How much welfare does homeowner lose by participating in the pension fund

by participating in the pension fund

Limin Zhou

Supervisor: Dr. Servaas van Bilsen

Facaulty of Economics and Business

University of Amsterdam

This dissertation is submitted for the degree of

Master of Science

in Actuarial Science and Mathematical Finance

List of tables v 1 Introduction 1 1.1 Motivation . . . 1 1.2 Literature Review . . . 2 1.3 Research Questions . . . 3 1.4 Findings . . . 3 2 Model 5 2.1 Theoretical Model . . . 5

2.2 Optimal Consumption Choice . . . 7

2.3 Optimal saving amount of renter and homeowner . . . 9

2.4 Consumption comparison based on different β . . . 14

2.5 Extension: Optimal saving amount of renter and homeowner due to dynamic consumption . . . 15

3 Results 21 3.1 Utility Comparison of Sub-optimal and Optimal Consumption . . . 21

3.2 Extension Part: Consumption Utility Comparison based on different β . . . 23

3.3 Welfare Loss Analysis . . . 26

4 Conclusion 29 4.1 General Conclusion . . . 29

4.2 Discussion . . . 30

References 31

3.1 Consumption . . . 21

3.2 Optimal Consumption utility, rh=0.05 . . . 22

3.3 Sub-optimal Consumption Utility . . . 22

3.4 The sub-optimal consumption . . . 23

3.5 Optimal Consumption Utility . . . 23

3.6 Comparison between Optimal Consumption and Sub-Optimal Utility . . . . 23

3.7 Optimal Consumption Utility Y =30000,rh=0.05, H=150000, i=0.01 . . . . 24

3.8 Sub-Optimal Consumption Utility for Owner rh=0.05,Y =30000, i=0.01,m=6566 24 3.9 Optimal Consumption Utility for Owner rh=0.05,Y =30000, i=0.01,m=6566 25 3.10 Optimal VS Sub-Optimal Consumption Utility m=6566, i=0.01,γ=6 . . . . 25

3.11 Welfare Loss of different rent fee and mortgage rate (γ=5,i=0.02,β =_1+i1 ) . . 26

3.12 Welfare Loss of different interest rate and mortgage fee (γ=5,β =_1+i1 ) . . . . 27 3.13 Welfare Loss of different risk aversion and mortgage fee (i=0.015,β =_1+i1 ) . 27

Introduction

1.1

Motivation

Due to the financial crisis since 2008, the economy of the world especially for European has become more important and needy. And in today’s financial market, housing become a really important investment choice. However, it is not easy to answer the question of whether to buy a house or to continue renting. People may find buying a house appealing because they see it as a great investment that is highly leverage and relatively illiquid, as well as an important addition to their personal assets. However, when buying a house one must also take into consideration that a house can be far more expensive based upon the simple increase in square footage between renting a small apartment and buying a house. Unlike an apartment, once one has found a house they are satisfied with, they must complete the intensive process of obtaining a mortgage loan. In all, there is a lot consideration involved when making the decision to buy a house or not.

Thus, it is important for us to study the optimal consumption of each situation. How-ever, insurance companies do not consider their policyholder’s preferences when setting up premiums which leads to the different consumption of each policyholder. As a result, it is not entirely fair to have the same premium for all policyholders. Due to the fact that policyholders have different preferences, providing them with the same premiums may result in one policyholder being more disadvantaged, and with unequal life satisfaction than others of the same premium. As a result, the insurance company should help the policyholder catch up to the optimal consumption over their life cycle.

1.2

Literature Review

The works of Merton and Samuelson and other authors have already studied optimal and portfolio choice over life time in a wide range of settings. Preferences are normally expressed by expected utility with constant relative risk aversion (CRRA) model which is normally used by a lot of researchers (ex. Yao and Zhang).

In this thesis, we are aiming to provide the closed-form solutions of the saving amount for people who have different preferences so as to meet their optimal consumption, and derive explicit solutions to life cycle utility maximization problems involving risky assets, different time preferences, and risk aversion. After having all these solutions, then we are able to further investigate how welfare does homeowner loss by participating pension fund if insurance companies only used the same premium rate with renters. For some further strategies are also discussed by Kraft„o, Dow and da Costa Werlang.

We assume the individual receives the fixed, risk-free labor income until a fixed retirement date. And for each consumption we assume it is constant and executed per year. After deriving optimal consumption choices, depending on different time preferences and risk aversion, I mainly examine the constant optimal consumption, and deterministic and stochastic optimal consumption which will also be discussed in the extension part. In all cases, we will try to examine the sensitivity of expected utility of all possible types of consumption in terms with different variables. Then we can have a better insight on how welfare loss of homeowner will be and have an overview understanding on how important to have different strategies based on different preferences.

After having the optimal consumption choice, same as what Bank and Riedel did for choosing the optimal consumption choice( detailed see [1]), I further break down all the consumptions during a lifetime for homeowners and renters. In order to have optimal consumption, I derive the specific saving amount for both renters and homeowners. Then, by tuning different interest rates, mortgage rates, and other variables, I further compare the utility differences between sub-optimal and optimal consumption which are using the same saving amount on all individuals, and using specific saving amount of different situations. Afterwards, we analyze the welfare loss of homeowners which is our final goal. In the end, by tuning different variables involved, we analysis the sensitivity of welfare loss in terms of different variables such as interest rates, mortgage rates and risk aversion.

1.3

Research Questions

• 1. How does individual preference affect the consumption stream? – Is the consumption stream of homeowners and renters different? – Whose consumption stream is more complicated?

• How does individual preference affect the optimal consumption – How do we choose the optimal choice?

– What factors are related with optimal choice?

– How much saving amount should we choose for satisfying the optimal consump-tion and are they different for homeowners and renters?

• How does individual preference affect the expected utility of optimal consumption? – Except individual preferences, what else can also affect the expected utility of

optimal consumption?

– How sensitive is the expected utility of optimal consumption with respect to other factors such as interest rates, mortgage rates and risk aversion?

• How much welfare does homeowner lose if applying the same saving amount rate with renters?

– How sensitive is the welfare loss of homeowners with respect to different factors? – For real market, does the welfare loss still have the same loss range?

1.4

Findings

My results explores consumption and portfolio choice under reference-dependent preferences which indicate that it is not appropriate to use the same saving amount for renters and homeowners because applying renters saving amount on homeowners may lead to two different levels of consumptions which is satisfied on different interest rates, mortgage rates, risk aversion and time preferences. And in the modeling part we can obviously see the different consumption stream as well as utility of consumption due to different individual preferences.

In chapter 3, we analysis the sensitivity of utility of optimal consumption with respect to interest rates, mortgage rates and risk aversion in section 3. It shows that the higher risk

aversion and the interest rate, the more utility will be and not really related with mortgage rate. Also, for the welfare loss of homeowners is related with risk aversion, interest rates and mortgage rates. The higher interest rates and risk aversion, the more welfare loss will be, and the less interest rates, the less welfare loss will be. And by the tables in the results part, we are able to see how much welfare loss of homeowners will be by participating in the pension fund.

Model

2.1

Theoretical Model

First, we investigate two different situations to discuss: Agent A who always rent the house and Agent B who finances the house with a loan.

