Formation of a Bubble and Its Background

Abstract

Japan experienced a formation of a huge bubble in the 1980s, which turned out to be a prelude to the subsequent two-decade-long economic stagnation. In this chapter, we will look at what happened to the asset price (land and stocks) in Japan in the 1980s (Section 1), and why it happened and, what could (should) have been done to contain the bubble (Section 2). In examining causes of the bubble, we will look at two factors: economic policy management in the 1980s and bank behaviors under financial liberalization from the latter half of the 1970s.

Supplement

1.1 Theoretical Price of an Asset

1.1.1 Definition of an Asset Price Bubble

A asset price bubble is a phenomenon wherein the price of an asset rises to a level that is not based on fundamentals.

1.1.2 Fundamental Price and Its Derivation

1.1.2.1 Fundamental Price

The price based on fundamentals (=theoretical price) is expressed as below:

If the asset has no risk and the return is fixed:

$$ \mathrm{P}=\mathrm{R}/\mathrm{i}, $$

(2.1)

where R = return (office or housing rent, interest, etc.) to an asset, P = price of asset, i = long-term, risk-free interest rate (typically, long-term government bond yield in a secondary market).

If the asset has a risk and the return to the asset grows each period at a fixed rate of g, the fundamental price is rewritten as follows:

$$ \mathrm{P}=\mathrm{R}/\left(\mathrm{i}+\alpha -\mathrm{g}\right), $$

(2.2)

where α is the risk premium (an additional return that is required for a risk asset to be held, on top of the return to a risk-free asset)

1.1.2.2 Derivation

1.1.2.2.1 Risk-free Case

Derivation of theoretical price (price based on fundamentals)

Assumption: There is no difference in risk profile across assets

Formulation: The rate of return will be equalized across different assets via arbitrage, so that we have

$$ \mathrm{R}/\mathrm{P}=\mathrm{i} $$

(2.3)

Note: Arbitrage may be defined as a simultaneous purchase and sale of assets to make profits from price discrepancy.

Then, we have:

$$ \mathrm{P}=\mathrm{R}/\mathrm{i} $$

(2.4)

1.1.2.2.2 Risk Asset Case

If the asset has a risk, then its rate of return has to be higher than that of risk-free assets (this additional part is called the risk premium (α) as explained before). Then, instead of (2.4), we have

$$ \mathrm{P}=\mathrm{R}/\left(\mathrm{i}+\alpha \right) $$

(2.5)

The aforementioned assumes that the asset price is constant. If the asset price changes, then, the formulation changes as follows.

The rate of return for buying the asset in period 1 for P₁ and selling it in period 2 for P₂ is shown in the left-hand side of Equation (2.6). and it will be equal to the rate of return of the risk-free asset on the right-hand side of the equation.

$$ \left(\mathrm{R}+\left({\mathrm{P}}_2-{\mathrm{P}}_1\right)\right)/{\mathrm{P}}_1=\mathrm{i} $$

(2.6)

Or,

$$ {\mathrm{P}}_1=\left(\mathrm{R}+{\mathrm{P}}_2\right)/\left(1+\mathrm{i}\right) $$

(2.7)

In the same way, the asset price in period 2 (P₂) may be expressed as follows.

$$ {\mathrm{P}}_2=\left(\mathrm{R}+{\mathrm{P}}_3\right)/\left(1+\mathrm{i}\right) $$

(2.8)

By inserting (2.8) into (2.7), we have

$$ {\mathrm{P}}_1=\left(\mathrm{R}+\left(\mathrm{R}+{\mathrm{P}}_3\right)/\left(1+\mathrm{i}\right)\right)/\left(1+\mathrm{i}\right) $$

(2.9)

Or,

$$ {\mathrm{P}}_1=\mathrm{R}/\left(1+\mathrm{i}\right)+\mathrm{R}/{\left(1+\mathrm{i}\right)}^2+{\mathrm{P}}_3/{\left(1+\mathrm{i}\right)}^3 $$

(2.10)

By repeating this, we obtain;

$$ {\mathrm{P}}_1=\mathrm{R}/\left(1+\mathrm{i}\right)+\mathrm{R}/{\left(1+\mathrm{i}\right)}^2+\mathrm{R}/{\left(1+\mathrm{i}\right)}^3+\cdots, $$

(2.11)

which means that P₁ is equal to the discounted present value of future returns

1.1.2.2.3 Growth of Return

If we assume that the return (R) grows every period at a fixed rate of g, then we have

$$ {\mathrm{P}}_{11}=\mathrm{R}/\left(1+\mathrm{i}\right)+\mathrm{R}\left(1+\mathrm{g}\right)/{\left(1+\mathrm{i}\right)}^2+\mathrm{R}{\left(1+\mathrm{g}\right)}^2/{\left(1+\mathrm{i}\right)}^3+\cdots $$

(2.12)

As this is the sum of geometrical series, assuming g < i, it can be written as:

$$ {\mathrm{P}}_1=\mathrm{R}/\left(\mathrm{i}-\mathrm{g}\right) $$

(2.13)

If the asset has a risk, then we use i+α instead of i:

$$ {\mathrm{P}}_1=\mathrm{R}/\left(\mathrm{i}+\alpha -\mathrm{g}\right) $$

(2.14)

Note: The above explanation is from Chapter 2 of Nenji Keizai Hokoku (Annual Economic Report) (1991).

1.2 Interest Rate-Adjusted PER and the Ratio of the Actual to Theoretical Stock Price

The theoretical price of stock is expressed by Eq. (2.1) (i.e., P = R/(i + α − g)).

If we assume no risk and no growth in return, the theoretical stock price is expressed by Eq. (2.2) (i.e., P = R/i).

Assuming the return (R) in Eq. (2.2) can be replaced with profit per one unit of stock (ρ), then the theoretical stock price (P) is expressed as

$$ \mathrm{P}=\rho /\mathrm{i} $$

(2.15)

The interest rate- adjusted PER (i × (Actual stock price/profit per one unit of stock (ρ))) corresponds to the ratio of Actual stock price to Theoretical stock price:

$$ {\displaystyle \begin{array}{l}\mathrm{iPER}=\mathrm{i}\times \left(\mathrm{Actual}\ \mathrm{stock}\ \mathrm{price}/\rho \right)=\mathrm{Actual}\ \mathrm{price}/\\ {}\left(\rho /\mathrm{i}\right)=\mathrm{Actual}\ \mathrm{stock}\ \mathrm{price}/\mathrm{Theoretical}\ \mathrm{stock}\ \mathrm{price}\end{array}} $$

(2.16)

The theoretical price rises if the risk premium decreases, and/or the expected growth rate of return increases

Note: PER = price/earning ratio, that is, stock price divided by profit per one unit of stock