2.1 Individual optimization problem of the agents

Next, we formulate the individual optimization problems of the agents. We suppose that there are I agents who have log utility on K discrete-time consumption with heterogeneous views on the Brownian motion W.

Let and be and valued -progressively measurable processes satisfying , , , representing the position of the agent i on the N + 1 securities and the money market, respectively. Also, let be the wealth process of agent i, i.e., the total value of agent i’ portfolio. In detail, agent i invests of the wealth in the N + 1 securities and the rest in the money market account and continuously balances its position. That is,

where is an N + 1 dimensional column vector whose elements are 1.

In addition to the investments on N + 1 securities, at K discrete times , agent i consumes where are - measurable nonnegative random variables satisfying .

We consider the following admissible sets for the consumption so that agent i’ wealth is always nonnegative, i.e.,

(3)

Namely, noting that the wealth process satisfies the following SDE,

(4)

where

(5)

we assume the following set of consumption processes

(6)

where the probability density is of the form

where is an -valued -progressively measurable process satisfying a weak version of Novikov’s condition (e.g., Corollary 3.5.14 in Karatzas and Shreve [12]) with and satisfying

(7)

Remark 1. The above admissibility of the consumption process derives from the condition in (3) where the agent i does not go bankrupt, i.e., its wealth is always nonnegative. Since

multiplying both sides by the state price density for t = T, applying Ito’s formula, and taking expectation yields

(8)

In detail, noting that

Therefore

(9)

Since is a local martingale bounded from below, thus a supermartingale,

we have

Remark 2. In this study, we discretely set the consumption timing to align with the timing of the dividends paid, where the dividends are assumed to be paid as consumption goods. In particular, we consider the clearing of consumption goods at each discrete time, which indicates that the total dividend amount is consumed by the agents at each time. The matched timing between consumption and dividends and the clearing condition make it possible to obtain the state price density process in equilibrium.

For dividend processes, we model them as discrete-time processes to represent the feature of periodically paid dividends. The dividend reflects supply change of the securities such as outstanding notional changes by purchase of government bonds by the central bank and redemption, and new issuance of the bonds by the government at planned timings.

For the utility function, we assume that each agent i has the following log utility on consumption with subjective belief on the Brownian motion W with heterogeneous time preferences .

In detail, the agent maximizes the sum of its expected utility on the discrete consumption with discounting for time preference , where . The probability density for subjective belief of agent i, is defined as

where is an -valued -progressively measurable process satisfying a weak version of Novikov’s condition (see, e.g., Corollary 3.5.14 in Karatzas and Shreve [12]). Here, represents the subjective views of the agent i on the Brownian motion W. Namely, for the probability measure defined as

(10)

by Girsanov’s theorem, defined as is a -Brownian motion and indicates agent i’ bias on the Brownian motion W.

For related studies dealing with subjective views by change of probability measure and their estimation, Nakatani et al. [13] estimated the market sentiments represented by a way of Girsanov’s measure transformation in the Japanese government bond market.

Also, Kizaki et al. [4] further dealt with subjective views in the form of Girsanov’s transformation of the probability measure as the inf-sup and sup-sup problem, where the conservative view is expressed as taking infimum on the objective function with respect to the change of the probability measure, and the aggressive view is expressed as taking supremum instead. For example, in Kizaki et al. [4], is considered for the conservative sentiment of the agent, while is considered for the aggressive sentiment.

The alternative approach is a robust control for portfolio optimization (e.g., Hansen and Sargent [14]) in which the belief is considered as a model uncertainty and the minimum is taken to be conservative about the model risk.

Then, we describe individual optimization problem of the agent i as follows.

1. Individual optimization problem

Maximize

(11)

with respect to , where is defined in (6).

This individual optimization problem indicates that the agent i aims to maximize its expected utility in its consumption at discrete times with the time preference by choosing the consumption amount while continuously trading N + 1 securities and the money market account.

