4.3.2 Duration and Modified Duration
In the realm of fixedincome securities, understanding the sensitivity of bond prices to interest rate changes is crucial for investors. Duration and modified duration are key concepts that provide insights into this sensitivity. This section delves into the intricacies of these measures, their calculations, applications, and limitations.
Understanding Duration
Duration is a measure of the weighted average time until a bond’s cash flows are received. It provides a single number that summarizes the bond’s interest rate risk. There are two primary types of duration: Macaulay duration and modified duration.
Macaulay Duration
Macaulay duration is named after Frederick Macaulay, who introduced this concept in 1938. It is the weighted average time to receive the bond’s cash flows, where the weights are the present values of the cash flows. The formula for Macaulay duration is:
$$
\text{Macaulay Duration} = \frac{\sum_{t=1}^{n} \left( \frac{t \cdot C_t}{(1+y)^t} \right)}{P}
$$
Where:
 \( t \) = time period
 \( C_t \) = cash flow at time \( t \)
 \( y \) = yield to maturity (YTM)
 \( P \) = current bond price
Macaulay duration is expressed in years and provides a measure of the bond’s price sensitivity to interest rate changes.
Modified Duration
Modified duration adjusts the Macaulay duration to account for changes in yield. It is a more practical measure for estimating the percentage change in a bond’s price given a change in yield. The formula for modified duration is:
$$
\text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{y}{m}}
$$
Where:
 \( y \) = yield to maturity
 \( m \) = number of compounding periods per year
Modified duration is a critical tool for bond investors as it provides a direct estimate of how much a bond’s price will change with a 1% change in interest rates.
Calculating Modified Duration
To calculate modified duration, follow these steps:
 Calculate Macaulay Duration: Use the formula provided to determine the Macaulay duration of the bond.
 Adjust for Yield: Divide the Macaulay duration by \( 1 + \frac{y}{m} \) to obtain the modified duration.
Let’s illustrate this with an example.
Example Calculation
Consider a bond with the following characteristics:
 Face value: $1,000
 Coupon rate: 5%
 Yield to maturity: 4%
 Maturity: 5 years
 Annual coupon payments
Step 1: Calculate Macaulay Duration
First, calculate the present value of each cash flow and the weighted average time to receive these cash flows.
Year 
Cash Flow 
Present Value of Cash Flow 
Weight (PV / Price) 
Weighted Time 
1 
$50 
$48.08 
0.0481 
0.0481 
2 
$50 
$46.23 
0.0462 
0.0924 
3 
$50 
$44.45 
0.0445 
0.1335 
4 
$50 
$42.73 
0.0427 
0.1708 
5 
$1,050 
$860.66 
0.8607 
4.3035 
Total present value (price) = $1,042.15
Macaulay Duration = \( \frac{0.0481 + 0.0924 + 0.1335 + 0.1708 + 4.3035}{1.04215} = 4.53 \) years
Step 2: Calculate Modified Duration
Modified Duration = \( \frac{4.53}{1 + \frac{0.04}{1}} = 4.35 \)
This means that for a 1% increase in yield, the bond’s price will decrease by approximately 4.35%.
Practical Applications of Modified Duration
Modified duration is widely used in bond portfolio management for several reasons:

Estimating Price Changes: Modified duration allows investors to estimate the percentage change in a bond’s price for a given change in yield. This is crucial for understanding interest rate risk.
$$
\Delta \text{Price} \approx \text{Modified Duration} \times \Delta \text{Yield}
$$

Portfolio Immunization: By matching the duration of assets and liabilities, investors can immunize their portfolios against interest rate changes.

Risk Management: Modified duration helps in assessing the interest rate risk of a bond or a bond portfolio, enabling better risk management strategies.
Illustrating Price Change Estimation
Let’s revisit our example bond with a modified duration of 4.35. Suppose the yield increases by 1% (0.01), the estimated price change would be:
$$
\Delta \text{Price} \approx 4.35 \times 0.01 = 0.0435 \text{ or } 4.35\%
$$
If the bond’s current price is $1,042.15, the new price would be approximately:
$$
\text{New Price} = 1,042.15 \times (1  0.0435) = 997.75
$$
Limitations of Duration as a Risk Measure
While duration is a valuable tool for measuring interest rate risk, it has limitations:

Linear Approximation: Duration assumes a linear relationship between bond prices and yield changes, which is only accurate for small yield changes.

