+ r(t − u) − 1 2 Z t u Vsds + ρ Z t u p VsdZs(1), σ(u, t) = s (1 − ρ2) Z t u Vsds. Step 2
In this step we describe the simulation from a compound Poisson process, in step 2a the number of jumps is simulated, step 2b gives the moment of the jump, step 2c the size of the jump and step 2d generates the share price process, ¯St for t = 1, ..., mT , of the CSVJ model.
5.2 Discrete stochastic volatility model with jumps in the share price
In the discrete model, we use the Euler approximation as in Equation 5 and 6. Broadie and Kaya (2006) partition the time interval into n equal segments of length ∆t = 1/n, then tj = j/n for
we in Step 2a set Jj ∼ dU [1, BmT ], 1/ne where d, 1/ne rounds up to a number that is a multiple of
1/n. We can apply step 1 and step 2 as discussed to obtain a sample of quarterly observations, ¯St
6
Results
In the methodology we want to report on different aspects of these strategies. Simulation from the CSVJ model is found to be time consuming, we use Matlab version 2016a for both simulation models. The supercomputer of the University of Groningen, Peregrine2, allowed to complete the simulations in time. We will first give the parameter settings of the models and discuss the main results after that. All tables that are referred to in this section can be found in Appendix C. Throughout, µ = 6.61% annually, see Table 1 and we assume g = 5% annually. Furthermore, final target wealth L = 10, 000, target date T = 5 years and m = 4 for quarterly investments. To give some idea on how time consuming the calculation are, the simulations of the CSVJ model would require approximately a month to complete on a normal computer.
Since we use these assumptions for both the historical evaluation and the simulations, we first comment on the parts that are fixed for both the evaluation and simulation. We find C = 374.04 and Cd = 437.36 and Table 4 presents Vt∗ and Itd. Furthermore, total discounted costs of DCA
equal:
mT
X
j=1
Ijde−µj/M = 8387.53.
Next, we provide the simulation settings and describe the resulting share price processes. After that we provide the comparison of the performance measures.
6.1 Simulation settings
Due to extensive computation time required we will use two sets of parameters for our models, those estimated by Duffie et al. (2000) and Eraker (2004). A more elaborate parameter sensitivity analysis could be useful, but is not possible within the time frame of this research. Broadie and Kaya (2006) use the parameters of Duffie et al. (2000), who estimates the parameters in a study, which includes about a hundred option prices for different maturities at November 2, 1993. They give us the following annualized parameters:
µ = 0.0451 ρ = −0.79 λ = 0.11 µs= −0.1391 σs= 0.15
κ = 3.99 θ = 0.014 σ = 0.27 V0 = 0.014.
Jessen (2010) uses the parameters as estimated by Eraker (2004), who uses daily prices of the S&P 500 from 1987 till 1990 and 22,500 option prices over this period. Jessen (2010) gives us the following annualized parameters:
µ = 0.066 ρ = −0.586 λ = 0.504 µs= −0.004 σs= 0.066
κ = 4.788 θ = 0.042 σ = 0.512 V0= 0.042.
Based on these parameters we generate 10,000 simulations for the 5 year period. For the DSVJ model we use steps of size 0.01 day and assume 250 trading days per year, step sizes ranging from 1 day to 0.0001 day do not influence the distribution function of the log returns, only the fraction of simulations that have to be discarded reduces as the step size decreases.
In Figure 2 the empirical distribution of the quarterly log returns are presented based on the 224 historical observations and 200,000 observations of the simulation models. Table 2 presents the
sample average, sample error, skewness and kurtosis for each of the models, for the CSVJ model daily observations are not calculated due to the limited available time. The CSVJ model yields quarterly log returns more volatile than the historical data and based on Table 2 the DSVJ model with Jessen parameters fits best. Figure 2 confirms these observations. Lastly, Figure 3 shows the empirical distribution of the share price after 5 years. First, one can observe a log shaped line for the CSVJ model and a S shape for the DSVJ model and the historical data. In this paper we will not investigate this difference in shape, we leave this open for further research.