Remark 3. Although we may adopt exponential utility or power utility, in this study, we adopt log utility for the following reasons. Firstly, in individual optimization, because the problem is in an incomplete market setting, we take infimum concerning the possible state price density processes parametrized with orthogonal to the volatility vectors of the underlying asset price processes, while taking supremum on the consumption and trading strategies on expected utility. This infimum part of the individual optimization in an incomplete market setting is solved in the case of the log-utility.

While it remains unsolved in the power utility and exponential utility cases, in the log-utility case, thanks to the explicit expression of the consumption process for the individual agent after solving the individual optimization problem in an incomplete market setting.

Moreover, by incorporating the views on fundamental risks for the agents in the log utility case, we can obtain the equilibrium state price density process that reflects the agents’ views on the Brownian motion.

2.2 Optimal consumption for the individual optimization

This section provides an optimal consumption process for this individual optimization problem (11). In the following, we consider the following primal problem for the individual optimization.

(12)

To solve this primal problem, we first solve the dual problem and later confirm that the obtained solution is optimal.

Remark 4. The primal optimization problem describes the individual optimization problem (11) for the following reasons. If the given does not satisfy the budget constraint in (6) for some , the inf part is by taking any large value. Thus, the sup part indicates that the sup is taken with respect to in in (6). Hence, the primal problem expresses the individual optimization problem (11) where taking supremum on the expected utility with respect to satisfying the budget constraint in (6). The dual problem is described and solved as follows.

Dual problem

(13)

Proposition 1. , , set as

attain the inf-sup dual problem (13), where

(14)

(15)

Here, is an orthogonal part of to the linear space spanned by , i.e., , .

Proof. See Appendix A.1. □

Then, by a convex duality approach, we confirm that the solution obtained , , and , of the dual problem (13) is also a solution of the primal problem (12) as follows.

Theorem 1. The solution of the dual problem (13), , , obtained as

(16)

also attains the sup-inf of the primal problem (12).

Proof. See Appendix A.2. □

2.3 Optimal wealth and portfolio processes of the agents

Finally, for the optimal consumption obtained as a solution to the primal problem, the corresponding portfolio process along with the optimal wealth, which is nonnegative, is obtained as follows.

Theorem 2. Under the assumption that for , the optimal wealth process and the portfolio process corresponding to the optimal consumption in (16), i.e., that generates wealth satisfying the condition of the nonnegative wealth process in (3), for the individual optimization problem (11) are given as follows.

(17)

(18)

and

(19)

Proof. See Appendix A.3.□

Remark 5. First, in (17) corresponding to in (16), (18) and (19) satisfies Since is the optimal consumption in the admissible set , in which all the consumption processes in (4), (5) with nonnegative condition (3) are included as shown in Remark 1, it follows that is the optimal consumption and portfolio among triplets for a problem subject to (4), (5), and the nonnegative wealth condition (3). In other words, we solve the individual optimization of the candidates in the admissible set , which is a larger set that includes the consumption and portfolio processes with which the wealth process never goes bankrupt. Thus, the optimal solution is found in the broader set of consumption and portfolio processes. Then we confirm that with the optimal solution found in the broader set of consumption and portfolio processes, the wealth process is positive, that is, based on the expression of the optimal wealth process.

Thus, the obtained consumption and portfolio processes are optimal among the initial intended set, i.e., the strategies with which the wealth process is always nonnegative.

3.1 State-price density process and yield curves in equilibrium

First, we obtain the following expressions for the interest rate r and the density process , which represents the common part of the risk-neutral probability measure and incorporates the market price of risk in equilibrium.

Theorem 3. In equilibrium, in (15) is expressed as

for and for ,

(21)

Also the associated state price density process H is given by

(22)

Moreover, supposing that the interest rate process r is piece-wise -measurable random variable between the discrete times , r is expressed as

(23)

By expression of the state price density process (22), the price of the zero coupon bond and the yield for maturity at t = 0 are described as follows.

