###### Calibrating a Binomial Interest Rate T ...

The following steps should be followed when calibrating binomial interest rate trees to... **Read More**

Consider an investor investing in a default-free bond. Further, assume that there is no inflation. The investor would still be entitled to compensation for not consuming today. This compensation is the opportunity cost of consuming today. The investor can either invest \(P_0\) today in a default-free bond that pays one monetary unit in the future (after \(t\) periods) or buy goods worth the same amount.

The decision to purchase the bond depends on the willingness of the investor to substitute today’s consumption with future consumption. This is measured by the marginal utility of consumption in the future, relative to the marginal utility of consumption today. The **intertemporal rate of substitution** \((\bf{ {m}_{t}})\) is given by:

$$ m_t=\frac{u_t}{u_0} $$

Where:

- \(u_t\) is the marginal utility of consumption after t periods.
- \(u_0\) is the marginal utility of consumption today.

The intertemporal rate of substitution is usually less than one since investors prefer today’s consumption to future consumption.

Investors demand compensation for delaying consumption. If they consume less today relative to their average consumption, they demand a higher reward in the form of interest rates. Therefore, the price of a zero-coupon bond is equal to the inter-temporal rate of substitution. That is:

$$ P=E\left(m_t\right) $$

The one-period risk-free rate of a default-free bond that will pay one monetary unit after one period is given by:

$$ R=\frac{1}{E\left(m_1\right)}-1 $$

**Notes to candidates:**

- Real interest rates are high when the expected consumption growth is high (intertemporal substitution) or when risk is low (precautionary saving).
- During good economic times, current consumption is high since individuals have relatively high current income. This makes the marginal utility of consumption low today.
- As wealth increases, investors’ marginal utility of consumption declines since they have already satisfied their basic needs. Investors would, therefore, benefit more from an asset that pays off more in bad economic times relative to one that pays off in good economic times.

A risk-averse individual is one who demands compensation for uncertainty. An investor’s absolute risk aversion decreases with an increase in wealth or income. In particular, an investor invests more in risky assets as their wealth or income increases. This decreases the marginal utility of holding risky assets.

Consequently, the risk premium is lower for wealthier individuals, which implies that wealthier individuals are more willing to bear a given risk relative to more indigent individuals. However, when the markets are at equilibrium, all investors, regardless of whether they are wealthy or poor, have the same willingness to hold risky assets.

The value of non-default-free assets is established relative to the value of the default-free bond. For instance, assume that an investor is holding a security for only one period and its current price is \(P_0\). In this case, the bond has value \(P_1^\prime\) in one period.

If the bond is sold before maturity at market price, the price in one period is unpredictable.

The following is the pricing relationship:

$$ P_0=\frac{E(P_1^\prime)}{1+R}+Cov(P_1^\prime,m_1) $$

Where \(R\) is the risk-free rate.

GDP growth forecast is usually high since we expect more goods in the future than today. An increase in real GDP growth leads to an increase in the real default-free interest rate. Consequently, investors’ willingness to substitute across time falls. Subsequently, there will be less saving and more borrowing. The end result of all these is an increment in the real default-free rate of interest.

Remember, however, that it is not easy to correctly predict GDP growth from one period to another. Under these uncertainties, the interest rates are positively related to the expected GDP growth rate, as well as the expected volatility of GDP growth.

In conclusion, we expect a higher average level of real short-term interest rate in an economy with high and volatile growth, all else being equal. The converse is also true. We can deduce that due to a higher risk premium, the real short-term interest rate is positively correlated with the volatility in GDP growth.

## Question

Which of the following statements is

most likelycorrect?

- An increase in real GDP growth implies the real default-free rate of interest will decrease.
- An increase in real GDP growth leads to an increase in the real default-free interest rate.
- An increase in real GDP growth implies less borrowing and more savings.
## Solution

The correct answer is B.An increase in real GDP growth leads to an increase in the real default-free interest rate. Consequently, investors’ willingness to substitute across time falls, and as a result, there will be less saving and more borrowing.

As a result, the real default-free rate of interest will increase. However, it is not easy to correctly predict the GDP growth from a period to another. Under these uncertainties, the interest rates will still be positively related to the expected GDP growth rate, as well as the expected volatility of real GDP growth.

Reading 44: Economics and Investment Markets

*LOS 44 (c) Explain the relationship between the long-term growth rate of the economy, the volatility of the growth rate, and the average level of real short-term interest rates.*