Result 4 The DSVJ model is not influenced by step sizes ranging from one per day till 10000 per day. The CSVJ model is computational intense and we observe a different shape when we consider the distribution of the share price in 5 years. Also the distribution of quarterly returns has a higher spread then in the DSVJ model and the historical data.
Figure 2: Empirical distribution of the quarterly log returns
Table 2: Summary on the log returns Historical
Period average sample error skewness kurtosis day 0.0003 0.0101 -0.9991 29.8694 quarter 0.0163 0.0782 -0.7218 5.6030
year 0.0661 0.1405 -0.5669 2.8921 CSVJ Jessen
Period average sample error skewness kurtosis quarter 0.0016 0.2200 -2.5606 17.4843 year 0.0064 0.4334 -1.2900 6.1940
CSVJ Broadie
Period average sample error skewness kurtosis quarter -0.0037 0.1685 -3.1954 25.4421 year -0.0149 0.3348 -1.6485 8.2003
DSVJ Jessen
Period average sample error skewness kurtosis day 0.0002 0.0133 -0.1701 8.4356 quarter 0.0105 0.1063 -0.7544 4.8476 year 0.0424 0.2151 -0.7141 4.2280
DSVJ Broadie
Period average sample error skewness kurtosis day 0.0001 0.0087 -8.4675 366.4135 quarter 0.0053 0.0686 -1.6794 9.7863
year 0.0214 0.1392 -1.2088 5.6242
6.2 Comparison of DCA, VA and VM
Table 5 shows the down side risk of DCA and also the multiplicative constant αd such that if we
multiply Itd by αd then P (WTd < L) < 0.05. If this downside risk is acceptable it would raise the
Figure 3: Empirical distribution of S5.
parameters DCA has lower discounted costs than VA and VM. However, of these cases where DCA has the lowest discounted costs in only four of all our simulations and evaluations WTd> L.
Result 2 states that according to the Black-Scholes model there are no time diversification effects. We compare the sum of periodic payments per year to find that for both VA and VM the same average payment but a lower sample error for the sum of payments. For example consider year five of the VM strategy for which the annual payments give an average of 1905.18 with a sample error of 1287.03. Summing the individual payments yields an average of 1903.40 and a sample error of 1559.01, which is a lot higher. For the simulation models we observer the same effects.
underfunding do not differ much. However, the expected shortfall is a bit smaller for VA and VM. Furthermore, when comparing to Table 8 the average IRR is only a bit smaller than those of an investor with unlimited funds, while the sample errors of the IRR are in the same range for each of the datasets. We summarize the results of limiting the investment capabilities of the investor in Result 5.
Result 5 An investor with limited capability to invest large amounts per period and per rolling year faces the risk of underfunding when investing in VA or VM. However, this risk is no larger than when investing in DCA, Furthermore the expected shortfall is slightly reduced while the costs of investing measured by the internal rate of return is significantly higher for VA and VM. In addition the money account does not give any real benefits except for slightly reducing the probability of underfunding.
Lastly, we consider a the time horizon of twenty instead of five years. Table 12 presents us the main results for the historical simulations and the DSVJ model with both Broadie and Jessen parameters. The time horizon is increased by a factor four, therefore we tighten the limits of investor A and B and divide them by four which gives: 625 and 375 per period and 1250 and 750 per year, respectively. We observe quite different results compared to the five year period. The risk of underfunding is quite severe for VA and VM while zero for DCA in the historical evaluation. In the simulation models the probability of underfunding is also 10 to 20 percentage points higher than in the historical evaluation. However, the expected shortfall is small compared to the decrease in expected discounted costs for VA and VM, certainly when considering the discounted expected shortfall. In addition in the simulations, the expected shortfall of DCA is even a bit larger than that of investor B following VA or VM.
Result 6 For the longer time horizon we find considerably increased probabilities of underfunding for both VA and VM as compared to DCA. However, the expected shortfall is limited to the reduction in discounted costs. In addition the money account in VM provides a reduction in the probability of underfunding for small increased discounted costs.
