Convexity

The duration measure indicates that regardless of whether interest rates increase or decrease, the approximate percentage price change is the same. However, while for small changes in yield the percentage price change will be the same for an increase or decrease in yield, for large changes in yield this is not true. This suggests that duration is only a good approximation of the percentage price change for small changes in yield.

For example, using a 5% 20-year bond selling to yield 4% with a duration of 13.09. For a 10 basis point change in yield, the estimate was accurate for both an increase or a decrease in yield. However, for a 200 basis point change in yield, the approximate percentage price change was off considerably.

The reason for this result is that duration is in fact a first (linear) approximation for a small change in yield. The approximation can be improved by using a second approximation. This approximation is referred to as “convexity.” The use of this term in the industry is unfortunate because the term convexity is also used to describe the shape or curvature of the price/yield relationship. The convexity measure of a security can be used to approximate the change in price that is not explained by duration.

Break-even Inflation Rate

The break-even inflation rate (BEI) is a market-based measure of expected inflation. It is the difference between the yield of a nominal bond and an inflation-linked bond of the same maturity.

BEI is comprised of two elements: expected inflation (\pi) and risk premium for uncertainty in inflation (\theta).

The fundamental difference between the pricing formula as applied to, for example, a three-month T-bill and its application to, for example, a default-free zero-coupon bond relates to their investment horizons. The relative certainty about the real payoff from a three month T-Bill and thus the relative certainty about the amount of consumption that the investor will be able to undertake with the payoff means that the investment in the T-Bill will be a good hedge against possible bad consumption outcomes. In other words, the payoff, in real terms, from a three month T-Bill is highly unlikely to fall if the investor loses his or her job during the T-Bill’s three month investment horizon. The low, probably zero, correlation between the T-Bill’s payoff with bad consumption outcomes will mean that the risk premium needed to tempt an investor to invest in the T-Bill will be close to zero.

However, it is unlikely that the same level of certainty would apply, for example, to a 20-year default-free conventional government bond. For such a bond, it would seem reasonable to assume that the risk premium would be higher than that related to a one- or three-month T-Bill. Note that the cash flow in Equation 10 is still certain, but only in nominal terms. Because investors will naturally have less confidence in their ability to form views about future inflation over 20 years relative to their abilities to form those views over three months, the greater uncertainty about the real value of the bond’s payoff will cause investors to demand a premium in compensation for this uncertainty, represented by πt,s.

The difference between the yield on, for example, a zero-coupon default-free nominal bond and on a zero-coupon default-free real bond of the same maturity is known as the break-even inflation (BEI) rate. It should be clear from the discussion earlier that this break-even inflation rate will incorporate the inflation expectations of investors over the investment horizon of the two bonds, θt,s, plus a risk premium that will be required by investors to compensate them predominantly for uncertainty about future inflation, πt,s. Although the evolution of real zero-coupon default-free yields over time should be driven mainly by the inter-temporal rate of substitution, the evolution of their nominal equivalents will, in addition, be driven by changing expectations about inflation and changing perceptions about the uncertainty of the future inflation environment. We can see this evolution by plotting the constant maturity zero-coupon break-even inflation rates over time.

Modigliani–Miller Theorem

The Modigliani–Miller theorem is a set of two propositions on corporate capital structure. It was first proposed by Franco Modigliani and Merton Miller in 1958.

Two Propositions (No Taxes)

Modigliani and Miller made some very serious assumptions. The most important two are that there are no taxes and no costs of financial distress. Additional assumptions will be discussed in the next section. The two propositions are still true when the no taxes assumption is relaxed.

Proposition I : the market value of any firm is independent of its capital structure.

This means a firm cannot change its total value just by splitting its cash flows into different streams: The firm’s value is determined by its real assets, not by how it is financed. Thus capital structure is irrelevant as long as the firm’s investment decisions are taken as given.

Firms can not create value simply by changing the company’s capital structure.

Proposition II: the cost of equity is a linear function of the company’s debt/equity ratio.

More specifically, expected return on equity = expected return on assets + (expected return on assets – expected return on debt) * debt-equity ratio.

The mathematical representation (which can be derived from the WACC formula) is:

r_E = r_0 + (r_0 - r_D)(D/E)

According to this proposition, as the company increases its use of debt financing, the cost of equity rises. We know from MM Proposition I that the value of the company is unchanged and the weighted average cost of capital remains constant if the company changes its capital structure. What Proposition II then means is that the cost of equity increases in such a manner as to exactly offset the increased use of cheaper debt in order to maintain a constant WACC.