Suppose both of them have exactly the same risk-free labor income Y ∈ [0, R] die at the same date of death TD. They both live in the same house worth H euros. The rent fee in this

paper is the house value times rent fee rate which is H · r_h∀ t ∈ [0,TD]. The saving amount

per year is equal to s ∀ t ∈ [0,R] which means they will put s amount of money into their own defined contribution DC plan. And we assume they do not invest risky investments, then the wealth is only accumulated by interest rate r. Thus the pension wealth is:

W_t+1= Wt· (1 + r) + s (2.1)

Also, we assume the pension write payment P will be paid annually (easier to calculate). The present value of total pension payments WR should be equal to the pension wealth at

retirement. W_R= P( 1 1 + i+ 1 (1 + i)2+ ... + 1 (1 + i)TD−R−1) (2.2)

Pension payment P then will be easily derived from above function (2.2): P= iWR

1 − vTD−R (2.3)

Now, we investigate the consumption of both A and B. For agent A (Renter):

Since A always rents the house, the rent will be paid until death. Then, the consumption is labor income minus rent fee and saving amount before retirement, and after retirement it will be equal to pension payment minus rent fee.

C_tA= Y − rh· H − s (0 < t ≤ R) (2.5)

C_tA= P − r_h· H (R < t < TD) (2.6)

For agent B (Homeowner):

Agent B is slightly different than A, because B only need to pay the loan Tm (Tm<R)

years. The mortgage rate is denoted as im.

For simplicity, we assume the loan is paid annually. The annual mortgage payment m is then: H = m · (1 + 1 1 + im + 1 (1 + i_m)2+ ... + 1 (1 + i_m)Tm−1) (2.7) = m ·1 · (1 − v Tm m) 1 − vm (2.8) v_m= 1 1 + im (2.9) m=H· (1 − vm) 1 − vTm m (2.10) The mortgage fee is supposed to pay off before retirement. The consumption of agent B is labor income minus mortgage fee and saving amount before paying off mortgage Tm.

And, normally the mortgage is supposed to pay off before retirement. After paying off all the debt and before retirement, the consumption is becoming labor income minus saving amount only, During retirement, the consumption is simply pension payment. The mathematical description is right below:

C_tB= Y − m − s (0 < t ≤ Tm) (2.11)

C_tB= Y − s (Tm< t ≤ R) (2.12)

C_tB= P (t > R) (2.13) Our main goal is to keep the consumption of both homeowner and renter constant during the lifetime. The reason will be discussed in the following section.

2.2

Optimal Consumption Choice

In economics, a utility function is normally a simple device for summarizing the information contained in the consumer’s preference relation. And it is an important term that measures the welfare or satisfaction of a customer when consuming certain amounts of goods or services. Here we also need utility to measure individual consumption satisfaction during a whole lifetime. (also further discussed by Issler, Bodie et al.)

The expected lifetime utility is given by (assume it is discrete) U0= E[

TD−1

∑

k=0

βk· u(c_k)] 0 ≤ β ≤ 1 (2.14) β above stands for the coefficient of time preference.

u(·) = 1 1 − γ · c

1−γ

k (2.15)

Meanwhile, the static budget constraint which is the initial wealth shall be greater than the present value of total consumption during all lifetime should be satisfied

V₀≤W0 (2.16) TD

∑

k=0 1 (1 + i)kck≤W0 (2.17)

In order to have the maximum utility of consumption, we apply the Dynamic Lagrange method to find the optimal consumption choice.

The Lagrange function is given by(the second equality is from 2.16 L = E[

_∑

TD i=0 βi· u(ct)] − y(V0−W0) (2.18) = E[ TD

∑

k=0 βk· u(ck) − y · 1 (1 + i)kck] + y ·W0 (2.19)

yis the Lagrange multiplier.

By setting the first derivative equal to 0 we have: βkc−γ_k = y 1 (1 + i)k (2.20) c−γ_k = y· β −k (1 + i)k (2.21)

β beta is the time preference parameter. In this paper, we mainly discuss the case where β = (1 + i)−1. In this case, optimal consumption will be:

c_k= y−1γ (2.22)

which is constant. That is why the individuals’ aim is to achieve a constant consumption stream over time.

(Extension Part)

However, β may slightly change due to different situations or circumstances. In this extension part, we discuss the situation of optimal consumption in which beta is not equal to (1 + r)−1

Situtation 1: β is larger than the (1 + i)−1 β1=

1 α1+ i

(0 ≤ α1< 1) (2.23)

Then the optimal consumption will be

c_k= y−1γ · (α1+ i

1 + i )

−k

γ (2.24)

Situation 2: β is smaller than the (1 + r)−1 β2=

1 α2+ i

(1 < α2< ∞) (2.25)

Then the optimal consumption will be

c_k= y−1γ · (α2+ i

1 + i )

−k

γ (2.26)

Since the Lagrange multiplier y has already been fixed for each of situation and it is easy to calculate, then it is clear that the optimal consumption of both situation 1 and 2 have their

own specific deterministic rate. Further details about consumption based on different β will be given in the section 2.5.

In general,the optimal consumption is as below: c_k= y−1γ · (α + i

1 + i)

−k

γ (2.27)

And In our case, α is simply equal to 1.

2.3

Optimal saving amount of renter and homeowner

It is clear that in order to keep the consumption constant, we must evaluate the savings amount, known as the premium paid by the policyholder. First, let us discuss Agent A. By combing 2.5 and 2.6, we have a savings amount that is equal to the labor income minus the pension payment.

s= Y − P (2.28) By deriving (2.1), and W1= s, then WRis

WR= s(1 + (1 + i) + (1 + i)2+ ... + (1 + i)R−1) (2.29)

= s(1 + i)

R_{− 1}

i (2.30)

Subsituting (2.30) into (2.3) and (2.28), we have s= Y − s (1+i)R₋₁ i · i 1 − vTD−R (2.31) (1 +(1 + i) R_{− 1} 1 − vTD−R )s = Y (2.32) s∗= Y · 1 − v TD−R (1 + i)R_{− v}TD−R (2.33)

In our case, we assume that the fixed labor income of both Agent A and Agent B is 30,000 euros (which is the average wage in the Netherlands), R is 50 years which represents the duration of time between the first premium payment and retirement, and TDis 70 years

which represents the duration of time between retirement and death. By tuning different interest rates i from 0.005 to 0.025 (the normal interest rate change in Europe), the following

image shows the trend of the optimal saving amount and payment with respect to different interest rates.

Clearly, the write payment is increasing if the interest rate is going up, and meanwhile the saving amount will be decreasing.

Below is the saving rate graph which more clearly shows what percent of the labor income should be invested into the pension fund.

For example, if our interest rate is 5%, we would like to ask the individual to save 25% of their labor income in their pension wealth. The higher the interest rate is, the smaller

the saving amount rate will be. This is because the accumulated wealth grows more rapidly when the interest rate is higher.

Now, we investigate the situation of the homeowner. Let us assume that the interest rate is 1%. Suppose the insurance company will use the same saving amount rate for people who are homeowners. Then, by using the optimal saving amount we have above, we would like to save 21.87% of our labor income. Here we assume that the house value H is 150,000 euros, The mortgage period Tmis 30 years, and the mortgage rate is 2%. Calculated by the equation

2.10, we can see that the annual mortgage payment m will be around 6566 euros. Then, below we see the consumption of each year:

As can be seen, our consumption is not optimal, but rather it has a sudden elevation after paying out the mortgage debt at time Tm. That is because the consumption of homeowners

will be changed after paying off their mortgage debt at Tm.