(24)

As (24) indicates, the yield curve shifts in accordance with the total dividend. In detail, as the total dividend increases, which is equivalent to supply of consumption goods by the market clearing condition, through the expression of the state-price density process, the price of the zero-coupon bond decreases and the yield increases and vice versa.

3.2 Equilibrium between market value processes and wealth processes

Next, we present the equilibrium expression for the market value of security j.

Proposition 2. The market value associated with the dividend process in equilibrium is expressed as

(25)

Proof. See Appendix A.5.

In the following, we show that the total wealth of the I agents is equal to the total of the market value of the N + 1 securities in equilibrium. We denote the total market value of N + 1 securities by and the aggregate wealth process of agents I by X, that is,

and

Proposition 3. The total of the market values of N + 1 securities in equilibrium has the following expression

(26)

where

In particular, at ,

Moreover, , the total of the market values of N + 1 securities, equals X, the aggregate wealth process of agents I.

Proof. See Appendix A.6. □

Moreover, the market clearing condition also derives the clearing equations for each security and the money market account as follows.

Proposition 4. Suppose , in (1) are linearly independent vectors. Then, the market clearing for every risky asset

and the market clearing for the money market account , hold.

Proof. See Appendix A.7. □

3.3 Remark on recursive structure of volatility processes of the securities

Finally, we remark on the recursive structure in the expression of in (25).

The j-th market value process in (25) is expressed with in (14) that includes , the orthogonal projection of on the space spanned by . Since , , represent the volatilities of the security prices , the pricing expression (25) exhibits a recursive structure, and we must specify in a manner consistent with this recursion when simulating the model.

We note that this recursive structure in (25) arises from the incorporation of differences in the agents’ views on the fundamental risks represented by Brownian motions, which is associated with some factors that cannot be hedged with these N + 1 securities. This incorporation of heterogeneous views provides the model with an incomplete market feature, leading to different state price density processes of agents in their individual optimization problems in (11).

In the following, we provide two particular cases to specify in which the relation (25) holds.

First, when is given as a realization of a factor process y, that is, , assuming that is proportional to its volatility , we can verify that as in the following.

Proposition 5. Suppose that are given as a realization of a factor process y satisfying

That is, . If is proportional to , that is,

(27)

where is a -valued -progressively measurable process, then .

Proof. See Appendix A.8.

Remark 6. We note that the volatility processes is that of in (25), which has the expression that includes and , and, in general, as long as these random variables are specified, the volatility of is obtained.

In this case, with the explicit expression of , we can easily check whether are linearly independent, which is required for the clearing condition of the market for securities and the money market account, as we observed in Proposition 4.

Remark 7. If is specified as as in (27), , are obtained via (25). Then, is decomposed as , where .

Next, when is the realization of a factor process , and is given as a linear combination of , we have the following expression of and confirm .

Proposition 6. Suppose that is the realization of a factor process that satisfies the following SDEs. That is, , , and

If is a linear combination of , then is also a linear combination of as follows.

(28)

where , .

Moreover, if are linearly independent, .

Proof. See Appendix A.9. □

Remark 8. The expression of in (28) indicates that are linear combinations of , which means that the space spanned by is included in the space spanned by . Since are linearly independent, both spaces have the same dimension N + 1, indicating that they are the same spaces. Thus, if we denote the space spanned by by , the decomposition of is described as follows.

where and .

4.1 Interpretation of the model

In this numerical example, we provide a concrete interpretation of the dividend processes , , and the market value processes , .

First, the dividend processes, which represent the supply of consumption goods in the model, are interpreted here as the aggregate coupon payments from government bonds and the dividends from stocks. In this numerical setting, the aggregate coupon payments reflect changes in the outstanding notional amounts of government bonds in the secondary market. When the outstanding notional amount increases, the total coupon payments and therefore the dividend processes increase proportionally, and when the outstanding notional amount decreases, the dividend processes decrease. Thus, the dividend processes capture the effect of changes in the outstanding notional amounts of government bonds.

