Learning Module 13: Curve-Based and Empirical Fixed-Income Risk Measures
Fixed Income
Effective Duration (EffDur)
\[ EffDur = \frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Curve) \times (PV_0)}. \]
- Effective duration and effective convexity are useful for gauging the interest rate risk of bonds whose future cash flows are uncertain.
- Effective duration and effective convexity can be used to estimate the percentage change in a bond’s full price for a given shift in the benchmark yield curve.
- Calculating effective duration (\(EffDur\)) is very similar to calculating approximate modified duration, as shown in Equation 1.
- The differences are that \(\Delta Curve\) is in the denominator—since effective duration is a curve duration statistic, it measures interest rate risk in terms of a parallel shift in the benchmark yield curve—and that \(PV_{-}\) and \(PV_{+}\) are calculated using option pricing models.
- These models are covered in more detail in later modules but include such inputs as
- the length of the call protection period,
- the length of the call protection period,
- the schedule of call prices and call dates,
- the schedule of call prices and call dates,
- an assumption about credit spreads over benchmark yields (which also includes any liquidity spread),
- an assumption about credit spreads over benchmark yields (which also includes any liquidity spread),
- an assumption about future interest rate volatility, and
- an assumption about future interest rate volatility, and
- the level of market interest rates (e.g., the government par curve). The analyst holds the first four inputs constant and then raises and lowers the fifth input (i.e., parallel shifts) to derive \(PV_{+}\) and \(PV_{-}\), respectively.
- The differences are that \(\Delta Curve\) is in the denominator—since effective duration is a curve duration statistic, it measures interest rate risk in terms of a parallel shift in the benchmark yield curve—and that \(PV_{-}\) and \(PV_{+}\) are calculated using option pricing models.
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### Effective Duration (EffDur)
$$
EffDur = \frac{(PV_{-}) - (PV_{+})}{2 \times (\Delta Curve) \times (PV_0)}.
$$
- Effective duration and effective convexity are useful for gauging the
interest rate risk of bonds whose future cash flows are uncertain.
- Effective duration and effective convexity can be used to estimate the
percentage change in a bond’s full price for a given shift in the benchmark
yield curve.
- Calculating effective duration ($EffDur$) is very similar to calculating
approximate modified duration, as shown in Equation 1.
- The differences are that $\Delta Curve$ is in the denominator—since
effective duration is a curve duration statistic, it measures interest
rate risk in terms of a parallel shift in the benchmark yield curve—and
that $PV_{-}$ and $PV_{+}$ are calculated using option pricing models.
- These models are covered in more detail in later modules but include such
inputs as
- (1) the length of the call protection period,
- (2) the schedule of call prices and call dates,
- (3) an assumption about credit spreads over benchmark yields (which also
includes any liquidity spread),
- (4) an assumption about future interest rate volatility, and
- (5) the level of market interest rates (e.g., the government par curve).
The analyst holds the first four inputs constant and then raises and
lowers the fifth input (i.e., parallel shifts) to derive $PV_{+}$ and
$PV_{-}$, respectively. Effective Convexity (EffCon)
\[ EffCon = \frac{[(PV_{-}) + (PV_{+})] - [2 \times (PV_0)]}{(\Delta Curve)^2 \times (PV_0)} \tag{2} \]
- Effective duration and effective convexity are useful for gauging the interest rate risk of bonds whose future cash flows are uncertain.
- Effective duration and effective convexity can be used to estimate the percentage change in a bond’s full price for a given shift in the benchmark yield curve.
- The formula for calculating effective convexity (\(EffCon\)) is also very similar to the formula for approximate convexity, as shown in Equation 2.
- The differences are that \(\Delta Curve\) is in the denominator—since effective duration is a curve duration statistic, it measures interest rate risk in terms of a parallel shift in the benchmark yield curve—and that \(PV_{-}\) and \(PV_{+}\) are calculated using option pricing models.
- These models are covered in more detail in later modules but include such inputs as
- the length of the call protection period,
- the length of the call protection period,
- the schedule of call prices and call dates,
- the schedule of call prices and call dates,
- an assumption about credit spreads over benchmark yields (which also includes any liquidity spread),
- an assumption about credit spreads over benchmark yields (which also includes any liquidity spread),
- an assumption about future interest rate volatility, and
- an assumption about future interest rate volatility, and
- the level of market interest rates (e.g., the government par curve). The analyst holds the first four inputs constant and then raises and lowers the fifth input (i.e., parallel shifts) to derive \(PV_{+}\) and \(PV_{-}\), respectively.
- The differences are that \(\Delta Curve\) is in the denominator—since effective duration is a curve duration statistic, it measures interest rate risk in terms of a parallel shift in the benchmark yield curve—and that \(PV_{-}\) and \(PV_{+}\) are calculated using option pricing models.