In order to keep the constant consumption, We decide to set the different savings amount s1 and s2 at different periods which are s1 before Tm and s2 after Tm and before R, the

consumption will be formulated as below:

C_tB= Y − m − s1 (0 < t ≤ Tm) (2.34)

C_tB= Y − s2 (Tm< t ≤ R) (2.35)

By deriving function 2.34, 2.35 and 2.36, we have:

s₂= s1+ m (2.37)

s1= Y − m − P (2.38)

The pension wealth is also changed as:

WR= s1· (1 + (1 + i) + ... + (1 + i)R−1) + (s2− s1) · (1 + (1 + i) + ... + (1 + i)R−Tm−1) (2.39) = s₁(1 + i) R_{− 1} i + m · (1 + i)R−Tm_{− 1} i (2.40)

By plugging function 2.39 into function 2.3, the payment P is P= i· (s1 (1+i)R−1 i + m · (1+i)R−Tm−1 i ) 1 − vTD−R (2.41) = (s1· ((1 + i) R_{− 1) + m · ((1 + i)}R−Tm_{− 1)} 1 − vTD−R (2.42)

Putting function 2.42 back into function 2.38, we arrive at the optimal s∗₁: s1= Y − m − (s1· ((1 + i)R− 1) + m · ((1 + i)R−Tm− 1) 1 − vTD−R (2.43) s₁· (1 +(1 + i) R_{− 1} 1 − vTD−R ) = Y − m − m· ((1 + i)R−Tm_{− 1)} 1 − vTD−R (2.44) s₁·(1 + i) R_{− v}TD−R 1 − vTD−R = Y − m − m· (1 + i)R−Tm_{− 1)} 1 − vTD−R (2.45) s∗₁= 1 − v TD−R (1 + i)R_{− v}TD−R· (Y − m − m· ((1 + i)R−Tm_{− 1)} 1 − vTD−R ) (2.46) s∗₁= 1 − v TD−R (1 + i)R_{− v}TD−R· (Y − m · (1 + i)R−Tm_{− v}TD−R 1 − vTD−R ) (2.47) s∗₁= 1 − v TD−R (1 + i)R_{− v}TD−R·Y − (1 + i)R−Tm_{− v}TD−R (1 + i)R_{− v}TD−R · m (2.48)

By function 2.37, the optimal saving amounts∗₂is

s∗₂= s∗₁+ m (2.49) By tuning the interest rate from 0.005 to 0.025, the graph of s1, s2and write payment P

The trend of saving amount s1and s2 is as same as that of renters instead one saving

amount replaced by two saving amounts. And the saving amount s1and s2is paralleled withe

each other, the gap between two saving amounts is the mortgage fee, which is reasonable since saving amount s2is aiming to fill the gap after paying off the mortgage fee.

Also, we may need to look at the saving amount rate of s1and s2

This means that, when the interest rate is 0.5%, it is better to invest 13.5% of our labor income into the pension fund. Then, once the house loan has been paid off at Tm, it is better

to invest 35.4% of our labor income into the pension fund. However, as we mentioned above, the insurance company may have only one saving amount to policy holders, then it may lead to welfare loss of homeowners.

2.4

Consumption comparison based on different β

For the convenience of comparison, we analyze two specific α to see different trends. Since the optimal consumption is constant when β is equal to (1 + i)−1 for both renters and homeowners, it is easier to analyze and calculate the consumption of renters. Then, we pick up (i = 0.015,rh= 0.06,γ = 5) as our control group.

For situation 1: we set α1as 0.95 which means β is greater than (1 + i)−1.

Graphed by excel, we can see how the optimal consumption changes.

The consumption where β is greater than real interest discount is increasing by the time. It guides us to know that we may need to increase our consumption annually if our time preferences are more sensitive than interest rate discount.

Reversely, the β of consumption which is less than the real interest rate discount is decreasing by the time. And the consumption should be decreased at a certain level annually based on the less sensitivity of time preference.

However, by observing at the trend of consumption in both situations, it is easy to find that the α which is related with β , the time preferences. It shall not be far away from real interest rate discount, otherwise the consumption will be increasing or decreasing hugely without satisfying the budget constraint. Then it is hard to find the optimal saving amount. In reality, this situation happened rarely, since the time preference always depend on real interest rate discount leading to only a slightly difference.

2.5

Extension: Optimal saving amount of renter and

home-owner due to dynamic consumption

In section 2.4, we have a overview on how optimal consumption goes with the different time preference. Since we are also trying to further investigate how to reach up to optimal consumption more reality. Then, this section we are going to derive the more specific optimal saving amount for renters and homeowners.

Situation of Renters:

As discussed in the section 2.1, the consumption equation is 2.4 and 2.5, and the optimal saving amount will be deterministic. By replacing the consumption function with 2.26, the new consumption equation is as below:

y−1γ · (α + i 1 + i) −t γ = Y − r h· H − s(t) (0 < t ≤ R) (2.50) y−1γ · (α + i 1 + i) −t γ = P − r h· H (R < t < TD) (2.51)

Our aim is to reach an optimal saving amount of s∗, because the pension wealth will be equal to the amount of saving that we invest into the pension fund. More precisely, the present value of pension wealth at time R will be equal to the present value of accumulated total amount of saving that the policyholder invested at time R. Then, we have the following equation: R

∑

k=1 s(k) · (1 + i)R−k= TD−R

∑

k=1 P· (1 + i)−k (2.52) In order to calculate the optimal saving amount, we need to utilize the Lagrange multiplier y.

R

∑

k=1 (Y − r_h· H − y−1γ · (α + i 1 + i) −k γ) · (1 + i)R−k= TD−R

∑

k=1 (y−1γ · (α + i 1 + i) −k+R_γ + r_h· H) · (1 + i)−k (2.53) y− 1 γ = ∑ R k=1(Y − rh· H)(1 + i)R−k− ∑Tk=1D rh· H · (1 + i)−k ∑Rk=1(α +i1+i) −k γ(1 + i)R−k+ ∑TD−R k=1 ·( α +i 1+i) −k+R γ (1 + i)−k (2.54) y− 1 γ =∑ R k=1Y· (1 + i)R−k− ∑ TD k=1rh· H · (1 + i)R−k ∑T_k=1D (α +i_1+i)− k γ · (1 + i)R−k (2.55) Then, we have the optimal saving amount s(t):

s(t) = Y − r_h· H −∑ R k=1Y· (1 + i)R−k− ∑ TD k=1rh· H · (1 + i)R−k ∑T_k=1D (α +i_1+i)− k γ · (1 + i)R−k · (α + i 1 + i) −_γt (2.56) Suppose that α is 1.01,γ is 5, i is 0.01, and the rest of parameters are consistent with the ones utilized above. Then, by using formula 2.56, the saving amount and consumption are shown as below:

The consumption is going down as we proved in section 2.4, and the saving amount will be stopped after retirement.

First, we look at the homeowner consumption when applying the same saving amount as the renter which results in suboptimal consumption.

