Question

Consider a consol bond, i.e. a bond which will forever pay one unit of cash at \(t=1,2, \ldots\) Suppose that the market yield \(y\) is constant for all maturities. (a) Compute the price, at \(t=0\), of the consol. (b) Derive a formula (in terms of an infinite series) for the duration of the consol. (c) Use (a) and Proposition \(15.11\) in order to compute an analytical formula for the duration. (d) Compute the convexity of the consol.

Short answer

In summary: (a) The price of the consol at t=0 is \(\frac{1+y}{y}\). (b) The infinite series formula for the consol bond's duration is \(-\frac{1}{y^2}\). (c) The analytical formula for the consol bond's duration using (a) and Proposition 15.11 is \(\frac{1}{y}(1+y)\). (d) The convexity of the consol is \(\frac{2(1+y)}{y^2}\).

Step-by-step solution

(Compute the price at t=0)

To find the price of the consol, we need to calculate the present value of all future cash flows. Since the bond pays one unit of cash perpetually and the market yield (y) is constant for all maturities, we can express the present value as an infinite geometric series: \(\text{Price} = \sum_{t=1}^{\infty} \frac{1}{(1+y)^t} = \frac{1}{(1+y)} + \frac{1}{(1+y)^2} + \frac{1}{(1+y)^3} + \cdots\) We know the formula for a geometric series converging to a finite value is: \(\text{Sum} = \frac{1-\text{common ratio}}{\text{first term}}\) Here, the common ratio is \(\frac{1}{(1+y)}\), and the first term is also \(\frac{1}{(1+y)}\). Therefore, \(\text{Price} = \frac{1}{1 - \frac{1}{(1+y)}} = \frac{1+y}{y}.\)

(Derive the infinite series formula for the consol bond's duration)

To find the duration of the consol bond, we will need to calculate the present value of the cash flow, weighted by time: \(\text{Duration} = \sum_{t=1}^{\infty} \frac{t}{(1+y)^t}\) We can differentiate the sum of the geometric series to find this value: \(\frac{d}{dy} \sum_{t=1}^{\infty} \frac{1}{(1+y)^t} = \frac{d}{dy} \frac{1}{y} = -\frac{1}{y^2}\) Using the rule for differentiating a power series term by term, we can write: \(-\frac{1}{y^2} = \sum_{t=1}^{\infty} \frac{-t}{(1+y)^{t+1}} = \sum_{t=1}^{\infty} \frac{t}{(1+y)^t}\) Thus, the infinite series formula for duration can be written as: \(\text{Duration} = \sum_{t=1}^{\infty} \frac{t}{(1+y)^t} = -\frac{1}{y^2}\)

(Compute an analytical formula for the duration using (a) and Proposition 15.11)

According to Proposition 15.11 (not provided but presumably relates to duration calculations), we can calculate duration as: \(\text{Duration} = \frac{-\frac{d}{dy} (\text{Price})}{\text{Price}}\) Using the price formula from (a), we can calculate the derivative of the price wrt y: \(\frac{d}{dy} (\text{Price}) = \frac{d}{dy} \left(\frac{1+y}{y}\right) = -\frac{1}{y^2}\) Therefore, \(\text{Duration} = \frac{-\frac{d}{dy} (\text{Price})}{\text{Price}} = \frac{-(-\frac{1}{y^2})}{\frac{1+y}{y}} = \frac{1}{y} (1+y)\)

(Compute the convexity of the consol)

To calculate the convexity of the consol bond, we will need to calculate the second derivative of the price with respect to y: \(\frac{d^2}{dy^2} (\text{Price}) = \frac{d}{dy} \left(-\frac{1}{y^2}\right) = \frac{2}{y^3}\) Convexity is defined as the ratio of the second derivative of the price with respect to y and the price: \(\text{Convexity} = \frac{\frac{d^2}{dy^2} (\text{Price})}{\text{Price}}\) So, \(\text{Convexity} = \frac{(\frac{2}{y^3})}{\frac{1+y}{y}} = \frac{2(1+y)}{y^2}\)

Key concepts

Understanding Market Yield

Market yield is a crucial concept when dealing with bonds, including consol bonds, which are perpetual in nature. It reflects the return that investors require to invest in a bond, given its risk level. Think of it as a benchmark interest rate that tells you how much money you can expect to make from your bond investment. When we speak of a consol bond paying one unit of cash forever, it's the market yield that will determine how much the series of these payments is worth today.

With consol bonds, because they never mature, market yield has an enduring impact on the bond's price. If the yield goes up, the price of the bond goes down. This inverse relationship is vital in understanding bond pricing and reflects the opportunity cost of holding a bond compared to other potential investments in the market. Remember, when calculating the present value of a consol bond's perpetual payments, it's the market yield that is used to discount these future cash flows.

Present Value of Cash Flows

The present value of cash flows is the sum of payments a bond will make in the future, discounted back to their value today. This concept is based on the principle of the time value of money, which posits that a dollar today is worth more than a dollar in the future due to its potential earning capacity.

For a consol bond, which has an infinite stream of cash flows, you compute the present value by treating it as an infinite geometric series. This series converges when the market yield is greater than zero, providing us with a neat formula for the bond's price. This calculation ensures that you're paying a fair price for the bond based on the current market yield.

Duration of a Bond

Duration is a measure of sensitivity of a bond's price to changes in interest rates, which offers a view of the bond's interest rate risk. Think of it as a weighted average time until the bondholder receives the bond's cash flows. In the case of a consol bond, the duration can be significant as it denotes the time frame over which the investor needs to wait to recoup the bond's price, considering the market yield.

As we have performed in Step 2 of the solution, calculating the duration of a consol involves the sum of the everlasting cash flows, weighted by their respective time periods. The longer the duration, the more sensitive the bond is to changes in interest rates. For consol bonds, the duration provides an interesting insight because it manifests the long-term commitment an investor makes considering the perpetual nature of the bond.

Convexity of a Bond

Convexity is a measure of the curvature of how the bond's duration changes as the yield changes. Bonds with higher convexity will have their prices less affected by interest rate changes than bonds with lower convexity. This is vital to consider for a consol bond due to its perpetual characteristics.

The convexity of a consol bond, as shown in Step 4, is derived from the second derivative of the price with respect to the yield. It takes into account how the bond's price will move given a change in yields, above and beyond the price change predicted by the duration alone. With convexity, investors are better prepared to understand the bond's interest rate risk profile comprehensively. This measure is especially important for consol bonds because of their higher sensitivity to interest rate changes.