The input variables for determining Macaulay and modified yield duration of fixed-rate bonds are:
By changing one of the above variables while holding others constant, we can analyze the properties of bond duration, which in turn, helps us assess the interest rate risk. We will use the formula for Macaulay duration to understand the relationship between each variable and duration:
The fraction of the coupon period that has gone by (t/T)
First, let us consider the relationship between fraction of time that has gone by (t/T) and duration. There is no change to the expression in braces {} as time passes by. Fraction of time (t/T) increases as time passes by. Assume T is 180. If 50 days have passed, then t/T = 0.277. If 90 days have passed, then t/T = 0.5. If 150 days have passed, then t/T = 0.83. As t/T increases from t = 0 to t = T with passing time, MacDuration decreases in value. Once the coupon is paid, t/T becomes zero and MacDuration jumps in value. When time to maturity is plotted against MacDuration, it creates a saw tooth pattern as shown graphically below:
Time-to-Maturity
The relationship between Macaulay Duration and time-to-maturity for four types of bonds (zero-coupon, discount, premium, and perpetuity) at t/T = 0 are shown.
The above points are summarized below:
Relationship between bond duration and other input parameters | |
Bond parameter | Effect on Duration |
Higher coupon rate | Lower |
Higher yield-to-maturity | Lower |
Longer time-to-maturity | Higher for a premium bond. Usually, holds true for a discount bond, but there can be exceptions. Exception: low coupon (relative to YTM) bond with long maturity. |
Example 11: Calculating the approximate modified duration
A mutual fund specializes in investments in sovereign debt. The mutual fund plans to take a position on one of these available bonds.
Bond | Time-to-Maturity | Coupon Rate | Price | Yield-to-Maturity |
(A) | 5 years | 10% | 70.093879 | 20% |
(B) | 10 years | 10% | 58.075279 | 20% |
(C) | 15 years | 10% | 53.245274 | 20% |
The coupon payments are annual. The yields-to-maturity are effective annual rates. The prices are per 100 of par value.
Solution to 1:
Calculate PV_{+} and PV_{–}; then calculate modified duration.
Bond A:
PV_{0 }= 70.093879;
PV_{+} = 69.977386
70.210641
= 70.210641
ApproxModDur = = 3.33.
The approximate modified duration of Bond A is 3.33.
Bond B:
PV_{0 }= 58.075279
PV_{+} = = 57.937075
PV_{+} = 57.937075
PV_{–}= = 58.213993
PV_{–}= 58.213993
ApproxModDur = = 4.77
The approximate modified duration of Bond B is 4.77.
Bond C:
PV_{0 }= 53.245274
PV_{+} = = 53.108412
PV_{+} = 53.108412
PV_{–} = = 53.382753
PV_{–} = 53.382753
ApproxModDur = = 5.15.
The approximate modified duration of Bond C is 5.15.
Solution to 2:
Bond C with 15 years-to-maturity has the highest modified duration. If the yield to maturity on each is decreased by the same amount – for instance, by 10bps, from 20% to 19.90% – Bond C would be expected to have the highest percentage price increase because it has the highest modified duration.
Interest Rate Risk Characteristics of Callable Bond
A callable bond is one that might be called by the issuer before maturity. This makes the cash flows uncertain. So, the YTM cannot be determined with certainty. The exhibit below plots the price-yield curve for a non-callable/straight bond and a callable bond. It also plots the change in price for a change in the benchmark yield curve. Bond price is plotted on the y-axis and the benchmark yield on the horizontal axis.
Interest Rate Risk Characteristics of Putable Bond