As was discussed in section 2.3, we assume the house value H is 150,000 euros, Tmis 30

years, and the mortgage rate is 2%. By using equation 2.9, we can have the annual mortgage payment m at around 6,566 euros, and the rest of parameters remain the same as the renters discussed above. The consumption of the homeowner is:

C_tB= Y − m − s(t) (0 < t ≤ Tm) (2.57)

C_tB= Y − s(t) (Tm< t ≤ R) (2.58)

C_tB= P (t > R) (2.59) After substituting the saving amount we derive for renters, the graph of sub-optimal consumption of homeowner is shown as below:

Since the consumption of homeowner has one step which we discussed in section 2.3, not surprisingly, the trend will remain if we only apply one saving amount rate on homeowner.In order to have the optimal consumption, as discussed for the constant time preference β , we set-up two different deterministic saving amounts. The interest rate i we set up is 0.01 here, and the rest of parameters will remain the same. Then the consumption should be hold equal

to the following equations: y−1γ · (α + i 1 + i) −t γ = Y − m − s 1(t) (0 < t ≤ Tm) (2.60) y− 1 γ · (α + i 1 + i) −t γ = Y − s 2(t) (Tm< t ≤ R) (2.61) y−1γ · (α + i 1 + i) −t γ = P (t > R) (2.62)

Same derivation way as 2.52

Tm

∑

k=1 s₁(k) · (1 + i)R−k+ R

∑

k=Tm+1 s₂(k) · (1 + i)R−k= TD−R

∑

k=1 P· (1 + i)−k (2.63) y−1γ ₌ ∑ Tm k=1(Y − m) · (1 + i)R−k+ ∑Rk=Tm+1Y(1 + i) R−k ∑Rk=1(α +i1+i) −k γ(1 + i)R−k+ ∑TD−R k=1 ·(α +i1+i) −k+R γ (1 + i)−k (2.64) y−1γ = ∑ R k=1Y(1 + i)R−k− ∑ Tm k=1m(1 + i)R−k ∑T_k=1D (α +i_1+i)− k γ · (1 + i)R−k (2.65) Then, we have optimal saving amounts s1(t) and s2(t):

s1(t) = Y − m − ∑Rk=1Y(1 + i)R−k− ∑ Tm k=1m(1 + i)R−k ∑T_k=1D (α +i_1+i)− k γ · (1 + i)R−k · (α + i 1 + i) −_γt (2.66) s2(t) = s1(t) + m (2.67)

We found the saving amount s2is greater than s1with the amount of mortgage fee. Which

makes sense, because we need to fill the consumption gap after paying off the mortgage fee. After using the optimal saving amount, then the consumption is turned into the optimal consumption. This can be seen in the graph below:

This proves that by using two deterministic saving amount we can make the consumption of homeowners optimal. Furthermore, it is also really important for us to have a more clear insight how much welfare loss does we compensate if applying our two deterministic saving amounts for homeowners.Then we are able to realize if it is worth enough for insurance companies to pay more attention to people who has different preferences so that have more actions on protecting welfare loss for individual. In the next chapter, we will further discuss the comparison between optimal and sub-optimal consumption utility.

Results

3.1

Utility Comparison of Sub-optimal and Optimal

Con-sumption

In Chapter 2, we derive the optimal saving amount for renters and homeowners in different situation of time preference β , so as to reach up to the optimal consumption of individual has different preference of investing. In this Chapter 3, we will analysis the sensitivity of the consumption utility with respect to different factors and then have a clearer image on how large between sub-optimal and optimal consumption utility.

Here we discuss about the situation of β = (1 + i)−1.

For the individuals who prefer a lifetime of renting a house, we begin by tuning different interest rates i and rent fee rates rh. Then, we look at how the optimal consumption changes.

Table 3.1 Consumption i\rh 0.04 0.05 0.06 0.07 0.08 0.09 0.005 16469 14969 13469 11969 10469 8969 0.01 17439 15939 14439 12939 11439 9939 0.015 18331 16831 15331 13831 12331 10831 0.02 19140 17640 16140 14640 13140 11640 0.025 19864 18364 16864 15364 13864 12364

For interest rate i we set the range from 0.0 5 to 0.025 which is normally the interest rate range during recent decades in Europe. So is rent fee rate r_h, which is also almost covering all the possible situations in recent years.

The table shows that the consumption will be increased if the interest rate goes up and decreased when the rent fee goes up. This is truly convincing, because if the interest rate goes

up the money will be cumulated faster and also if rent fee charges a lot, then the consumption will be decreased for sure.

Table 3.2 Optimal Consumption utility, r_h=0.05

γ \i 0,005 0,01 0,015 0,02 0,025

2 -3.957236e-03 -3.179072e-03 -2.602509e-03 -2.168311e-03 -1.836220e-03 4 -5.887229e-12 -4.171331e-12 -3.062413e-12 -2.322803e-12 -1.814973e-12 6 -1.576531e-20 -9.851934e-21 -6.486461e-21 -4.478947e-21 -3.229148e-21 8 -5.025908e-29 -2.770059e-29 -1.635583e-29 -1.028158e-29 -6.839535e-30 10 -1.744658e-37 -8.480863e-38 -4.490770e-38 -2.569968e-38 -1.577425e-38 Above, we calculate the optimal consumption utility respectively by using the function 2.14, if we keep the rhas 0.05. As we seen the utility is always negative, which is nothing

more than a way to present a preference relationship. The larger the utility means the greater the optimal consumption.

Afterwards, we look at the situation of the individuals who prefer buying house. First, we look at the consumption utility if the same saving amount is applied as the people who prefer renting a house by tuning different loan rates. Then the optimal consumption becomes to be the sub-optimal consumption. Here we keep the risk aversion γ as 6, and interest rate i as 0.01. Since all the situations have the same way to analysis, it is more convenient for us to pick only one situation to observe the difference.

Table 3.3 Sub-optimal Consumption Utility

i_m 0,02 0,03 0,04 0,05 0,06 m 6566 7430 8341 9293 10281 C(1-30) 16873 16009 15098 14146 13158 C(30-70) 23439 23439 23439 23439 23439

Utility -4.508063e-21 -5.65381e-21 -7.341212e-21 -9.899953e-21 -1.391234e-20 Since, the utility only stands for the preference relationship and the negative cannot give us more clearer insight how large gap or how bad the suboptimal consumption will lead to. In order to compare the constant consumption most effectively, we convert two different consumptions into one consumption which has the same utility by using R. Then, the difference will not be utility anymore, instead we have the difference between consumption which is more straight forward for us to analysis. The sub-optimal consumption will be shown below:

Table 3.4 The sub-optimal consumption

Utility -4.5081e-21 -5.654e-21 -7.3412e-21 -9.8999e-21 -1.3912e-20 Sub-Optimal Consumption 18636.25 17811 16904.52 15923.18 14875.67

As expected, the sub-optimal consumption is decreasing with the increase of mortgage rate. Then, we look at how consumption utility changes by applying the constant consumption which is taking two different levels of savings amount s1and s2.

Table 3.5 Optimal Consumption Utility

i_m 0.02 0.03 0.04 0.05 0.06 s1 3373 2953 2511 2049 1569

s₂ 9939 10383 10852 11342 11850 Consumption 20061 19617 19148 18658 18150 Utility -3.1191e-21 -3.4888e-21 -3.9370e-21 -4.4817e-21 -5.1449e-21

It can clearly be seen that the utility of the constant consumption is larger than the utility of multiple consumptions. This can also be verified by the consumption of Table 2.4 and Table 2.5. For further clarification, the comparison table is shown as below:

Table 3.6 Comparison between Optimal Consumption and Sub-Optimal Utility im 0.02 0.03 0.04 0.05 0.06

Sub-Optimal Consumption 18636.25 17811 16904.52 15923.18 14875.67 Optimal Consumption 20061 19617 19148 18658 18150 The difference between sub-optimal and optimal consumption is related with the welfare loss which will be further explained in the section 4.3 Welfare Loss Analysis.

3.2

Extension Part: Consumption Utility Comparison based

on different β

In order to have more comprehensive insight on how consumption utility changes, by using the saving amount rates calculated in the model part (Section 2), we can now analyze the consumption utility of renters and homeowners based on different time preference β .

First, we look at the consumption utility of renters by tuning different α and risk aversion γ .