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### Effective Convexity (EffCon)
$$
EffCon = \frac{[(PV_{-}) + (PV_{+})] - [2 \times (PV_0)]}{(\Delta Curve)^2
\times (PV_0)} \tag{2}
$$
- Effective duration and effective convexity are useful for gauging the
interest rate risk of bonds whose future cash flows are uncertain.
- Effective duration and effective convexity can be used to estimate the
percentage change in a bond’s full price for a given shift in the benchmark
yield curve.
- The formula for calculating effective convexity ($EffCon$) is also very
similar to the formula for approximate convexity, as shown in Equation 2.
- The differences are that $\Delta Curve$ is in the denominator—since
effective duration is a curve duration statistic, it measures interest
rate risk in terms of a parallel shift in the benchmark yield curve—and
that $PV_{-}$ and $PV_{+}$ are calculated using option pricing models.
- These models are covered in more detail in later modules but include such
inputs as
- (1) the length of the call protection period,
- (2) the schedule of call prices and call dates,
- (3) an assumption about credit spreads over benchmark yields (which also
includes any liquidity spread),
- (4) an assumption about future interest rate volatility, and
- (5) the level of market interest rates (e.g., the government par curve).
The analyst holds the first four inputs constant and then raises and
lowers the fifth input (i.e., parallel shifts) to derive $PV_{+}$ and
$PV_{-}$, respectively. Estimate the percentage change in a bond’s full price
\[ \%\Delta PV^{Full} \approx (-EffDur \times \Delta Curve) + \left[\frac{1}{2} \times EffCon \times (\Delta Curve)^2 \right] \tag{3} \]
- Just as with yield-based interest rate risk measures, effective duration and effective convexity can be used to estimate the percentage change in a bond’s full price for a given shift in the benchmark yield curve (\(\Delta\) Curve), as shown in Equation 3.
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### Estimate the percentage change in a bond’s full price
<!-- Not sure on the name for equation 3 -->
$$
\%\Delta PV^{Full} \approx (-EffDur \times \Delta Curve) +
\left[\frac{1}{2} \times EffCon \times (\Delta Curve)^2 \right] \tag{3}
$$
- Just as with yield-based interest rate risk measures, effective duration and
effective convexity can be used to estimate the percentage change in a
bond’s full price for a given shift in the benchmark yield curve
($\Delta$ Curve), as shown in Equation 3. Key rate duration (or partial duration)
\[ KeyRateDur_k = -\frac{1}{PV} \times \frac{\Delta PV}{\Delta r_k} \tag{4} \]
\[ \sum_{k=1}^{n} KeyRateDur_k = EffDur \tag{5} \]
- Key rate duration (or partial duration) is a measure of a bond’s sensitivity to a change in the benchmark yield at a specific maturity. Such a measure is important to isolate the price responses of bonds to changes in the rates of key maturities on the benchmark yield curve
- Key rate durations define a security’s price sensitivity over a set of maturities along the yield curve, with the sum of key rate durations being equal to the effective duration, as shown in Equation 4 and Equation 5:
Where:
- \(r_k\) represents the \(k\) th key rate.
- In contrast to effective duration, key rate durations help identify “shaping risk” for a bond—that is, a bond’s sensitivity to changes in the shape of the benchmark yield curve (e.g., the yield curve becoming steeper or flatter or twisting).
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### Key rate duration (or partial duration)
$$
KeyRateDur_k = -\frac{1}{PV} \times \frac{\Delta PV}{\Delta r_k} \tag{4}
$$
$$
\sum_{k=1}^{n} KeyRateDur_k = EffDur \tag{5}
$$
- Key rate duration (or partial duration) is a measure of a bond’s sensitivity
to a change in the benchmark yield at a specific maturity. Such a measure is
important to isolate the price responses of bonds to changes in the rates of
key maturities on the benchmark yield curve
- Key rate durations define a security’s price sensitivity over a set of
maturities along the yield curve, with the sum of key rate durations being
equal to the effective duration, as shown in Equation 4 and Equation 5:
Where:
- $r_k$ represents the $k$ th key rate.
- In contrast to effective duration, key rate durations help identify
“shaping risk” for a bond—that is, a bond’s sensitivity to changes in the
shape of the benchmark yield curve (e.g., the yield curve becoming steeper
or flatter or twisting). Expected Estimated Price Change
\[ \frac{\Delta PV}{PV} = -KeyRateDur_k \times \Delta r_k \tag{6} \]
- Equation 6 rearranges terms from Equation 4 to solve for \(\Delta PV/PV\) (or \(\%\Delta PV^{Full}\)):
View Markdown Source
### Expected Estimated Price Change
<!-- Not sure on the name for equation 6 -->
$$
\frac{\Delta PV}{PV} = -KeyRateDur_k \times \Delta r_k \tag{6}
$$
- Equation 6 rearranges terms from Equation 4 to solve for $\Delta PV/PV$
(or $\%\Delta PV^{Full}$):