Convexity: For larger yield changes, the curvature of the priceyield relationship (convexity) must be considered. Convexity adjustment provides a more accurate estimate of price changes for significant yield movements.

Assumptions: Duration assumes that all cash flows are reinvested at the same yield, which may not be realistic in a changing interest rate environment.

Callable Bonds: Duration may not accurately reflect the interest rate risk of callable bonds, as their cash flows can change with interest rates.
Conclusion
Duration and modified duration are essential tools for bond investors, providing insights into interest rate risk and price sensitivity. While they offer valuable information, investors must be aware of their limitations and consider additional factors like convexity for more accurate risk assessment.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is Macaulay Duration?
 [x] The weighted average time until cash flows are received.
 [ ] The measure of a bond's yield to maturity.
 [ ] The time until a bond's maturity.
 [ ] The interest rate risk of a bond.
> **Explanation:** Macaulay duration is the weighted average time until a bond's cash flows are received, providing a measure of interest rate risk.
### How is modified duration calculated?
 [x] Modified Duration = Macaulay Duration / (1 + Yield per Period)
 [ ] Modified Duration = Macaulay Duration × Yield per Period
 [ ] Modified Duration = Yield per Period / Macaulay Duration
 [ ] Modified Duration = Macaulay Duration + Yield per Period
> **Explanation:** Modified duration adjusts Macaulay duration for yield changes, calculated as Macaulay Duration divided by (1 + Yield per Period).
### What is the primary use of modified duration?
 [x] Estimating the percentage change in bond price for a change in yield.
 [ ] Calculating the bond's yield to maturity.
 [ ] Determining the bond's maturity date.
 [ ] Assessing the bond's credit risk.
> **Explanation:** Modified duration is used to estimate the percentage change in a bond's price for a given change in yield, helping investors understand interest rate risk.
### What is a limitation of using duration as a risk measure?
 [x] It assumes a linear relationship between bond prices and yield changes.
 [ ] It accurately predicts bond prices for large yield changes.
 [ ] It considers the convexity of the bond.
 [ ] It is only applicable to zerocoupon bonds.
> **Explanation:** Duration assumes a linear relationship between bond prices and yield changes, which is only accurate for small yield changes.
### How does convexity relate to duration?
 [x] Convexity adjusts for the curvature of the priceyield relationship.
 [ ] Convexity is the same as duration.
 [ ] Convexity is unrelated to bond pricing.
 [ ] Convexity measures the bond's credit risk.
> **Explanation:** Convexity adjusts for the curvature of the priceyield relationship, providing a more accurate estimate of price changes for significant yield movements.
### What happens to bond prices when yields increase?
 [x] Bond prices decrease.
 [ ] Bond prices increase.
 [ ] Bond prices remain unchanged.
 [ ] Bond prices are unpredictable.
> **Explanation:** When yields increase, bond prices typically decrease due to the inverse relationship between bond prices and yields.
### Why is modified duration important for portfolio immunization?
 [x] It helps match the duration of assets and liabilities.
 [ ] It predicts future interest rates.
 [ ] It calculates the bond's coupon payments.
 [ ] It measures the bond's credit quality.
> **Explanation:** Modified duration helps in matching the duration of assets and liabilities, allowing investors to immunize their portfolios against interest rate changes.
### What is the impact of a 1% increase in yield on a bond with a modified duration of 4.35?
 [x] The bond's price decreases by approximately 4.35%.
 [ ] The bond's price increases by approximately 4.35%.
 [ ] The bond's price remains unchanged.
 [ ] The bond's price decreases by approximately 1%.
> **Explanation:** A 1% increase in yield results in an approximate 4.35% decrease in the bond's price, as indicated by its modified duration.
### What assumption does duration make about cash flows?
 [x] All cash flows are reinvested at the same yield.
 [ ] Cash flows are reinvested at varying yields.
 [ ] Cash flows are not reinvested.
 [ ] Cash flows are reinvested at the bond's coupon rate.
> **Explanation:** Duration assumes that all cash flows are reinvested at the same yield, which may not be realistic in a changing interest rate environment.
### True or False: Duration is only applicable to zerocoupon bonds.
 [ ] True
 [x] False
> **Explanation:** Duration is applicable to all types of bonds, not just zerocoupon bonds, as it measures the interest rate risk of a bond's cash flows.