Investment advice for the small investor
7
Conclusion
We examine three investment strategies: DCA, VA and VM, that are designed for the needs of small investors. They provide the investor with a clear set of rules which dictate the number of shares to buy or sell. These rule based strategies are attractive from a behavioural economics perspective but deemed irrational from the neo-classical finance perspective. We use historical evaluation of the S&P 500 and Treasury bill rates as well as simulation from the CSVJ and the DSVJ. Due to computational constraints we use two parameter sets, one as Broadie and Kaya (2006) use and the other from Jessen (2010).
Our analytical exploration of a relative simple model shows the complexity of these strategies and the need for exploration numerically or by simulation, Result 1, 2 and 3. The simple model leads us to Result 2, the sum of consecutive periods does not reduce the risk of the costs over these periods. This does not apply for the historical evaluation nor for the more advanced models we use for our simulation.
The DSVJ model and CSVJ model give quite different distributions for the share price in five years. While the shape for the DSVJ model follows the historical data the CSVJ model presents a log shaped distribution (Result 4). We did not examine this difference and leave this for further research. Hayley (2010) criticizes the internal rate of return since it assumes that the investor is able to invest any amount required in every time period. For small investors and households this critiscm holds true. Therefore, we consider VA and VM in which the investments are limited per period and per rolling year. Result 5 summarizes that the IRR is slightly reduced and for a time horizon of five years the probability of underfunding is no larger than for DCA, while the expected shortfall is smaller.
Result 6 shows that the probability of underfunding increases considerable with a time horizon of twenty years. However, the expected shortfall is small when we compare with the reduction in total discounted costs. Hence, VA and VM can still be attractive. To conclude on this we recommend further research to investigate the following to consider: What would be the implications of a raise in the value path on the probability of underfunding, expected shortfall and expected discounted costs in this setting? Another twist could be to consider that in the VM strategy the minimum or target value of the money account is increased from zero to for example five to ten percent of L. This should decrease the probability of underfunding while we expect the IRR still to outperform DCA.
The money account provides no real benefit in the five year horizon (Result 5). However, for the twenty year investment horizon the money account reduces the probability of underfunding. A more sensitive analysis into the investment horizon for which the money account provides benefits to the investor could be interesting for households and small investors.
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Appendix A: Examples of the strategies
Table 3: Example of the strategies in different market environments Declining Market
VA DCA
Period Price Value req. Shares req. Shares bought Invest Invest Shares bought
1 20 600 30 30 600 600 30 2 15 1200 80 50 750 600 40 3 12 1800 150 70 840 600 50 4 10 2400 240 90 900 600 60 Total 240 3090 2400 180 Final wealth 2400 1800
Average Price 14.25 Average Cost 12.88 Average Cost 13.33
IRR -18.73% IRR -18.95%
Rising Market
VA DCA
Period Price Value req. Shares req. Shares bought Invest Invest Shares bought
1 8 600 75 75 600 600 75 2 10 1200 120 45 450 600 60 3 12 1800 150 30 360 600 50 4 20 2400 120 (30) (600) 600 30 Total 120 810 2400 215 Final wealth 2400 4300
Average Price 12.50 Average Cost 6.75 Average Cost 11.16
IRR 39.37% IRR 40.66%
Fluctuating Market
VA DCA
Period Price Value req. Shares req. Shares bought Invest Invest Shares bought
1 12 600 50 50 600 600 50 2 10 1200 120 70 700 600 60 3 15 1800 120 0 0 600 40 4 12 2400 200 80 960 600 50 Total 200 2260 2400 200 Final wealth 2400 2400
Average Price 12.25 Average Cost 11.3 Average Cost 12
IRR 4.23% IRR 0.00%
Appendix B: Theorems and definitions Section 5
Theorem 2 According to Cox (1985) the square root process, dXt= κ(θ − Xt)dt + σ p XtdZt, implies Xt|Xu∼ σ2(1 − e−κ(t−u)) 4κ χ 2 nc " 4κθ σ2 , 4κXue−κ(t−u) σ2(1 − e−κ(t−u)) # , Hence, Xt given Xu is distributed as a constant, σ
2(1−e−κ(t−u))
4κ , times the noncentral chisquare
distribution with 4κθσ2 degrees of freedom and 4κXue −κ(t−u)
σ2(1−e−κ(t−u)) the noncentrality parameter.
