Bond Pricing and Yield Curve Analysis

Yield to Maturity and Returns

Yield to Maturity (YTM)

The Yield to Maturity is the internal rate of return of a bond. It is the interest rate that equates the present value of all future cash flows to the current market price. It assumes that all coupons are reinvested at that same rate.

Holding-Period Return (HPR)

While YTM is a proxy for the average return if held to maturity, the Holding-Period Return (HPR) measures the actual return over a specific period, which may be shorter than the bond's life.

$$\text{HPR} = \frac{\text{Sell Price} - \text{Buy Price} + \text{Coupon(s)}}{\text{Buy Price}}$$

Exam Tip: Remember the inverse relationship between price and yield. When market interest rates rise, bond prices fall, and vice versa.

The Yield Curve: Pure vs. On-the-Run

The yield curve is a graphical representation of the relationship between yield to maturity and time to maturity. Investors use it to gauge market expectations for future interest rates.

There are two primary ways to view the yield curve:

1. Pure Yield Curve: This refers to the curve constructed from stripped, zero-coupon Treasury securities.
2. On-the-Run Yield Curve: This refers to the yields of recently issued coupon bonds that are currently trading at or near par value. This is the version most commonly published in financial media.

The shape of the curve can vary: it can be flat, rising (upward sloping), inverted (downward sloping), or hump-shaped. An inverted yield curve is often viewed as a signal of an impending recession.

Spot Rates, Short Rates, and Forward Rates

To analyze the term structure deeply, we must distinguish between different types of interest rates.

Spot Rates and Short Rates

• Spot Rate: The yield to maturity of a zero-coupon bond for a specific maturity.
• Short Rate: The interest rate applicable to a specific time interval (e.g., a 1-year rate).

Forward Rates

When we move from a world of certainty to one of uncertainty, we use forward rates to describe the interest rate for a future period. A forward rate is essentially a forecast of a future short rate.

The relationship between the spot rate ($y$) and the forward rate ($f$) is defined by the following equation:

$$\begin{aligned} (1 + y_n)^n &= (1 + y_{n-1})^{n-1} \times (1 + f_n) \end{aligned}$$

For example, if the 3-year spot rate is 7% and the 4-year spot rate is 8%, the implied forward rate for the fourth year ($f_4$) is:

$$\begin{aligned} 1 + f_4 &= \frac{(1.08)^4}{(1.07)^3} \\ 1 + f_4 &= 1.1106 \\ f_4 &= 11.06\% \end{aligned}$$

Theories of the Term Structure

Why does the yield curve take its shape?

• Expectations Hypothesis: Suggests that forward rates are simply the market's consensus expectation of future short rates.
• Liquidity Preference Theory: Argues that investors demand a "liquidity premium" to hold longer-term bonds because they are riskier. This theory predicts an upward-sloping yield curve even if short rates are expected to remain constant.
• Market Segmentation Theory: Suggests that different maturity bonds trade in separate, unconnected markets, and the curve is determined by the supply and demand in each segment.