Second, we interpret as the market value of security j, which is the present value of its future dividends. For , represents the market value of government bonds with time to maturity belonging to zone , and for j = N + 1, it represents the market value of the stock index.

Changes in the outstanding notional amounts arise from central bank purchases and from government issuance and redemption within each maturity zone. Thus, the process is treated as an exogenous input reflecting these notional changes in zone caused by the activities of the central bank and the government.

4.2 Parameter setting and factor processes

We specify the multi-agent model parameters and the factor processes that drive the dividends and mortality rates used for insurance pricing.

The terminal time is set to T = 100 years, and we consider N + 1 = 6 securities, consisting of five government bond maturity sectors and a stock price index. Specifically, government bonds are categorized into five maturity sectors: 1–3 years, 3–5 years, 5–10 years, 10–25 years, and beyond 25 years, together with a representative stock price index. We denote these sectors by j = 1, 2, 3, 4, 5 and the stock price index by j = 6.

We set I = 3 agents, each with an initial wealth , representing three types of institutional investors. For example, the first type may correspond to younger life-cycle investors, the second to older investors, and the third to investors such as banks and insurance companies. Alternatively, these types may be interpreted as domestic private investors, domestic institutional investors, and foreign institutional investors.

We consider 100 discrete time points , , for dividend and consumption timings. The dimension of the Brownian motion W is set to d = 8, which exceeds the number of securities N + 1 = 6. The drift of the dividend processes depends on an economic factor Y¹, driven by the Brownian motion component W₇.

The mortality rate defined below is influenced by both the economic factor Y¹ and the public health deterioration factor Y², driven by W₈. The factors satisfy the following SDEs:

(29)

We assume that Y¹ and Y² are Gaussian processes with positive initial values and positive drifts, representing an economic growth factor and a public health deterioration factor, respectively. The public health deterioration factor negatively affects the drift of Y¹ through l < 0. Although Gaussian processes may occasionally take negative values, the parameters , , , and are chosen so that Y¹ and Y² remain positive in nearly all simulated sample paths. The mortality rate is defined by

(30)

so that as the economy grows, the mortality rate decreases, and as disasters occur, the mortality rate increases.

Specifically, for insurance products for individuals of age t₀, we consider the base mortality rate corresponding to that of a population of age t₀ + t. In the following numerical examples, we assume t₀ = 20, covering the mortality rate for ages 20–120.

The discrete dividend processes for and are realizations of the following continuous factor processes for , characterized by the drift and diffusion coefficients and :

where

(31)

The parameter represents the absolute value of the volatility for the factor process , and describes the correlation between the Brownian motions driving the economic factor and the j-th factor process . Thus, the dividend processes, corresponding to government bond coupon payments reflecting the outstanding notional amounts of government bonds or dividends from the stock index, have drift components linked to the economic factor and diffusion components correlated with it.

4.3 Base case parameters for factor processes

As base case parameters for the factor processes, we set , , , , for , , for , , for , , for , , and for , where is the initial value of the total dividend defined by

(32)

We set for because the effect of economic conditions on government bond issuance is not one-directional. In a strong economy, monetary tightening may increase the effective supply of government bonds available in the market, while higher tax revenues may reduce the need for new issuance. Since these effects can offset each other, we set the sensitivity parameter to zero in the base case.

Three agents, denoted by i = 1, 2, 3, are considered, each with an initial wealth of and time preference parameters , , and . The parameters representing their views on fundamental risks, for in (27), are set to a¹ = 1.0, a² = 0, and . For example, the first agent has aggressive views with a lower time preference, placing more emphasis on future spending, the second agent has neutral views with moderate time preference, and the third agent has conservative views with a high time preference, prioritizing near-term spending.

Next, by shifting parameters in the model, we examine the impact of changes in bond supply in the market, agents’ expectations about future bond supply, and public health conditions on the long-term discount rate and insurance pricing.

