# Intertemporal budget constraint notion

Let $$W$$ denote the investor's wealth $$B$$ is the the investment in the risk free asset (suppose that the risk free rate is taken to be zero) and $$X$$ is the investment on the risky security which has price $$P$$. Se the investors total wealth is: $$W=B+PX$$ The capital gain that is generated in some some time interval $$[t-1,t]$$ is about $$X_{t-1}(P_t-P_{t-1})$$ and the portfolio rebalancing then occurring to the intertemporal budget constrained is: $$B_t+P_tX_t=B_{t-1}+P_tX_{t-1}$$ I wonder why does the last equation hold? Does this come from the intertemporal budget constraint and how?

In the sequel it refers to the wealth change that is: $$W_{t}-W_{t-1}=X_{t-1}(P_{t}-P_{t-1})$$

$$\underline{Note:}$$ It is from Kerry Back 1992 paper

## 1 Answer

At time $$(t-1)$$, the investor buys some risk free bond, $$B_{t-1}$$ and some risky asset $$X_{t-1}$$ at price $$P_{t-1}$$, such that the budget constraint holds, i.e. $$W_{t-1} = B_{t-1} + P_{t-1}X_{t-1}$$ At period $$t$$, one unit of risk free bond pays off one unit, so $$B_{t-1}$$ units of risk free bonds pays off $$B_{t-1}$$ units of wealth at $$t$$.

For a risky asset, one unit of $$X$$ is now worth $$P_t$$. Since the agent holds $$X_{t-1}$$ units of the risky asset, she effectively has wealth $$P_tX_{t-1}$$ from risky assets.

Adding the two gives us the budget at period $$t$$, i.e. $$W_{t} = B_{t-1} + P_t X_{t-1}$$ And period $$t$$ portfolio choice satisfies $$W_t = B_t + P_t X_t$$ Combining the two gives us the desired equation.

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