Table 3.7 Optimal Consumption Utility Y=30000,r_h=0.05, H=150000, i=0.01

γ \α 0.98 0.99 1 1.01 1.02

2 -3.307849e-03 -3.210558e-03 -3.179072e-03 -3.209939e-03 -3.302770e-03 4 -4.422491e-12 -4.232916e-12 -4.171331e-12 -4.232670e-12 -4.420364e-12 6 -1.050817e-20 -1.001316e-20 -9.851934e-21 -1.001337e-20 -1.050959e-20 8 -2.963326e-29 -2.817597e-29 -2.770059e-29 -2.817785e-29 -2.964786e-29 10 -9.088546e-38 -8.630449e-38 -8.480863e-38 -8.631278e-38 -9.095104e-38 Then, we apply the same saving amount rate of renters on the homeowners to calculate the sub-optimal consumption utility. we use the same annual mortgage payment 6566 euros as was discussed above.

Table 3.8 Sub-Optimal Consumption Utility for Owner r_h=0.05,Y =30000, i=0.01,m=6566

γ \α 0.98 0.99 1 1.01 1.02

2 -2.628122e-03 -2.550824e-03 -2.525807e-03 -2.553478e-03 -2.543297e-03 4 -2.879661e-12 -2.629752e-12 -2.445761e-12 -2.316237e-12 -2.232741e-12 6 -5.632436e-21 -4.997788e-21 -4.507876e-21 -4.134753e-21 -3.857684e-21 8 -1.357967e-29 -1.184062e-29 -1.046945e-29 -9.390984e-30 -8.548815e-30 10 -3.636615e-38 -3.138259e-38 -2.741249e-38 -2.424256e-38 -2.171150e-38 For both optimal and sub-optimal consumption utility, if the time preference is more closer to real interest rate discount, the more utility will be. This might because if the time preference of the individual is equal to real interest rate discount, then it has more precise consumptions in terms with time. However, individual may not such expert at the time preference as banks or real world does, thus it is also important for us to look at how different the utility within these human bias. Also the utility will be increased if the risk aversion is higher. This is more persuasive that people who has higher risk aversion are willing to have constant consumption which will lead to better utility.

After determining the sub-optimal consumption utility, now we will investigate the optimal consumption utility for homeowners by applying the optimal saving amounts, which are two deterministic saving amounts. After having the utility of optimal consumption which are supposed to be lower than sub-optimal consumption. Then we can further analysis the gap between optimal consumption utility and sub-optimal consumption utility. In this case, we set the rent fee rate rhas 0.05, the risk-free labor income Y as 30000 euros per year, and

Table 3.9 Optimal Consumption Utility for Owner r_h=0.05,Y =30000, i=0.01,m=6566

γ \α 0.98 0.99 1 1.01 1.02

2 -2.790579e-03 -2.671641e-03 -2.594588e-03 -2.550331e-03 -2.524086e-03 4 -2.218026e-12 -2.122947e-12 -2.092060e-12 -2.122824e-12 -2.216959e-12 6 -3.326796e-21 -3.170081e-21 -3.119038e-21 -3.170147e-21 -3.327248e-21 8 -5.922136e-30 -5.630899e-30 -5.535896e-30 -5.631274e-30 -5.925053e-30 10 -1.146550e-38 -1.088760e-38 -1.069889e-38 -1.088864e-38 -1.147377e-38 From Table 3.9, it is obvious that optimal consumption utility is greater than the sub-optimal consumption utility. And it has the same results with Table 3.7 and Table 3.8. All of them has the greatest utility when the α is equal to 1, which the time preference β is equal to real interest rate discount.

For convenience, I have only selected risk aversion γ as 6 to compare the difference between optimal and sub-optimal consumption. The table is shown as below:

Table 3.10 Optimal VS Sub-Optimal Consumption Utility m=6566, i=0.01,γ=6

α 0.98 0.99 1 1.01 1.02

Sub-Optimal Consumption -5.632e-21 -4.998e-21 -4.508e-21 -4.135e-21 -3.858e-21 Optimal Consumption -3.327e-21 -3.170e-21 -3.119e-21 -3.170e-21 -3.327e-21

In conclusion, no matter how risk aversion γ and time preferences coefficient β (here we use α to express) change, we can always see that the sub-optimal is always worse than optimal consumption. Thus, it is necessary for insurance companies to decide different saving amount rates for individuals who have different preferences. In the following section, we are going to investigate the welfare loss by applying sub-optimal and optimal consumption. This will allow us a clearer understanding of how optimal consumption will benefit people. (in our case the idea is to use two different saving rates for homeowners rather than using the same saving amount with renters).

3.3

Welfare Loss Analysis

Welfare loss analysis is a tool to assess the impact of alternative strategies on the agent’s well-being which is also further studied by Feldstein By checking the welfare loss, which is a relative decline in certainty equivalent consumption. The welfare loss function is shown as below:

W L= ce

∗_{− ce}

ce∗ (3.1)

where ce∗ is the certainty equivalent consumption level corresponding to the optimal consumption strategy.[12]

Here, we discuss the consumption utility with sub-optimal and optimal consumption. This indicates that individuals who are choosing the same amount of saving amount that insurance companies normally offer or who prefer choosing different saving amounts of money depending on different time intervals. This is the goal of this project which, to reiterate, is to try to keep the same consumption amount over the course of a lifetime.

By using R, we first calculate the utility of the sub-optimal consumption, and then, we use the "uniroot" function to search for the certainty equivalent consumption level of constant consumption which is ce∗. The code is attached in the appendix.

To begin, let us look at the welfare loss in terms of different rent fees and mortgage fees which are the indications of the policy pricing of mortgage agencies or lenders (normally is bank). Here, we pick up risk aversion γ as 5, interest rate i as 0.02 and β is equal to the real interest rate discount.

Table 3.11 Welfare Loss of different rent fee and mortgage rate (γ=5,i=0.02,β =_1+i1 ) r_h\im 0,005 0,01 0,015 0,02 0,025 0,1 3.18% 3.68% 4.25% 4.88% 5.58% 0,3 3.18% 3.68% 4.25% 4.88% 5.58% 0,5 3.18% 3.68% 4.25% 4.88% 5.58% 0,7 3.18% 3.68% 4.25% 4.88% 5.58% 0,9 3.18% 3.68% 4.25% 4.88% 5.58%

As can be seen, the data is consistent across the table which would assume that welfare loss is only considered for homeowners, and that welfare loss is not relevant to renters and rent fees. Because our setting is to use the same saving amount of renters on homeowners, then the rent fee is not related since they will change on homeowners as well. The mortgage

fee does influence the welfare loss because the higher mortgage rate, the more welfare loss of homeowners occur. And the range of welfare loss is between 3.18% and 5.58% which is quite large for individual all lifetime.

Then, because there is no need to consider the rent fee as we discussed above, and thus we now tune the interest rate i and mortgage rate im. The range of both variables are tuned in

the acceptable situations during the research on the real-estate and bank historical data which setting from 0.5% to 2.5% for interest rate and also the same range for mortgage rate. This range is covering most of cases in recent few decades.

Table 3.12 Welfare Loss of different interest rate and mortgage fee (γ=5,β =_1+i1 ) i\im 0,005 0,01 0,015 0,02 0,025 0,005 4.44% 5.16% 5.98% 6.89% 7.91% 0,01 3.97% 4.61% 5.34% 6.14% 7.04% 0,015 3.56% 4.12% 4.76% 5.48% 6.27% 0,02 3.18% 3.68% 4.25% 4.88% 5.58% 0,025 2.84% 3.29% 3.79% 4.35% 4.97%

From Table 3.12, the indication is quite obvious that the higher the interest rate, the lower the welfare loss will be. However, this is like to the previous mortgage rate to welfare loss correlation, which determines that the higher mortgage rate, the higher the welfare loss. Since the interest rate in Europe is normally low, and the mortgage rate is rather higher, we can see the maximum welfare loss of homeowners in out table is even up to 7.91% which is pretty high. And it is too much unfair that only due to different preferences will lead too much welfare loss to homeowners.