Theorem 3 According to Johnson et al. (1994), for d > 1, the χ2nc can be written as: χ2nc[d, λ] = χ2nc[1, λ] + χ2[d − 1]
Theorem 4 According to Broadie and Kaya (2006) for 0 < d ≤ 1 and N a Poisson random variable with mean 12λ:
χ2nc[d, λ] = χ2[d + 2N ]
Theorem 5 Girsanov (1960), for process dXt= µ(t, x)dt + σ(t, x)dZt under probability measure P
and equivalent probability measure ˜P and filtration F . We have: E[f (t, x)|F ] = E[f (t, x)U˜ t|F ]
˜ E[Ut|F ]
where Ut is the Radon Nikodym derivative, dP/d ˜P . Furthermore, E ( ˜E) is the expectation under
probability measure P ( ˜P ), respectively.
Theorem 6 Pitman and Yor (1982) give us for a square Bessel process, X(s), with infinitesimal generator 2xdxd22 + η d dx: ¯ E exp −b 2 2 Z t 0 X(s)ds X(0) = x, X(t) = y = 2bt ebt− e−bt exp x + y 2t 1 − bte bt+ e−bt ebt− e−bt Iν 2√xyb ebt− e−bt Iν √xy t ,
where ν = η/2 − 1, Iν denotes the modified bessel function of the first kind, and ¯E the expectation
with respect to the law of the squared Bessel process.
Theorem 7 Abate and Whitt (1992) give us the trapezoidal rule for the distribution, F (x) with characteristic equation Φ(u):
F (x) = hx π + 2 π ¯ N X j=1 sin hjx j Re[Φ(hj)] − ed(h) − eT( ¯N ),
where h is the grid size, ed(h) the discretization error and eT( ¯N ) the truncation error. Abate and
Theorem 8 Hayley’s method for solving f (x) = 0 gives us: xn+1= xn− 2f (xn)f0(xn) 2f02(x n) − f (xn)f00(xn) , with an initial guess x0.
Definition 7 let χ2[d] define the chisquare distribution, where d is the degrees of freedom. Next, χ2nc[d, λ] defines the noncentral chisquare distribution, where d is the degrees of freedom and λ the noncentrality parameter. For d = 1,
χ2nc[1, λ] = (Z +√λ)2,
where Z ∼ N (0, 1), standard normally distributed.
Definition 8 The infinitesimal generator of process X(t) = µ(t)dt + σ(t)dZt equals
Appendix C: Tables Section 6
Table 4: Overview of the value path and the DCA investments per period t Vt∗ Itd 1 379.51 442.86 2 770.11 448.43 3 1172.05 454.07 4 1585.58 459.79 5 2010.94 465.57 6 2448.41 471.43 7 2898.23 477.36 8 3360.68 483.36 9 3836.04 489.44 10 4324.57 495.60 11 4826.57 501.83 12 5342.32 508.14 13 5872.11 514.53 14 6416.26 521.01 15 6975.06 527.56 16 7548.82 534.20 17 8137.87 540.91 18 8742.53 547.72 19 9363.13 554.61 20 10000.00 561.58
Table 5: Risk measures for DCA and VM Data P (WTd< L) E[WTd|Wd T < L] E[WTd|WTd≥ L] αd Historical 0.17 8901.50 12580.64 1.21 CSVJ Jessen 0.41 7172.57 15608.99 2.15 CSVJ Broadie 0.45 7645.14 13191.32 1.87 DSVJ Jessen 0.31 8430.36 13014.88 1.40 DSVJ Broadie 0.33 8823.38 11633.30 1.28
