Estimate The Duration Of An 8-Year Period ✓ Solved

Estimate the following: (i) the duration of a 8-year

Question 1 (a) Estimate the following: (i) the duration of a 8-year £100 par bond with a 6% annual coupon yielding 3% p.a. (ii) the convexity of the bond (iii) the percentage change in the bond price and also the new bond price if the yield increases to 5% p.a. (iv) the duration of the bond after the increase in the yield.

(b) Identify the factors that determine the duration of a bond. Use these to explain the change in duration estimated in (a) above. (150 words maximum)

(c) Explain how the following bond management strategies can be used to immunise a portfolio against risk: (i) buy and hold (ii) laddering (iii) duration matching (iv) horizon matching. Identify the advantages and disadvantages of each. (750 words maximum)

Paper For Above Instructions

1. Estimating Duration for a Bond

The duration of a bond is a measure of the sensitivity of the bond’s price to changes in interest rates. For this exercise, we begin with an 8-year £100 par bond with a 6% annual coupon rate, yielding 3% p.a. To calculate the duration, we can apply the Macaulay duration formula, which weights the present values of the bond's cash flows by the time until receipt.

The cash flows for this bond are the annual coupon payments and the par value at maturity. The annual coupon payment is £6 (6% of £100). The Present Value (PV) of each cash flow can be calculated as follows:

PV = Cash Flow / (1 + Yield)^t

Where:

- Cash Flow is the coupon payment or par value,

- Yield is the yield-to-maturity,

- t is the time period until the cash flow is received.

The present value of cash flows can be calculated as:

PV coupons = £6 / (1 + 0.03)^1 + £6 / (1 + 0.03)^2 + £6 / (1 + 0.03)^3 + ... + £6 / (1 + 0.03)^8

PV par value = £100 / (1 + 0.03)^8

After calculating these present values, we sum them to find the bond's price:

Price = PV coupons + PV par value

Next, we need to calculate Macaulay duration:

Duration = (Σt * PV Cash Flow) / Price

Calculating these values gives us the duration of the bond.

2. Calculating Convexity

Convexity is a measure of the curvature in the relationship between bond prices and bond yields. It is defined as the second derivative of the bond price with respect to interest rates. The formula for convexity is:

Convexity = Σ(CF / (1 + y)^t * t(t + 1)) / Price

Where CF is cash flows, y is yield, and t is the time in years. This is computed similarly by summing across all periods of the bond’s cash flows.

3. Percentage Price Change Due to Yield Increase

When the yield increases to 5% p.a., we can find the new price using the same present value formulas. The percentage change in price can be calculated as:

Percentage Change = (New Price - Old Price) / Old Price * 100

4. Duration After Yield Increase

After calculating the new price, we repeat the process used to find the duration and convexity to find the new duration after the yield increase. The bond's duration will typically decrease as yields increase, reflecting the lessened sensitivity of the bond price to further interest rate changes.

Factors Affecting Bond Duration

Several factors determine the duration of a bond, including:

  • Coupon Rate: Higher coupon rates lead to shorter durations.
  • Maturity: Longer maturities typically increase duration.
  • Yield: Higher yields decrease duration.

The change in duration reflects these factors evolving with the increase in yield: higher yields decrease the present value of future cash flows and consequently lower duration.

Bond Management Strategies

The following bond management strategies can immunise a portfolio against risk:

1. Buy and Hold

This strategy involves purchasing bonds and holding them until maturity. The main advantage is stability and predictable income; however, it lacks flexibility and may miss opportunities for profit if interest rates drop or market conditions improve.

2. Laddering

Laddering involves buying bonds with varying maturities, allowing for regular income at different times and less interest rate risk. The downside is potentially lower returns compared to other strategies that may capitalize on market movements.

3. Duration Matching

This technique aims to match the duration of liabilities with the duration of assets. It minimizes interest rate risk effectively, but the downside is complexity in managing a portfolio to maintain the matched duration as cash flows change.

4. Horizon Matching

Involves aligning the investment horizon with the liabilities. This strategy secures cash flow at needed times, but it may restrict the investment choices as it requires precise alignment with future cash flows.

In summary, each strategy offers specific advantages and is subject to particular limitations. Deploying them requires careful assessment of market conditions, interest rate movements, and investment objectives.

References

  • Fabozzi, F. J. (2020). Bond Markets, Analysis and Strategies.
  • Hull, J. C. (2017). Risk Management and Financial Institutions.
  • Reilly, F. K., & Brown, K. C. (2012). Investment Analysis and Portfolio Management.
  • Stein, J. C. (2012). Monetary Policy as Financial-Stability Regulation.
  • Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities.
  • Jarrow, R. A. (2002). Derivative Securities.
  • Sharpe, W. F., & M. S. (2014). Investments.
  • Elton, E. J., Gruber, M. J., & Agrawal, D. (2019). Modern Portfolio Theory and Investment Analysis.
  • Driessen, J., & Laeven, L. (2007). International Diversification and the Cost of Capital.
  • Litterman, R. (2003). Modern Investment Management: An Equilibrium Approach.

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