Next, we are going to tune the mortgage rate imand the coefficient of risk aversion γ by

altering the interest rate i to 0.015 to see how the welfare loss changes.

Table 3.13 Welfare Loss of different risk aversion and mortgage fee (i=0.015,β =_1+i1 ) γ \im 0,005 0,01 0,015 0,02 0,025 2 1.52% 1.78% 2.07% 2.40% 2.78% 4 2.92% 3.40% 3.94% 4.54% 5.22% 6 4.15% 4.79% 5.51% 6.31% 7.19% 8 5.18% 5.94% 6.78% 7.70% 8.70% 10 6.03% 6.87% 7.78% 8.77% 9.83%

Clearly, we can see that the higher the risk aversion, the more welfare loss there will be. This shows people who prefer risk will have less welfare loss than people who do not like to take

risk. And the maximum of welfare loss in the Table 3.13 is up to 9.83%, which is relatively high for people who prefer buying house with higher risk aversion.

Conclusion

4.1

General Conclusion

In this thesis, we investigate the optimal consumption of homeowners and renters.

In chapter 2.1, we first model the consumption stream of both individuals during a life-cycle. The main difference is homeowner do not need to pay the extra fee for a house after paying off the mortgage, while renters still need to pay rent fees for the rest of their life. Then, In chapter 2.2 we discuss the optimal consumption, and verify that it is reasonable for us to keep the optimal consumption constant for a life-cycle. Also, we further investigate the deterministic optimal consumption due to different time preferences.

In chapter 2.3-2.5, we are aiming to provide the closed-form of the optimal saving amount rate to both renters and homeowners based on different optimal consumption choice. The main difference is that the saving amount of homeowners has two steps while renters have only one because the consumption of homeowners will be changed after paying off the mortgage. Meanwhile, we also have a better observation of how optimal deterministic consumption looks depending on different time preferences.

In real life, companies always set up the same premium for individuals regardless of their preferences. (see, e.g, Schroder and Skiadas). Suppose that the company is using the same saving amount rate for homeowners, then consumption will be sub-optimal. Thus, after the modeling part we analyze how large the utility difference is between sub-optimal consump-tion and optimal consumpconsump-tion

In 3.1-3.2, we have a clearer look on how consumption changes based on different mortgage rate and interest rate, and compare the utility difference of sub-optimal and optimal con-sumptions. In the end, it shows that the utility of optimal consumption is always greater than that of sub-optimal consumption, no matter what time preference it is. After comparing the utility, we further evaluate the welfare loss of homeowners to see how sensitive welfare loss

will be due to different parameters. It shows that the welfare loss will be increased with the increasing of a mortgage rate and risk aversion. However, the welfare loss of homeowners will be decreased if the interest rate goes up.

4.2

Discussion

In this thesis, our labor income is risk-free and the interest rates and mortgage rates are both constant. It will be nice if we further discuss stochastic factors instead of constant factors, and maybe add some other factors such as: inflation rate, as well. Then, we may have a better oversight of how optimal consumption, and provide more detailed and complete closed-form of optimal saving amount rates to policyholders. Also, due to the limitation of time, it is still interesting and valuable to calculate the welfare loss of homeowners in the real market and further analysis the sensitivity of welfare loss in terms with different factors mentioned in the thesis. Maybe by using the Monte Carlo Method, we can simulate the real market discount factor. Depending on that, we can simulate optimal consumptions of both renters and the optimal saving amount rate. Then, we simulate the sub-optimal and optimal consumption of homeowners to do further welfare analysis.

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R code

#initial setting Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value

#Optimal saving for people who perfer renting house i <- 0.01 #interest rate

v <- 1/(1+i)

s <- Y*(1-v^(T_D-R))/((1+i)^R-v^(T_D-R))

r_h <- 0.05 #renting house fee ratio depending on house value C_A <- Y-r_h*H-s #Optimal Consumption

savingA <- function(i){ v <- 1/(1+i) s <- 30000*(1-v^(70-50))/((1+i)^50-v^(70-50)) s } consumptionA <- function(i,r_h){ s <- savingA(i) C_A <- 30000-r_h*150000-s C_A }

i_m <- 0.02 #loan rate v_m <- 1/(1+i_m)

T_m <- 30 #pay off year

m <- H*(1-v_m)/(1-v_m^T_m) #mortgage fee s_1 <- Y*(1-v^(T_D-R))/((1+i)^R-v^(T_D-R)) -((1+i)^(R-T_m)*m-m*v^(T_D-R))/((1+i)^R-v^(T_D-R)) s_2 <- s_1+m C_B <- Y-m-s_1 savingB <- function(i,i_m){ v_m <- 1/(1+i_m) T_m <- 30 Y <- 30000 H <- 150000 R <- 50 T_D <- 70 m <- H*(1-v_m)/(1-v_m^T_m) #mortgage fee s_1 <- Y*(1-v^(T_D-R))/((1+i)^R-v^(T_D-R)) -((1+i)^(R-T_m)*m-m*v^(T_D-R))/((1+i)^R-v^(T_D-R)) s_2 <- s_1+m list(s_1,s_2) } savingB(0.01,0.02) consumptionB <- function(i,i_m){ v_m <- 1/(1+i_m) T_m <- 30 H <- 150000 m <- H*(1-v_m)/(1-v_m^T_m) Y <- 30000 C_B <- Y-as.numeric(savingB(i,i_m)[1])-m C_B } (sdfsd <- consumptionB(0.01,0.02))

#Optimal utility for individual prefer renting house beta <- 1/(1+i) gamma <- 0.5 uc_A <- rep(C_A^(1-gamma),T_D)/(1-gamma) U_CA <- 0 for(i in 1:T_D){ U_CA <- beta^(i-1)*uc_A[i]+U_CA } U_CA #----Utiltiy Function---# Utility <- function(i_m,ir,gamma,Consumption){ uc<- Consumption/(1-gamma) UC <- 0 beta <- 1/(1+ir) for(i in 1:T_D){ UC <- beta^(i-1)*uc[i]+UC } UC } #---#

#Optimal utility for individual prefer buying house beta <- 1/(1+i) gamma <- 0.5 i_m <- 0.02 T_M <- 30 uc_B <- c(rep((Y-s-m)^(1-gamma),T_M), rep((Y-s)^(1-gamma),T_D-T_M))/(1-gamma) U_CB <- 0 for(i in 1:T_D){ U_CB <- beta^(i-1)*uc_B[i]+U_CB } U_CB