The probability is approximated by the fraction of WTd< L, the expected shortfall is approximate by the average of
Wd
Table 6: Summary on the total discounted costs
Strategy minimum q0.10 average q0.90 maximum sample error
Historical: VA 3737.28 5341.45 7002.84 8506.14 10644.52 1334.74 VM 4331.14 5540.44 7143.90 8611.72 10644.52 1284.07 CSVJ Jessen: VA -9260.31 2909.90 6819.17 10898.02 17488.00 3153.78 VM 1530.15 4328.87 7611.66 11232.31 17488.77 2656.88 CSVJ Broadie: VA -1183.88 5021.87 7794.76 10914.04 17426.62 2343.09 VM 2358.35 5445.02 8058.82 11028.39 17708.94 2193.11 DSVJ Jessen: VA 1252.08 4996.94 7226.58 9603.75 15905.07 1833.83 VM 2922.51 5400.30 7473.74 9718.30 15906.00 1711.26 DSVJ Broadie: VA 3340.08 6439.86 7816.83 9362.77 13864.62 1170.02 VM 4545.79 6476.86 7847.64 9388.41 13864.62 1159.89
The total discounted costs of DCA are fixed and equal 8387.53. qα is the α percent quantile of the total discounted
costs.
Table 7: Summary on the final wealth distribution
Strategy minimum q0.10 average q0.90 maximum sample error
Historical: DCA 6933.83 9464.00 11949.42 14729.74 18552.91 2250.18 VM 10000.00 10000.00 10189.76 10618.47 12427.72 431.50 CSVJ Jessen: DCA 1431.78 5730.67 12086.78 19567.78 63038.63 5751.49 VM 10000.00 10000.00 11076.56 13190.91 28078.30 1603.09 CSVJ Broadie: DCA 1529.91 6355.47 10692.21 15223.24 34245.82 3567.60 VM 10000.00 10000.00 10359.33 11190.22 19583.57 735.77 DSVJ Jessen: DCA 3502.11 8051.66 11593.68 15404.09 26039.30 2939.33 VM 10000.00 10000.00 10336.80 11165.15 15163.99 597.54 DSVJ Broadie: DCA 4471.07 8485.28 10701.81 12842.29 19104.10 1694.67 VM 10000.00 10000.00 10042.13 10128.69 13171.13 147.15
Table 8: Summary on the annualized IRR in percentages per strategy Strategy minimum q0.10 average q0.90 maximum sample error
Historical: DCA -17.50 -2.45 6.59 15.90 24.48 7.93 VA -21.27 -1.67 7.26 16.45 25.44 8.23 VM -21.27 -1.58 7.21 16.36 24.93 8.15 CSVJ Jessen: DCA -194.17 -27.99 1.14 26.39 63.57 23.26 VA -420.40 -19.25 5.91 29.30 112.09 23.95 VM -333.74 -18.41 5.58 27.35 53.44 22.10 CSVJ Broadie: DCA -177.93 -22.15 -0.95 17.16 45.12 17.37 VA -457.91 -17.79 1.54 18.44 45.74 17.28 VM -316.2 -17.66 1.51 18.11 40.40 16.65 DSVJ Jessen: DCA -62.28 -9.99 4.42 17.61 36.22 11.21 VA -86.24 -8.56 5.73 18.59 37.51 11.27 VM -85.81 -8.48 5.65 18.28 36.21 11.10 DSVJ Broadie: DCA -43.68 -7.49 2.14 10.51 25.54 7.35 VA -54.79 -6.75 2.66 10.81 25.43 7.30 VM -54.79 -6.73 2.66 10.8 24.67 7.28
Table 9: Costs per year for VA