The risk of the equity depends on two factors: the risk of the company’s operations (business risk) and the degree of financial leverage (financial risk). Business risk determines the cost of capital, whereas the capital structure determines financial risk.

The expected rate of return on the common stock of a levered firm increases in proportion to the debt–equity ratio (D/E), expressed in market values; the rate of increase depends on the spread between r_A, the expected rate of return on a portfolio of all the firm’s securities, and r_D, the expected return on the debt.

Note that r_E = r_A if the firm has no debt.

Assumptions

  • Expectations are homogeneous. This means investors agree on the expected cash flows from a given investment. This means that all investors have the same expectations with respect to the cash flows from an investment in bonds or stocks.

  • Bonds and shares of stock are traded in perfect capital markets. This means that there are no transactions costs, no taxes, no bankruptcy costs, and everyone has the same information. In a perfect capital market, any two investments with identical cash flow streams and risk must trade for the same price.

  • Investors can borrow and lend at the risk-free rate.

  • There are no agency costs. This means that managers always act to maximize shareholder wealth.

  • The financing decision and the investment decision are independent of each other. This means that operating income is unaffected by changes in the capital structure.

  • No costs of asymmetric information

  • debtholders have prior claim to assets and income relative to equityholders, the cost of debt is less than the cost of equity

Two Propositions (With Taxes)

Weighted Average Cost of Capital

The weighted average cost of capital (WACC) is a weighted average of the after-tax required rates of return on a company’s common stock, preferred stock, and long-term debt, where the weights are the fraction of each source of financing in the company’s target capital structure.

Formula

r_{WACC} = \displaystyle\frac{D}{D+E} \times r_D \times (1-t) + \displaystyle\frac{E}{D+E} \times r_E

where

  • r_{WACC} is the WACC
  • r_D is the before-tax marginal cost of debt
  • r_E is the cost of equity
  • D denotes the market value of the shareholders’ outstanding debt
  • E denotes the market value of the shareholders’ outstanding equity
  • t is the marginal tax rate

Capital Structure

A company’s capital structure is the mix of debt and equity the company uses to finance its business. The goal of a company’s capital structure decision—the choice between how much debt and how much equity a company uses in financing its investments—is to determine the financial leverage or capital structure that maximizes the value of the company by minimizing the weighted average cost of capital. The weighted average cost of capital (WACC) is given by the weighted average of the marginal costs of financing for each type of financing used.

Portfolio

Definition

A portfolio is a specific weighting of assets.

Notation

Portfolio weight is often represented by greek letter \omega (omega).

Why Do We Need Portfolios

Diversification is protection against ignorance. It makes little sense if you know what you are doing. – Warren Buffett

The primary purpose of a portfolio is to reduce risk through diversification.

If you know with certainty that stock A is going to provide the highest return in the future, you can simply buy A. (This is technically also a portfolio with 100% weight on A and 0% weight on everything else.) But if you know two stock A and B are going to provide the highest and second highest return, but not which is which, then the best way to invest is to split your money between A and B. This is diversification.

In real life, if you can pick stocks like Warren Buffett, you can have a highly concentrated portfolio and no need to diversify. But it is extremely difficult to constantly pick good stocks.

What Makes A “Good” Portfolio

This is where different theories come in. We need to know what investors want and what they want to avoid.

One of the most influential theories is the modern portfolio theory (MPT).

Risk–Return Trade-Off

Investors invest for anticipated future returns, but those returns rarely can be predicted precisely. There will almost always be risk associated with investments. Actual or realized returns will almost always deviate from the expected return anticipated at the start of the investment period. For example, in 1931 (the worst calendar year for the market since 1926), the S&P 500 index fell by 46%. In 1933 (the best year), the index gained 55%. You can be sure that investors did not anticipate such extreme performance at the start of either of these years.

Naturally, if all else could be held equal, investors would prefer investments with the highest expected return. However, the no-free-lunch rule tells us that all else cannot be held equal. If you want higher expected returns, you will have to pay a price in terms of accepting higher investment risk. If higher expected return can be achieved without bearing extra risk, there will be a rush to buy the high-return assets, with the result that their prices will be driven up. Individuals considering investing in the asset at the now-higher price will find the investment less attractive. Its price will continue to rise until expected return is no more than commensurate with risk. At this point, investors can anticipate a “fair” return relative to the asset’s risk, but no more. Similarly, if returns were independent of risk, there would be a rush to sell high-risk assets and their prices would fall. The assets would get cheaper (improving their expected future rates of return) until they eventually were attractive enough to be included again in investor portfolios. We conclude that there should be a risk–return trade-off in the securities markets, with higher-risk assets priced to offer higher expected returns than lower-risk assets.