Valuation And Interest Rates

The value of any asset can be computed as the present value of its expected future cash flows discounted at an appropriate risk adjusted discount rate. The more uncertain the cash flows, the higher the discount rate.

Components of the discount rate are:

1. The real risk free rate (R). 2. Expected inflation (π).
2. A risk premium reflecting uncertainty about the cash flow (RP).

The value of an asset will change if either the cash flow forecasts change or any of the components of the discount rate changes. Risk premiums not only vary across assets (and asset classes), but also vary with changes in investors’ perception of risk. We will examine the decomposition of risk premiums for several asset classes in the remainder of this topic review.

The value of an asset depends on (1) its expected future cash flows and (2) the discount rate used to value those cash flows. As market participants receive new information, the timing and amounts of expected future cash flows are revised and valuations change as a result. The impact of new information will depend on its effect on current expectations so that an earnings report of 53% growth in earnings may have a positive or negative effect on the firm’s value, depending on whether expectations were for slower or more rapid growth.

Even in a world of no inflation, a default free bond has to compensate an investor for forgoing their current consumption. The investor evaluates the disutility of forgoing current consumption relative to the utility of obtaining future consumption.

The real risk free rate of interest derives from the inter temporal rate of substitution, which represents an investor’s trade off between real consumption now and real consumption in the future. Based on utility theory, we can represent this trade- off as:

For a given quantity of consumption, investors always prefer current consumption over future consumption (u0 > ut) and mt<1 as a result.

The current price (P0) of a zero coupon, inflation indexed, risk free bond that will pay $1 at time 1 can be expressed as:

P0 = E(m1)

in which case, the real risk free rate of return is:

Some key points to keep in mind:

• The higher the utility investors attach for current consumption relative to future consumption, the higher the real rate.
• Diminishing marginal utility of wealth means that an investor’s marginal utility of consumption declines as wealth increases. This suggests that marginal utility of consumption is higher during periods of scarcity, such as during economic contractions. If investors expect higher incomes in the future, their expected marginal utility of future consumption is decreased relative to current consumption.
• When investor expectations about the economy change to better economic times ahead, the expectation of higher incomes in the future will lead to an increase in current consumption and a reduction in savings. Investors will derive greater utility from current consumption relative to future consumption and would, therefore, save less. Conversely, investors expecting worse times ahead would prefer to increase future consumption by reducing current consumption and saving more.
• Investors increase their savings rate when expected returns are high or when uncertainty about their future income increases.

Risky Cash Flows and Risk Premiums

The risk aversion of investors can be explained by the covariance of an investor’s inter- temporal marginal rate of substitution and expected returns on savings. Our discussion so far was limited to risk free investments. However, if the underlying cash flows are uncertain, investors demand a risk premium for bearing the risk that comes with such uncertainty.

The investor’s expected marginal utility of a payoff is inversely related to the level of uncertainty of the payoff. Investors experience a larger loss of utility for a loss in wealth as compared to a gain in utility for an equivalent gain in wealth. This property is called risk aversion.

An investor’s absolute risk aversion declines with their wealth; wealthier investors are less risk averse and more willing to take risk relative to their poorer counterparts. However, the marginal utility of holding risky assets declines as an investor holds more risky assets in her portfolio. When the markets are in equilibrium, wealthy and poorer investors would have the same willingness to hold risky assets.

Consider a risk free, inflation indexed, zero coupon bond that an investor will sell prior to maturity. The uncertainty about the sale price gives rise to a risk premium. The price of the bond will be lower than the expected sale price discounted at the real risk free rate.

We can model this risk premium as:

The covariance between the expected future price of the bond and the investor’s inter- temporal rate of substitution can be viewed as a risk premium. Now imagine this relationship in the context of a risky asset (e.g., stocks). For risk averse investors, the covariance is negative; when the expected future price of the asset is high, the marginal utility of future consumption relative to current consumption is low.