#Tuning different gamma, interest rate and r_h i_r <- c(0.005,0.01,0.015,0.02,0.025)

r_h <- c(0.03,0.04,0.05,0.06,0.07) C_A <- matrix(,nrow=5,ncol=5) for(i in 1:5){ v <- 1/(1+i_r[i]) s <- Y*(1-v^(T_D-R))/((1+i_r[i])^R-v^(T_D-R)) for(k in 1:5){ C_A[i,k]<- Y-r_h[k]*H-s } } #---function---# savingA <- function(i){ v <- 1/(1+i) s <- 30000*(1-v^(70-50))/((1+i)^50-v^(70-50)) s } consumptionA <- function(i,r_h){ s <- savingA(i) C_A <- 30000-r_h*150000-s C_A } savingB <- function(i,i_m){ v_m <- 1/(1+i_m) T_m <- 30 Y <- 30000 H <- 150000 R <- 50 v <- 1/(1+i) T_D <- 70 m <- H*(1-v_m)/(1-v_m^T_m) #mortgage fee s_1 <- Y*(1-v^(T_D-R))/((1+i)^R-v^(T_D-R)) -((1+i)^(R-T_m)*m-m*v^(T_D-R))/((1+i)^R-v^(T_D-R)) s_2 <- s_1+m

as.numeric(list(s_1,s_2)) } consumptionB <- function(i,i_m){ v_m <- 1/(1+i_m) T_m <- 30 H <- 150000 m <- H*(1-v_m)/(1-v_m^T_m) Y <- 30000 C_B <- Y-as.numeric(savingB(i,i_m)[1])-m C_B } utility <- function(i,gamma,Consumption){ T_D <- 70 uc <- rep(Consumption^(1-gamma),T_D)/(1-gamma) # uc <-Consumption^(1-gamma)/(1-gamma) UC <- 0 beta <- 1/(1+i) for(i in 1:T_D){ UC <- beta^(i-1)*uc[i]+UC } UC }

#set up sub-optimal consumption stream Cstreamsub<- function(i,r_h,i_m){ s <- savingA(i) v_m <- 1/(1+i_m) T_m <- 30 Y <- 30000 H <- 150000 R <- 50 v <- 1/(1+i) T_D <- 70 m <- H*(1-v_m)/(1-v_m^T_m) C_S <- c(rep(Y-m-s,30),rep(Y-s,40))

C_S }

#calculate sub optimal CE

CE <- function(i,r_h,i_m,gamma){ Cstream <- Cstreamsub(i,r_h,i_m) U <- utility(i,gamma,Cstream) f <- function(x){ utility(i,gamma,rep(x,70))-U }

uniroot(f, lower=0.1, upper=100000)$root } #--Table 2.1---# T1 <- matrix(NA,nrow=5,ncol=5) for(i in c(0.005,0.01,0.015,0.02,0.025)){ for(r_h in c(0.04,0.05,0.06,0.07,0.08)){ T1[i/0.005,r_h*100-3] <- consumptionA(i,r_h) } } #--Table 2.2---#

T2 <- matrix(NA, nrow = 5, ncol = 5) for(gamma in c(2,4,6,8,10)){ for(i in c(0.005,0.01,0.015,0.02,0.025)){ coo <- consumptionA(i,0.05) T2[gamma/2,i/0.005] <- utility(i,gamma,coo) } } #--Table 2.3---# T3 <- matrix(NA,nrow = 1,ncol = 5) for(i_m in c(0.02,0.03,0.04,0.05,0.06)){ Cstream <- Cstreamsub(0.01,0.05,i_m) T3[i_m*100-1] <- utility(0.01,6,Cstream) }

#Table 2.4---# T4 <- matrix(NA,nrow=1,ncol=5) for(i_m in c(0.02,0.03,0.04,0.05,0.06)){ T4[i_m*100-1] <- CE(0.01,0.05,i_m,6) } #Table 2.6---# T6 <- matrix(NA,nrow=5,ncol=5) for(r_h in seq(0.1,0.9,by=0.2)){ for(i_m in seq(0.005,0.025,by=0.005)){ T6[(r_h*10+1)/2,i_m*200]<- (consumptionB(0.02,i_m) -CE(0.02,r_h,i_m,5))/consumptionB(0.02,i_m) } } #Table 2.7----# T7 <- matrix(NA,nrow=5,ncol=5) for(i in seq(0.005,0.025,by=0.005)){ for(i_m in seq(0.005,0.025,by=0.005)){ T7[i*200,i_m*200] <- (consumptionB(i,i_m) -CE(i,0.05,i_m,5))/consumptionB(i,i_m) } } data <- data.frame(seq(0.005,0.025,by=0.005) ,seq(0.005,0.025,by=0.005),T7) p <- plot_ly(data) #Table 2.8---# T8 <- matrix(NA,nrow=5,ncol=5) for(gamma in seq(2,10,by=2)){ for(i_m in seq(0.005,0.025,by=0.005)){ T8[gamma/2,i_m*200] <- (consumptionB(0.015,i_m) -CE(0.015,0.05,i_m,gamma))/consumptionB(0.015,i_m) } }

#initial setting Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value r_h <- 0.05 i <- 0.01 #interest rate gamma <- 5 alpha <- 1.01 sum <- 0 for (t in 1:50){ sum= sum+(Y-r_h*H)*(1+i)^(50-t) } sum2 <- 0 for (t in 1:70){ sum2= sum2+((alpha+i)/(1+i))^(-t/gamma)*(1+i)^(50-t) } (sum/sum2) saving <- array(70) for(t in 1:70){ saving[t] <-Y-r_h*H-(sum/sum2)*((alpha+i)/(1+i))^(-t/gamma) } sd <- array(50) for(k in 1:50){ sd[k] <- Y-r_h*H-(sum/sum2)*((alpha+i)/(1+i))^(-k/gamma) } #renters #initial setting Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value alpha <- 1.01

i <- 0.01 r_h <- 0.05 gamma <- 5 savingcoef <- function(Y,r_h,H,R,i,T_D,alpha,gamma){ sum1 <- 0 sum2 <- 0 sum3 <- 0 for(k in 1:R){ sum1 <- sum1+Y*(1+i)^(R-k) } for(k in 1:T_D){ sum2 <- sum2+r_h*H*(1+i)^{R-k} } for(k in 1:T_D){ sum3 <- sum3+((alpha+i)/(1+i))^{-k/gamma}*(1+i)^{R-k} } (sum1-sum2)/sum3 } savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) savingamount <- array(70) for(t in 1:50){ savingamount[t]=Y-r_h*H-savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } for(t in 51:70){ savingamount[t]= 0 } consumption <- array(70) for(t in 1:70){ consumption[t]=savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} }

savingamount library("ggplot2") ggplot()+geom_line(data=NULL,aes(c(1:70),savingamount, col=’savingamount’)) +geom_smooth(data=NULL,aes(c(1:70), consumption,col=’consumption’))+ labs( x=’years’, y=’’ )+ guides(color=guide_legend(title="Type")) ##house owner consumption2 <- array(70) for(t in 1:30){ consumption2[t]=Y-6566-savingamount[t] } for(t in 31:50){ consumption2[t]=Y-savingamount[t] } for(t in 51:70){ consumption2[t] <- consumption[t]+r_h*H } ggplot()+geom_line(data=NULL,aes(c(1:70), consumption2,col=’consumption’))+ labs( x=’years’, y=’’ )+ guides(color=guide_legend(title="Type")) #initial setting Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year

H <- 150000 #house value alpha <- 1.01 i <- 0.01 r_h <- 0.05 gamma <- 5 m <- 6566 T_m <- 30 savingcoef1 <- function(Y,m,R,i,T_m,alpha,gamma){ sum1 <- 0 sum2 <- 0 sum3 <- 0 for(k in 1:R){ sum1 <- sum1+Y*(1+i)^(R-k) } for(k in 1:T_m){ sum2 <- sum2+m*(1+i)^{R-k} } for(k in 1:T_D){ sum3 <- sum3+((alpha+i)/(1+i))^{-k/gamma}*(1+i)^{R-k} } (sum1-sum2)/sum3 } savingamount <- array(70) for(t in 1:30){ savingamount[t] <- Y-m-savingcoef1(Y,m,R,i,T_m,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } for(t in 31:50){ savingamount[t] <- Y-savingcoef1(Y,m,R,i,T_m,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } for(t in 51:70){ savingamount[t] <- 0 }

ggplot()+geom_line(data=NULL,aes(c(1:70), savingamount,col=’savingamount’)) consumption2 <- array(70) for(t in 1:30){ consumption2[t]=Y-6566-savingamount[t] } for(t in 31:50){ consumption2[t]=Y-savingamount[t] } for(t in 51:70){ consumption2[t] <- savingcoef1(Y,m,R,i,T_m,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } ggplot()+geom_line(data=NULL,aes(c(1:70),savingamount, col=’savingamount’))+geom_smooth(data=NULL,aes(c(1:70), consumption2,col=’consumption’))+ labs( x=’years’, y=’’ )+ guides(color=guide_legend(title="Type"))

