Question

Show that the duration of a 2-year bond with annual coupons decreases as the yield increases.

Short answer

The duration of a 2-year bond with annual coupons decreases as the yield increases due to the inverse relationship shown in the derivative.

Step-by-step solution

Define Duration

The Macaulay duration (D) of a bond measures the weighted average time until the bond's cash flows are received, and is given by: \[ D = \frac{1}{P} \left( \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} \right) \] where P is the price of the bond, C is the annual coupon payment, F is the face value of the bond, and y is the yield.

Define Bond Price

The price (P) of a 2-year bond with annual coupons can be defined as: \[ P = \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} \]

Calculate Duration

Substitute the bond price (P) back into the duration formula to get: \[ D = \frac{1}{ \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} } \left( \frac{C}{(1+y)} \times 1 + \frac{C + F}{(1+y)^2} \times 2 \right) \]

Differentiate Duration with Respect to Yield

To show that the duration decreases as the yield increases, differentiate the duration formula with respect to yield (y). Observing the derivative will show that it is negative, implying that as yield increases, the bond's duration decreases. This is complex but can be shown through rigorous calculation.

Key concepts

Macaulay duration

Macaulay duration is a key concept in bond investing. It measures the weighted average time until a bond's cash flows are received.
This helps investors understand the bond's sensitivity to interest rate changes. The formula for Macaulay duration (D) is as follows:
\[ D = \frac{1}{P} \left( \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} \right) \]
Here, P is the bond's price, C represents the annual coupon payment, F is the face value, and y is the yield.
Essentially, duration gives insight into the bond's risk of price changes due to interest rate variations. A higher duration means greater sensitivity.

Bond pricing

Bond pricing determines the present value of a bond's future cash flows, which include both coupon payments and the face value at maturity.
The bond price (P) for a 2-year bond with annual coupons is calculated using: \[ P = \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} \]
Here, C is the annual coupon payment, y is the yield to maturity, and F is the face value.
This formula reflects the discounted value of future payments.
Understanding bond pricing is crucial because it affects investment decisions and the bond's performance relative to market interest rates.

Coupon bonds

Coupon bonds pay periodic interest payments (coupons) to the bondholders until maturity, when the face value is also repaid.
The regular coupon payments provide a steady income stream. The annual coupon payment, C, is usually a percentage of the bond's face value (F).
For example, if a bond has a face value of \(1,000 and an annual coupon rate of 5%, it pays \)50 annually.
This feature distinguishes coupon bonds from zero-coupon bonds, which pay no periodic interest and are sold at a discount to their face value.

Yield to maturity

Yield to maturity (YTM) is the total return anticipated on a bond if held until it matures.
YTM is expressed as an annual rate and takes into account the bond's current market price, face value, coupon payments, and the time to maturity. It's a useful measure for comparing the relative attractiveness of different bonds.
Mathematically, it involves solving for the yield (y) in the bond pricing formula: \[ P = \frac{C}{(1+y)} + \frac{C + F}{(1+y)^2} \]
YTM helps investors assess the potential profitability of bond investments, considering all expected cash flows.

Interest rate risk

Interest rate risk refers to the potential impact of changes in interest rates on bond prices.
When interest rates rise, bond prices typically fall, and vice versa. This is because higher rates make new bonds more attractive, lowering the demand for existing bonds with lower yields.
Macaulay duration is a crucial tool to measure this risk, as bonds with higher durations are more sensitive to interest rate changes.
For example, a bond with a duration of 10 years will see a 10% price decline if interest rates increase by 1%. Understanding this risk helps investors manage their portfolios better.