Tobin’s Portfolio Diversification Model

Tobin’s Portfolio Diversification Model

Tobin developed a model to explain portfolio diversification that
did not require individuals to have an idea of a ‘normal’ interest
rate.

Basic idea:
This model is based on the idea that people are risk averse and
therefore choose to diversify their assets

Assumptions:
There is no ‘normal’ interest rate in Tobin’s model;
The returns on bonds = r + G.
Capital gains on bonds (G) can be positive or negative
The individual believes that gains and losses are equally
likely so that the expected capital gain is zero. Therefore the
best expectation of the total return on bonds is just the
interest rate, r. Bonds it must be noted are risky assets as
there is a possibility of capital loss when bond prices fall.

Money, on the other hand, is a safe asset by has zero
returns.

There is therefore a trade-off between bonds and money. By
holding more bonds, a person will enjoy higher returns but there
is also the risk of holding one’s wealth in bonds. Hence, a risk
averse person is willing to give up some of this possible return for
the security of holding some money.

Theory assumption

Let’s generalize what we have discussed. Assume that the
individual makes N trips to the bank over the period. On each trip
he will withdraw Y/N dollars and he then spends the money
gradually over the period. Money holdings therefore vary from Y/N
to zero and this averages to Y/(2N).

The question that we need to answer is what is the optimal choice
of N bearing in mind that the greater N is, the less money he holds
on average and therefore the less interest he forgoes. But at the
same time the greater N is, the more trips he has to make to the
bank, which increases his inconvenience.

Let’s assume that
the cost of going to the bank is a fixed amount of $F;
and
r, represent the interest foregone or the opportunity cost
of holding money.


For any N, the average amount of money held is Y/(2N). Therefore
the forgone interest is rY/(2N). The cost of making trips to the bank
is FN. Therefore, the total cost the individual bears is :
Total cost (C) = rY/(2N) + FN

The greater is N, the smaller is the cost of interest forgone but the
greater is the cost of the trips.


The individual holds more money:
if the fixed cost of going to the bank increases;
if expenditure (Y) is higher; and
when r is lower.

So contrary to the classical economist and Keynes, in Baumol’s
model, transaction demand for money is inversely related to interest
rate.

Baumol’s Inventory Approach Theory

This theory suggests that the level of inventory holding of money
depends on two factors:

the carrying cost of holding money, i.e. the interest
foregone by holding money and not bonds; and

the cost of making a transfer between money and bond
- transaction cost.

Alternatively the individual could invest a proportion of the initial
income payment in bonds and then sell the bonds when additional
money is needed for transactions. If the individual invested
his income payment in bonds at the beginning of the month,
the time profile of his money holdings will be as follows.


Money holdings at the beginning of the month is Y/2 and money
holdings are run down to zero by the midpoint of the period at a
uniform rate. Average money holding for the first half of the
period is therefore Y/4. At the mid point of the period, bonds are
sold causing money holdings to return to Y/2. This is then
spent at a uniform rate over the remaining half of the period. The
average money holding for the second half of the period is Y/4.
Thus the average money holding for the whole period is Y/4,
which is lower than Y/2 (the case where no bonds are held). The
advantage of this option is that he forgoes less interest since he is
holding less money on the average but the disadvantage is that he
has incurred some transaction costs.

Question

Question: How much money will an individual hold to make
payment for the things he buys?

Assuming that:
an individual receives an income of Y at the beginning of
the period (t=0 e.g. start of the month);
(ii) the individual spends this income at a constant rate such that he has exactly no money left at the end of the month (t=1);

Therefore, the average inventory money held during the
period = Y/2 , which is also the amount that will be held at the
midpoint of the period t/2.

Baumol’s Inventory Approach

Baumol’s Inventory Approach (Baumol-Tobin Model of
Cash Management)

Later economists further developed the Keynesian approach
to the demand for money. The transaction demand for money
according to them is also sensitive to interest rate.
The transaction demand for money arises because of the lack of
synchronization between receipts and payments. Hence, there is
a need to keep some money at hand to pay for the things we
buy.

Speculative Demand for money 3

If people are plungers and they believe that the normal interest
rate re, they will hold either all bonds or all money. Therefore the
speculative demand for money is discontinuous. A discontinuous
money demand function can only happen if people are plungers,
i.e. they will put all or none of their money into bonds.

As different individuals have different expectations regarding
interest rate and aggregation of the demand for money will yield a
demand for money curve that slopes continuously downwards.

Taking both his concepts of transaction demand and
speculative demand for money together, the Keynesian money
demand function can be represented as:
Md / P = f( r , Y)

Speculative Demand for money 2

Taken together, when interest rate is high, the high interest
payment foregone as well as a likely capital gain on bonds means
that demand for money (as a store of value) will be low. As
interest rate falls, the demand for money as an asset would increase.
Therefore according to Keynes, the demand for money as a store
of value is inversely related to interest rate. This is Keynes
speculative demand for money.

If current interest rate is higher than re, people would
rather hold all their extra money in bonds in the hope of making
capital gains when interest rate returns to the normal level. If
current interest rate is lower than re, people would prefer to hold
money in idle cash, so as to avoid capital losses when interest
rate returns to the normal level.