Financial planning for young households

Abstract

We analyze the financial planning problems of young households whose main decisions are how to finance the purchase of a house (liabilities) and how to allocate investments in pension savings schemes (assets). The problems are solved using a multi-stage stochastic programming model where the uncertainty is described by a scenario tree generated from a vector auto-regressive process for equity returns and interest rate evolution. We find strong evidence of the importance of taking into account the multi-stage nature of the problem, as well as the need to consider the asset and liability sides jointly.

Appendix: The full stochastic programming formulation

The following parameters and variables are used in the stochastic programming problem. (A “parameter” is something that can be specified/calculated/makes sense independently of any agent’s optimization problem. A “variable” is something that is chosen by the agent—however trivially—as part of the optimization problem.)

1.1 A.1 Parameters

minCVaR: the minimal conditional value-at-risk the agent is willing to accept;

p _n : probability of being at node n;

d _t : discount factor for time t viewed from time 0;

r _in : coupon rate for loan i at node n,

PP _in : the net present value of prepayments from one unit of loan i at node n including any retirement of the debt at prevailing market prices;

r _jtn : annual return on asset instrument j at node n, time t;

K _in : price of prepaying loan i at node n. We have that K _in =min{1,k _in } for FRMs and K _in =k _in for ARMs where k _in is the market price for the bond underlying loan i at node n;

I _t : the investor’s gross income at time t;

IA: the cost of buying the house;

P _in : price of loan i at node n;

c: variable transaction cost for issuing a loan;

c _l : fixed transaction cost for issuing a loan;

c _f : fixed cost associated with the mortgage origination;

admC: administration fee rates given as a percentage of outstanding debt for bond i;

γ: tax reduction rate from interest payments;

tx _l : low-bracket tax rate;

tx _h : high-bracket tax rate;

ba _h : basic allowance in high-bracket tax;

α: confidence (quantile) level for VaR and CVaR (∼5 % worst cases in our applications);

M: large constant (the big M);

cmin: minimum amount used for consumption;

tax _pal : tax rate on earnings from retirement investments;

β: fixed percentage of the household’s income that is used for contribution to retirement savings;

rd _tn : interest rate on savings in the bank account at time t and node n.

1.2 A.2 Variables

y _itn : units sold of loan i at node n, time t;

x _itn : units bought of loan i at node n, time t;

Z _itn : $\left\{ \begin{array}{@{}l@{\quad }l} 1 & \mbox{if any amount of loan } i \mbox{ is issued/sold},\\ 0 & \mbox{otherwise}; \end{array} \right.$

cref _tn : total costs of refinancing the loans at node n, time t;

LP _tn : $\left\{ \begin{array}{@{}l@{\quad }l} 1 & \mbox{if a loan is refinanced},\\ 0 & \mbox{otherwise}; \end{array} \right.$

A _itn : principal payment of loan i at node n, time t;

BT _tn : total payments on all loans at node n, time t, including interest and principal payments and price of prepaying the loans at horizon;

κ _jtn : percentage of income used to invest in instrument j at node n, time t;

VP _n : value of retirement investments before tax;

VH _n : horizon value of retirement savings after tax;

W _n : total wealth at horizon at node n;

C _tn : amount needed for consumption at node n, time t;

x1 _tn : part of gross income which is taxed with low-bracket tax rate;

x2 _tn : part of gross income which is taxed with high-bracket tax rate;

OD _itn : outstanding mortgage debt of loan i at node n, time t;

$\mathit{IR}^{+}_{tn}$: interest payments at node n, time t;

$\mathit{IR}^{-}_{tn}$: interest earnings at node n, time t;

$v^{+}_{tn}$: part of interest earnings taxed as income;

$v^{-}_{tn}$: part of interest payments deducted in tax;

CD _tn : deposit (or withdrawal) in the savings bank account at node n, time t;

CF _tn : balance of the savings bank account at node n, time t;

VaR: value-at-risk;

CVaR: conditional value-at-risk.

1.3 A.3 The liability (or mortgage) side

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1.4 A.4 The asset (or pension) side

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1.5 A.5 Taxes etc.

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