Chapter 10 / Saving Decisions

10.1 Interest Rate and Present Value

An important application of the theory of the consumer concerns decisions involving time, that is, saving and borrowing decisions. In the next section we will analyze the problem of an individul who lives and receives income in two periods (which we will call present and future, or period 0 and period 1) and must choose how much to consume in the present and in the future.

Interest Rate

An individual’s saving or borrowing decisions depend, among other things, on the prevailing conditions at which the individual can lend or borrow money. Lending an amount of money means making it available to the borrower, say a bank, for a certain period of time. On the contrary, borrowing an amount of money means having it available for a certain period of time, while owing it to the creditor, again say a bank. In both cases, at the end of the period the borrower must give back an amount of money greater than the capital borrowed. The difference is the interest, that is, the fee that the borrower must pay for the privilege of having the amount of money available during the period.

The interest that the borrower must pay is usually expressed as a fraction in percentage terms of the capital borrowed. This percentage is called the interest rate, denoted by $R$ in what follows. For an example, suppose that an individual saves 150 euros and the interest rate is 10% (that is, $R=0,1$). Then, in the future period, the individual will receive 165 euros, namely the capital, 150 euros, plus an interest equal to 10% of 150 euros, i.e. 15 euros.

More generally, if in the present period the individual saves $S$ euros, in the future period he will get back $(1+R)\times S$ euros.

Present Value

We have just seen that to compute the future value of a given amount of money we have to multiply that amount by $(1+R)$. Each euro saved in the present period allows us to have $1+R$ euros more in the future. Thus, saving $S$ euros today allows us to have $(1+R)S$ euros more in the future.

What if, on the contrary, we want to compute how much an individual must save today, in order to have a certain amount of money in the future? For this computation we simply do the inverse operation, that is, we divide that future amount by $(1+R)$. Suppose, for example, that the interest rate is 5% and the individual wants to save just enough to get back 210 euros in the future. The individual must then save $210/(1+R)=210/1.05=200$ euros.

Computing how much one needs to save (or borrow) today in order to have (or have to pay) a given amount of money in the future, is useful for calculating the Present Value of a flow of monetary payoffs. Suppose, for example, that the interest rate is 10% and you receive a stipend equal to $X_0=800$ euros in the present period and $X_1=1100$ euros in the future period. If in the present period you spend nothing and save your stipend entirely, in the future you will have 1980 euros:

$\begin{gathered} (1+R)\times X_0+X_1=1.1\times800+1100=1980 \end{gathered}$

The present value of the flow (denoted by $PV$) is the amount of money that, if saved in the present period, would have a future value exactly equal to the future value of the flow, namely 1980 euros:

$\begin{gathered} PV = \dfrac{(1+R)X_0 + X_1}{1+R} = \dfrac{1980}{1.1} = 1800 \end{gathered}$

The notion of present value is useful because it allows us to compare different flows of payoffs. From our discussion above, for example, we can conclude that if the interest rate is 10%, then for the individual the following scenarios are equivalent:

Receiving a stipend of 800 euros in period 0 and 1100 euros in period 1
Receiving a stipend of 1800 euros in period 0 and 0 euros in period 1
Receiving a stipend of 0 euros in period 0 and 1980 euros in period 1
Etc.

The following figure illustrates the concepts introduced in this section.

Next: Saving and Borrowing