t minimum q0.10 average q0.90 maximum sample error
Historical: 1 1278.99 1402.29 1541.47 1690.41 1933.30 118.09 2 812.77 1171.16 1608.24 2116.06 2853.75 371.96 3 100.50 915.50 1658.00 2585.30 3877.32 666.37 4 -710.41 596.84 1718.79 3093.52 5013.05 996.21 5 -1618.62 236.83 1809.68 3653.24 6270.69 1365.51 CSVJ Jessen: 1 -706.5 1245.04 1528.52 1870.36 2901.15 269.07 2 -3148.34 650.86 1568.47 2628.87 5110.95 824.67 3 -6598.84 -91.24 1624.02 3599.52 7955.78 1515.33 4 -8705.86 -918.78 1638.63 4568.94 12849.72 2265.49 5 -17398.7 -1784.36 1749.24 5943.26 16020.16 3200.84 CSVJ Broadie: 1 685.38 1357.67 1564.32 1826.26 2712.27 206.82 2 -1813.15 1060.52 1703.52 2478.71 5046.33 605.24 3 -3545.39 654.49 1846.40 3287.95 7249.93 1099.50 4 -5689.75 250.65 2014.03 4158.31 11071.95 1653.80 5 -8773.97 -231.75 2209.58 5318.62 15030.08 2309.41 DSVJ Jessen: 1 849.46 1371.38 1548.68 1746.49 2295.70 152.46 2 -23.56 1059.27 1630.88 2252.58 3940.55 477.60 3 -1193.50 682.53 1716.50 2841.10 6521.48 859.32 4 -2380.39 274.80 1817.91 3488.25 8932.36 1291.34 5 -476z6.43 -193.12 1904.16 4136.23 10414.60 1744.20 DSVJ Broadie: 1 1130.21 1466.32 1567.39 1690.31 2157.70 95.77 2 48.18 1381.92 1700.36 2092.80 3778.41 297.83 3 -187.31 1265.54 1860.94 2584.65 5846.06 546.48 4 -1322.83 1141.44 2030.45 3111.90 7224.89 824.23 5 -1789.05 989.66 2206.75 3663.10 8926.05 1099.87
Table 10: Costs per year for VM
t minimum q0.10 average q0.90 maximum sample error
Historical: 1 1278.99 1402.29 1541.47 1690.41 1933.30 118.09 2 811.55 1185.04 1613.44 2116.06 2853.75 368.47 3 90.84 944.48 1684.41 2585.30 3877.32 651.24 4 0 675.16 1775.09 3093.52 5013.05 957.02 5 0 427.15 1905.18 3631.76 6270.69 1287.03 CSVJ Jessen: 1 417.41 1249.91 1534.19 1870.36 2901.15 255.79 2 0 780.54 1631.28 2638.01 5110.95 746.29 3 0 333.50 1800.84 3598.25 7955.78 1301.29 4 0 0 1957.04 4538.08 12849.72 1834.44 5 0 0 2202.66 5705.78 15682.55 2467.38 CSVJ Broadie: 1 858.91 1358.25 1565.21 1826.26 2712.27 204.83 2 0 1092.57 1719.76 2480.79 5046.33 584.84 3 0 774.38 1903.20 3295.29 7249.93 1035.82 4 0 482.75 2118.36 4160.99 10864.09 1521.31 5 0 88.22 2372.2 5250.15 15239.47 2077.31 DSVJ Jessen: 1 888.96 1371.58 1548.78 1746.49 2295.70 152.20 2 119.82 1082.82 1641.16 2254.20 3940.55 467.82 3 0 806.18 1769.64 2849.98 6517.91 813.62 4 0 502.20 1914.76 3502.77 8932.36 1188.22 5 0 206.01 2064.3 4092.64 10414.6 1526.65 DSVJ Broadie: 1 1130.21 1466.32 1567.39 1690.31 2157.70 95.77 2 814.81 1382.69 1701.01 2093.80 3778.41 296.63 3 0 1268.84 1864.50 2585.50 5846.06 544.13 4 0 1155.79 2043.40 3123.70 7224.89 817.6 5 0 1032.53 2230.04 3665.63 8926.05 1084.06
