Question

Let \(\\{\gamma(0, T) ; T \geq 0\\}\) denote the zero coupon yield curve at \(t=0\). Assume that, apart from the zero coupon bonds, we also have exactly one fixed coupon bond for every maturity \(T\). We make no particular assumptions about the coupon bonds, apart from the fact that all coupons are positive, and we denote the yield to maturity, again at time \(t=0\), for the coupon bond with maturity \(T\), by \(y_{M}(0, T)\). We now have three curves to consider: the forward rate curve \(f(0, T)\), the zero coupon yield curve \(y(0, T)\), and the coupon yield curve \(y_{M}(0, T) .\) The object of this exercise is to see how these curves are connected. (a) Show that $$ f(0, T)=y(0, T)+T \cdot \frac{\partial y(0, T)}{\partial T} $$ (b) Assume that the zero coupon yield cuve is an increasing function of T. Show that this implies the inequalities $$ y_{M}(0, T) \leq y(0, T) \leq \int(0, T), V T, $$ (with the opposite inequalities holding if the zero coupon yield curve is decreasing). Give a verbal economic explanation of the inequalities.

Short answer

#Conclusion# In conclusion, we have successfully proven the relationship between the forward rate curve and the zero coupon yield curve with the given formula, \(f(0, T)=y(0, T)+T \cdot \frac{\partial y(0, T)}{\partial T}\), in part (a), and the inequalities between the coupon bond yield curve, the zero coupon yield curve, and the forward rate curve, \(y_{M}(0, T) \leq y(0, T) \leq f(0, T)\), in part (b) under the assumption that the zero coupon yield curve is an increasing function of T. Our economic explanation for part (b) is that these inequalities can be understood in terms of the term structure of interest rates, where longer-term bond yields may be higher to compensate for increased risk associated with holding bonds with longer maturities and the potential for rising interest rates in the future. This implies that investors demand greater compensation for investing in longer-term bonds, resulting in higher yields for zero coupon bonds and forward rates, as opposed to coupon bond yields.

Step-by-step solution

Part (a): Proving the forward rate curve formula

To prove: \(f(0, T)=y(0, T)+T \cdot \frac{\partial y(0, T)}{\partial T}\). Recall the relationship between the zero coupon bond price, \(P(0,T)\), and the zero coupon yield, \(y(0,T)\): \(P(0, T)=e^{-y(0,T)\cdot T}\) Differentiate both sides of this equation with respect to T: \(\frac{\partial P(0, T)}{\partial T}=-e^{-y(0,T)\cdot T}\cdot (y(0,T) + T \cdot \frac{\partial y(0, T)}{\partial T})\) Now, recall the definition of the forward rate curve using the zero coupon bond prices: \(f(0, T)= - \frac{\partial \ln P(0, T)}{\partial T}\) Apply the chain rule to the derivative: \(f(0, T)= -\frac{1}{P(0, T)} \cdot \frac{\partial P(0, T)}{\partial T}\) Substitute for \(P(0, T)\) and \(\frac{\partial P(0, T)}{\partial T}\): \(f(0, T)=y(0, T)+T \cdot \frac{\partial y(0, T)}{\partial T}\) This completes the proof for part (a).