This is because during good economic times, both investors’ labor incomes and most risky asset values are high. However, with higher future labor incomes, the marginal utility of future consumption is lower. The resulting negative covariance between the marginal utility of consumption and asset prices reduces the value of the asset for a given expected sale price, P1.

Everything else constant, the lower current price (P0) increases expected return. This higher expected return is due to a positive risk premium.

For a single period risk free bond, the covariance is zero as there is no uncertainty about the terminal value; there is no risk premium.

GDP Growth Rates

If GDP growth is forecasted to be high, the utility of consumption in the future (when incomes will be high) will be low and the inter temporal rate of substitution will fall; investors will save less, increasing real interest rates. Therefore, real interest rates will be positively correlated with real GDP growth rates. This is consistent with the existence of high real rates in rapidly growing developing economies such as those of India and China.

Interest rates are also positively correlated with the expected volatility in GDP growth due to higher risk premium.

So far we have not considered the implications of inflation in our analysis of the correlation between interest rates and GDP growth. Nominal risk free interest rates include a premium for expected inflation (π). However, actual inflation is uncertain.

This additional risk gives rise to an additional risk premium for the uncertainty about actual inflation (θ). This risk premium is higher for longer maturity bonds.

For short term risk free securities (e.g., T bills), the uncertainty about inflation is negligible and, therefore, the nominal interest rate (r) would be comprised of real risk- free rate (R) and expected inflation (π):

r(short term) = R + π

For longer term bonds, we add the risk premium for uncertainty about inflation, θ:

r(long term) = R + π + θ

Taylor Rule

Central banks are usually charged with setting policy rates so as to (1) maintain price stability and (2) achieve the maximum sustainable level of employment. The Taylor rule links the central bank’s policy rate to economic conditions (employment level and inflation):

Central banks can moderate the business cycle by making appropriate changes to the policy rate or can magnify the cycle by not responding appropriately to changing economic conditions (e.g., committing policy errors such as keeping rates too low).

Business Cycle and Slope of the Yield Curve

When the economy is in recession, policy rates tend to be low. Investors’ improving expectations about future GDP growth and increasing inflation as the economy comes out of recession leads to higher longer term rates. This results in a positively sloped yield curve.

Conversely, expectations of a decline in GDP growth results in a negatively sloped (inverted) yield curve. For this reason, an inverted yield curve is often considered a predictor of future recessions. Later stages of an economic expansion often are characterized by high inflation and high short term interest rates, while longer term rates tend to be low, reflecting investor’s expectation of decreasing inflation and GDP growth.

A term spread is the difference between the yield on a longer term bond yield and the yield on a short term bond. Evidence suggests that normal term spread is positive so the yield curve is upward sloping. Recall that the risk premium for uncertainty in inflation (θ) is higher for longer maturity bonds.

Positive term spreads can be attributed to increasing θ for longer periods.

The difference between the yield of a non inflation indexed risk free bond and the yield of an inflation indexed risk free bond of the same maturity is the break even inflation rate (BEI).

BEI = yield on non inflation indexed bond − yield on inflation indexed bond

Recall that for longer maturity bonds, the nominal rate is composed of the real rate, expected inflation, and a risk premium for inflation uncertainty. Therefore, BEI is composed of two elements: expected inflation (π) and a risk premium for uncertainty about actual inflation (θ).

BEI = π + θ

The required rate of return for bonds with credit risk includes an additional risk premium. This credit risk premium (credit spread) is the difference in yield between a credit risky bond and a default free bond of the same maturity.

required rate of return for credit risky bonds = R + π + θ + γ

where: γ = additional risk premium for credit risk = credit spread Credit spreads tend to rise during times of economic downturns and fall during expansions. Research has shown that defaults increase, and recovery rates decrease, during periods of economic weakness. Both effects result in greater credit losses during economic downturns.

When credit spreads narrows, credit risky bonds will outperform default free bonds. Overall, lower rated bonds tend to benefit more than higher rated bonds from a narrowing of credit spreads (their yields fall more). Conversely, when credit spreads widen, higher rated bonds will outperform lower rated bonds on a relative basis (because their yields will rise less).