Table 11: Implications limited investment VA, Investor A
Data P (WTv < L) E[WTv|Wv
T < L] average IRR s.e. IRR
Historical 0.02 8920.00 7.25 8.24 CSVJ Jessen 0.22 8086.17 5.36 24.33 CSVJ Broadie 0.16 8475.96 1.23 17.40 DSVJ Jessen 0.07 9153.89 5.69 11.31 DSVJ Broadie 0.03 9252.51 2.65 7.33 VA, Investor B Data P (WTv < L) E[WTv|Wv
T < L] average IRR s.e. IRR
Historical 0.18 8868.00 7.15 8.20 CSVJ Jessen 0.46 7647.77 4.62 24.08 CSVJ Broadie 0.42 8103.09 0.74 17.42 DSVJ Jessen 0.33 8833.52 5.51 11.37 DSVJ Broadie 0.25 9140.08 2.56 7.39 VM, Investor A Data P (WTvm< L) E[WTvm|Wvm
T < L] average IRR s.e. IRR
Historical 0.02 8955.88 7.21 8.17 CSVJ Jessen 0.17 8089.65 5.05 22.53 CSVJ Broadie 0.14 8463.12 1.18 17.14 DSVJ Jessen 0.06 9149.58 5.61 11.15 DSVJ Broadie 0.03 9235.94 2.65 7.32 VM, Investor B Data P (WTvm< L) E[WTvm|Wvm
T < L] average IRR s.e. IRR
Historical 0.18 8893.08 7.11 8.14
CSVJ Jessen 0.37 7624.82 4.23 22.85
CSVJ Broadie 0.39 8088.08 0.67 17.33
DSVJ Jessen 0.29 8815.11 5.43 11.24
DSVJ Broadie 0.25 9141.61 2.56 7.38
The probability is approximated by the fraction of WTs< L, the expected shortfall is approximate by the average of
Ws
Table 12: Time horizon adjusted to 20 years Historical P (WTv < L) E[WTv|Wv
T < L] s.a. IRR s.e. IRR E[
PT t=1e −µtIs t] s.e.( PT t=1e −µtIs t) DCA 0 - 7.61 3.26 4934.84 -VA 0 - 8.55 3.30 2061.09 930.84 VA - A 0.33 8759.99 8.46 3.24 1993.24 898.25 VA - B 0.50 8461.47 8.36 3.28 1930.34 892.63 VM 0 - 8.22 3.06 2617.51 799.84 VM - A 0.14 8824.42 8.20 3.06 2471.02 692.43 VM - B 0.26 8851.69 8.12 3.13 2369.39 676.65 DSVJ Jessen P (WTv < L) E[WTv|Wv
T < L] s.a. IRR s.e. IRR E[
PT t=1e−µtIts] s.e.( PT t=1e−µtIts) DCA 0.22 7687.63 3.55 5.31 4934.84 -VA 0 - 4.90 5.76 2957.45 1327.06 VA - A 0.57 8374.22 4.78 5.59 2707.67 1124.82 VA - B 0.76 7773.09 4.65 5.51 2496.52 975.31 VM 0 - 4.80 5.20 3489.22 1111.82 VM - A 0.31 8333.34 4.64 5.25 3224.95 911.07 VM - B 0.49 7758.38 4.48 5.28 2971.60 754.20 DSVJ Broadie P (WTv < L) E[WTv|Wv
T < L] s.a. IRR s.e. IRR E[
PT t=1e −µtIs t] s.e.( PT t=1e −µtIs t) DCA 0.25 8184.44 2.21 3.78 4934.84 -VA 0 - 2.73 3.91 3659.81 931.47 VA - A 0.47 8819.20 2.66 3.93 3493.42 805.07 VA - B 0.74 8290.17 2.58 3.90 3272.19 666.06 VM 0 - 2.77 3.80 3803.50 890.31 VM - A 0.36 8772.02 2.68 3.87 3623.76 758.03 VM - B 0.64 8272.90 2.59 3.88 3384.50 614.14
The probability is approximated by the fraction of WTs < L, the expection is approximate by the average, s.a.