After examining parameter shifts motivated by the historical events discussed below, we compute 100-year discount rates and insurance prices under the shifted parameters.

4.4 Supply change impact on the yield curve

First, we examine the impacts of changes in the supply of government bonds on yield curves by shifting the parameters , , and , the constant part of the drift in the dividend processes y³, y⁴, and y⁵.

Under the unconventional monetary policy conducted by the former BOJ governor Kuroda, in addition to the negative interest rate policy, large amounts of government bonds were purchased for monetary easing and the effective supply of government bonds available in the market decreased. We observed a 1.4% decrease in the 30-year discount rate from April 2013, when monetary easing began, to July 2016, before the introduction of yield curve control, in which an unlimited amount of bond purchase was attempted to keep the 10-year yield at zero (Ministry of Finance, Japan [16]). This change corresponds to decreasing the parameters , , and by 0.045.

As Fig 1 illustrates, the discount rate increases with tightening monetary policy, while it decreases with easing monetary policy. In addition, the yield curves become steeper than in the base case under both easing and tightening. This steepness arises because the parameters with larger values dominate the long-term behavior of the factor processes through their exponential form.

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Fig 1. Discount rate for the base, the easing, and the tightening cases.

Three cases of yield curves are plotted. for the base case, for the easing case, and for the tightening case.

https://doi.org/10.1371/journal.pone.0343055.g001

In the base case, all values are , while in the tightening case and in the easing case . As time passes, the total dividend becomes increasingly influenced by the factor processes with larger drift parameters, because their exponential growth rates dominate the others. As a result, both the easing and tightening scenarios display steeper yield curves.

4.5 Insurance prices

We assume an annual mortality rate in (30) that depends on two factors, namely an unhedgeable economic factor and a public health deterioration factor denoted by Y¹ and Y², respectively. Using the yield curve for discounting obtained in the previous subsection, we can price a death benefit that pays upon death at an exogenously given random time , and a life annuity that pays annually until time .

Following the approach of Chapter 8 in Bielecki and Rutkowski [17], we treat their default time as the death time and re-express the second equation in their Proposition 8.2.1 under the physical measure P. Their spot martingale measure , introduced at the beginning of Section 8.1.1, corresponds in our framework to a measure under which , , are martingales. For simplicity, we set the density process for the transformation from P to equal to obtained in the market equilibrium in (21).

Using the associated state price density process in (22), the initial values of the death benefit and the life annuity are given by:

Death benefit value:

Here, we set t = 0, , Z = V, and in the second equation of Proposition 8.2.1 in [17].

Life annuity value:

where we set t = 0, , , and in Proposition 8.2.1 of [17].

For simplicity, we set in both pricing formulas.

We define the base mortality rate as follows: 0.04% for , 0.06% for , 0.10% for 20 < t < 30, 0.24% for , 0.63% for , 1.7% for , 4.8% for , 15% for , 40% for , which roughly corresponds to the mortality rate for ages 20–100 and above for men in Japan (Ministry of Health, Labour and Welfare [15]). We set and k₂ = 0.01 in (30), implying that the economic factor decreases the mortality rate while the public health deterioration factor increases it.

First, in Table 1, the detailed data used for the base, easing, and tightening cases are provided in S1 Data, S2 Data, and S3 Data, respectively. If the drift parameters , , and of the factor processes y³, y⁴, and y⁵, which determine the long-term behavior of the dividend processes, shift from the base value 0.010 to , corresponding to a reduction in the effective market supply of government bonds due to large-scale central bank purchases, the yields decrease. As a result, the death benefit price increases from 0.093 to 0.165 and the life annuity price increases from 26.5 to 34.7, because a lower discount rate increases the present value of future payments.