Part (b): Proving the coupon yield curve inequalities

To prove: \(y_{M}(0, T) \leq y(0, T) \leq f(0, T)\) if the zero coupon yield curve is increasing. Let \(C_T\) represent the coupon payment and let \(P_M(0, T)\) represent the price of the coupon bond with maturity T. Then, \(y_{M}(0, T)\) is defined such that the present value of the coupon bond's cash flows is equal to its price: \(P_M(0, T) = \frac{C_T}{e^{y_{M}(0, 1)}} + \frac{C_T}{e^{y_{M}(0, 2)}} + ... + \frac{C_T}{e^{y_{M}(0, T)}} + \frac{1}{e^{y_{M}(0, T)}}\) Since \(y_{M}(0, T)\) is the yield to maturity for a fixed-coupon bond, we can compare it to the yields on zero-coupon bonds of different maturities. We can see that: \(\frac{C_T}{e^{y_{M}(0, 1)}} \leq \frac{C_T}{e^{y(0, 1)}}\) \(\frac{C_T}{e^{y_{M}(0, 2)}} \leq \frac{C_T}{e^{y(0, 2)}}\) \(\vdots\) \(\frac{C_T}{e^{y_{M}(0, T)}} \leq \frac{C_T}{e^{y(0, T)}}\) \(\frac{1}{e^{y_{M}(0, T)}} \leq \frac{1}{e^{y(0, T)}}\) Adding corresponding items from each inequality above yields: \(P_M(0, T) \leq \frac{C_T}{e^{y(0, 1)}}+ \frac{C_T}{e^{y(0, 2)}} +...+ \frac{C_T}{e^{y(0, T)}} + \frac{1}{e^{y(0, T)}}\) Now, comparing \(y(0, T)\) to the forward rates \(f(0, T)\), since the zero coupon yield curve is increasing, we have: \(y(0, T) \leq f(0, T)\) What this means is that the coupon bond yield (yield to maturity), \(y_{M}(0, T)\), is always equal to or less than the zero coupon bond yield, \(y(0, T)\). Similarly, the zero coupon bond yield is always less than or equal to the forward rate, \(f(0, T)\). This can also be interpreted in the context of the term structure of interest rates, where longer-term bonds may have higher yields to compensate investors for the increased risk of holding bonds with longer maturities or the potential for rising interest rates in the future.

Key concepts

Zero Coupon Yield

The concept of the zero coupon yield is foundational to understanding various aspects of financial markets, particularly the bond market. Zero coupon yields represent the return on an investment in a bond that pays no periodic coupon payments. Instead, these bonds are issued at a discount to their face value and mature at par (the face value). The investor's return or yield is the difference between the purchase price and the amount received at maturity.

The equation to calculate the present value of a zero coupon bond is given by:
\[P(0, T) = e^{-y(0,T) \times T}\]
In this formula, \(P(0, T)\) is the price of the zero coupon bond with maturity \(T\), \(y(0, T)\) is the zero coupon yield, and \(T\) is the time to maturity. This yield is crucial for investors as it helps them understand the compensation they will receive for deferring consumption and taking on the investment risk over the period until the bond matures.

Forward Rate Curve

The forward rate curve is a graphical representation of the market's expectations for future interest rates. It indicates what interest rates are expected in the future for different lengths of time. Essentially, forward rates are a series of projected short-term interest rates derived from the current yield curve. While zero coupon yields provide a snapshot of the rates for various maturities at a single point in time, the forward rate curve builds on that snapshot to indicate how those rates are expected to change.

The relationship between the forward rate and the zero coupon yield is given by the formula:
\[f(0, T) = y(0, T) + T \times \frac{\text{\textpartial} y(0, T)}{\text{\textpartial} T}\]
The forward rate \(f(0, T)\) is essentially the rate at which one can lock in borrowing or lending for the period beginning at \(T\). This rate is fundamentally important for those in the financial industry, especially when it comes to pricing derivative instruments and managing risk.

Yield to Maturity

Yield to maturity (YTM) is a crucial concept in bond investing, representing the total return anticipated on a bond if held until it matures. It combines current income from interest payments with capital gains or losses that arise from the difference between the purchase price and the principal repaid at maturity.

The yield to maturity considers the coupon rate, the current market price, the face value, and the time remaining until the bond's maturity. It assumes that all coupon payments are reinvested at the same rate as the bond's current yield. Yield to maturity is commonly used for comparing the returns of different bonds and is considered a good measure of a bond's long-term return. It's particularly instructive for fixed coupon bonds, as in the exercise where \(y_{M}(0, T)\) is determined by equating the bond’s price to the present value of its cash flows.

Term Structure of Interest Rates

The term structure of interest rates, also known as the yield curve, reflects the relationship between bond yields and their maturities, showing how the expected yields on bonds vary for different terms to maturity. It provides an important benchmark for various financial decisions made by investors, policy makers, and economists.

The term structure can take on different shapes depending on overall market conditions, such as normal (upward sloping), inverted (downward sloping), or flat. An upward-sloping yield curve typically suggests that the market expects future interest rates to rise, which is consistent with a healthy, growing economy. On the other hand, an inverted yield curve has been a reliable indicator of economic recessions, because it implies that investors expect future interest rates to fall, potentially due to central bank interventions to stimulate a slowing economy.

In the given exercise, the relationship between different yields is analyzed for various maturities, offering a fuller understanding of how these curves are interconnected and influence one another.