###consumption utility for renters tuing alpha and gamma table 2.7 Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value i <- 0.01 r_h <- 0.05 utility <- function(i,gamma,Consumption){ T_D <- 70 uc <- rep(Consumption^(1-gamma),T_D)/(1-gamma) # uc <-Consumption^(1-gamma)/(1-gamma) UC <- 0 beta <- 1/(1+i) for(i in 1:T_D){ UC <- beta^(i-1)*uc[i]+UC

} UC } U2_7 <- matrix(NA,nrow=5,ncol = 5) for(gamma in c(2,4,6,8,10)){ for(alpha in c(0.98,0.99,1,1.01,1.02)){ savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) consumption <- array(70) for(t in 1:70){ consumption[t] <- savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } U2_7[gamma/2,(alpha*100-97)]<- utility(i,gamma,consumption) } }

# sub optimal consumption utility for house owner Table2.8 m <- 6566 U2_8 <- matrix(NA,nrow=5,ncol = 5) for(gamma in c(2,4,6,8,10)){ for(alpha in c(0.98,0.99,1,1.01,1.02)){ savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) consumption <- array(70) for(t in 1:30){ consumption[t] <-r_h*H-m+savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } for(t in 31:50){ consumption[t] <-r_h*H+savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } for(t in 51:70){ consumption[t] <-r_h*H+savingcoef(Y,r_h,H,R,i,T_D,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma}

}

U2_8[gamma/2,(alpha*100-97)]<- utility(i,gamma,consumption) }

}

## optimal consumption utility for house owner Table 2.9 #initial setting Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value i <- 0.01 r_h <- 0.05 m <- 6566 T_m <- 30 savingcoef1 <- function(Y,m,R,i,T_m,alpha,gamma){ sum1 <- 0 sum2 <- 0 sum3 <- 0 for(k in 1:R){ sum1 <- sum1+Y*(1+i)^(R-k) } for(k in 1:T_m){ sum2 <- sum2+m*(1+i)^{R-k} } for(k in 1:T_D){ sum3 <- sum3+((alpha+i)/(1+i))^{-k/gamma}*(1+i)^{R-k} } (sum1-sum2)/sum3 } U2_9 <- matrix(NA,nrow=5,ncol = 5) for(gamma in c(2,4,6,8,10)){ for(alpha in c(0.98,0.99,1,1.01,1.02)){ savingcoef1(Y,m,R,i,T_m,alpha,gamma) consumption <- array(70)

for(t in 1:70){ consumption[t] <- savingcoef1(Y,m,R,i,T_m,alpha,gamma) *((alpha+i)/(1+i))^{-t/gamma} } U2_9[gamma/2,(alpha*100-97)]<- utility(i,gamma,consumption) } } #Simulation on renters ## Simulate Sample Paths ## ## define model parameters M0 <- 0

theta <- 0

k <- 0.015 #interest rate beta <- 0.2 #lambda

## simulate short rate paths

n <- 100 # MC simulation trials T <- 70 # total time

m <- 70 # subintervals

dt <- T/m # difference in time each subinterval

M <- matrix(0,m+1,n) # matrix to hold short rate paths M[1,] <- M0 for(j in 1:n){ for(i in 2:(m+1)){ dM <- -(k+.5*beta^2)*dt - beta*sqrt(dt)*rnorm(1,0,1) M[i,j] <- M[i-1,j] + dM } } #---c_t*M_t---# dC <- 0

C[1,] <- 0 for(j in 1:n){ for(i in 2:(m+1)){ dC <- -(k+.5*beta^2)*dt - beta*sqrt(dt)*rnorm(1,0,1) + 1/5*(k+.5*beta^2-log(1+k))*dt+beta/5*sqrt(dt)*rnorm(1,0,1) C[i,j] <- C[i-1,j] + dC } } ## plot paths t <- seq(0, T, dt)

matplot(t, exp(M[,1:100]), type="l", lty=1,

main="Stochastic Discount Factor M_t", ylab="M_t",ylim=c(0,5)) matplot(t, exp(C[,1:100]), type="l", lty=1,

main="Stochastic Discount Factor M_t", ylab="M_t") Y <- 30000 #labor income R <- 50 #retired year T_D <- 70 #Death year H <- 150000 #house value i <- 0.015 r_h <- 0.05 m <- 6566 #mortgage fee T_m <- 30 gamma <- 5

#----Lagrange Multiplier y---#

LM <- function (T_D,R,r_h,H,Y,M,i,gamma){ sum1 <- mean(colSums((Y-r_h*H)*exp(M[1:R,]))) -mean(colSums(r_h*H*exp(M[(R+1):T_D,]))) sum2 <- mean(colSums(exp(C[1:T_D,]))) (sum1/sum2) } #----consumption----#

C2 <- matrix(NA,nrow = 70, ncol = 100) C2<- LM(T_D,R,r_h,H,Y,M,i,gamma)*exp(C[t,]) t <- seq(1, T, dt)

matplot(t, C2[,1:100], type="l", lty=1, main="Renters Optimal Consumption", ylab="Consumption",ylim = c(0,100000)) #---saving amount----#

S <- Y-r_h*H-C2[1:R,1:100] t1 <- seq(1, R, dt)

matplot(t1, S[,1:R]/Y, type="l", lty=1, main="Renters Saving Amount Rate ",

ylab="saving amount rate" ,ylim = c(-1,1)) #---Consumption House owner----#

t <- seq(1, 50, dt) C_H <- Y- m -S[1:T_m,] D_H <- Y-S[(T_m+1):R,] r <- rbind(C_H,D_H)

matplot(t, r[,1:100], type="l", lty=1, main="House Owner Sub-Optimal Consumption", ylab="Consumption",ylim = c(0,40000))

#----Lagrange Multiplier y2---# LM2 <- function (T_D,R,m,Y,M,i,gamma){ sum1 <- mean(colSums((Y)*exp(M[1:R,]))) -mean(colSums(m*exp(M[(R+1):T_D,]))) sum2 <- mean(colSums(exp(C[1:T_D,]))) (sum1/sum2) }

#---Saving amount for house owner---# C3 <- matrix(NA,nrow = 70, ncol = 100)

C3<- LM2(T_D,R,m,Y,M,i,gamma)*exp(C[t,]) t <- seq(1, T, dt)

matplot(t, C3[,1:100], type="l", lty=1, main="House Owner Optimal Consumption", ylab="Consumption",ylim = c(0,100000)) S1 <- Y-m-C3[1:T_m,1:100]

S2 <- Y-C3[(T_m+1):R,1:100] S <- rbind(S1,S2)

t1 <- seq(1, R, dt)

matplot(t1, S/Y, type="l", lty=1,

main="House Owner Saving Amount Rate ", ylab="saving amount rate" ,ylim = c(-1,1))