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Table 1. Insurance pricing in the base, easing, and tightening cases.

https://doi.org/10.1371/journal.pone.0343055.t001

In contrast, if , , and increase from 0.010 to 0.055, corresponding to an increase in bond supply and therefore higher yields, the death benefit price decreases from 0.093 to 0.027 and the life annuity price decreases from 26.5 to 18.1, because a higher discount rate reduces the present value of future payments.

These insurance price changes are mainly driven by the change in the discount rate. In this scenario, the long-term discount rate for 100 years is 4.6% in the base case, 3.7% under easing, and 7.3% under tightening.

4.6 Market expectation change following the Bank of Japan’s announcement of monetary tightening

Finally, we examine how changes in market expectations regarding future bond supply affect long-term discount rates and insurance pricing. We investigate the impact of shifts in market expectations about the outstanding values of assets in the secondary market by adjusting a¹, a², and a³, the parameters representing agents’ views on fundamental risks defined as in (27).

Following the Bank of Japan’s announcement of its exit from yield curve control in March 2024, market expectations of a future reduction in the BOJ’s government bond purchases intensified. These expectations persisted until the official announcement of reduced bond purchases in July 2024. The heightened expectations led to lower bond prices and higher yields.

Specifically, we observed a 0.4% increase in the 30-year yield between March 2024, when the Bank of Japan exited the yield curve control policy, and July 2024, when the BOJ announced reduced bond purchases (Ministry of Finance, Japan [16]). In our model, an increase in the market outstanding, which is the aggregate value of tradable securities in the secondary market, corresponds to an increase in the dividend process, reflecting higher coupon or stock dividend payments.

This 0.4% increase in the 30-year yield corresponds to shifts of a¹, a², and a³ by 0.1 in (27), representing changes in market expectations regarding the expected return of the secondary market outstanding.

Table 2 shows the impact on insurance pricing via the discount rate resulting from these changes in agents’ views on fundamental risks. The detailed data used for Table 2 are provided in S4 Data.

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Table 2. Impact of aggressive agents on the 100-year discount rate and insurance prices. a¹ = 1.0, a² = 0, for the base case and a¹ = 1.1, a² = 0.1, for the aggressive case.

https://doi.org/10.1371/journal.pone.0343055.t002

If we set a¹ = 1.1, a² = 0.1, and , meaning that agents become more aggressive and expect the secondary market outstanding values to increase, the discount rate becomes higher and the death benefit price becomes lower.

4.7 Stability analysis and joint shock analysis

4.7.1 Stability analysis.

To examine the robustness of the model, we provide a stability analysis by perturbing several key parameters. Since the main policy implication of the paper is obtained in the easing case (), we evaluate the sensitivity of this case by slightly modifying the supply parameter around this value. For convenience, let . Specifically, we consider two close values, and , and compare the resulting long-term yields and insurance prices with those under .

As shown in Table 3, the detailed data used for this stability analysis are provided in S5 Data ( = −0.030) and S6 Data ( = −0.040). Thus, the qualitative implications of the easing case, namely lower long-term yields and higher insurance prices, are maintained under these small perturbations.

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Table 3. Stability analysis around the easing case.

https://doi.org/10.1371/journal.pone.0343055.t003

4.7.2 Joint shock analysis.

To further clarify the comparative statics of the model, we also consider scenarios in which the supply shock and the expectation shock occur simultaneously. We examine two economically meaningful joint cases.

First, we consider an easing scenario combined with more aggressive expectations, implemented as together with . Second, we examine a tightening scenario combined with more conservative expectations, implemented as together with .

These combinations correspond to realistic settings in which supply and expectations move together. The detailed data used for Table 4 are provided in S7 Data (tightening–conservative scenario) and S8 Data (easing–aggressive scenario). Table 4 shows that the interaction of supply and expectations produces changes that are consistent with the directions observed in the separate comparative statics analyses in Sections 4.5 and 4.6.

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Table 4. Base case, easing with aggressive expectations, and tightening with conservative scenarios.

https://doi.org/10.1371/journal.pone.0343055.t004