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Reading 46: Understanding Fixed-Income Risk and Return - Coggle Diagram
Reading 46: Understanding Fixed-Income Risk and Return
Return
Sources of Bond Return
Coupon and principal payments
Reinvestment interest on coupon payment
Assume reinvestment interest = YTM
Capital gain or loss
Relative to constant-yield price trajectory
= 0 if held to maturity
if sell prior to maturity
Capital gain (loss) = sale price - carrying value
Carrying value is the calculated PV of future value, based on constant YTM value
Horizon yield is the Interest rate that equates Future Gain and Present Price
Effect of Change in YTM
YTM changes after purchase but just prior to the first coupon date, coupon reinvestment rate = YTM
Case 1: Bond held to maturity
Rate of return = \( \left ( \frac{Coupon+Par+Revinvest.Income}{Purchase.Cost} \right )^{\frac{1}{N}}-1\)
Only reinvestment income is affected
(Reinvestment risk)
YTM UP → Reinvestment Rate UP → Realized Return UP
YTM DOWN → Reinvestment Rate DOWN → Realized Return DOWN
Case 2: Sell bond after 1 period
Rate of return = \( \left ( \frac{Coupon+Sale.Price}{Purchase.Cost} \right )-1\)
Only sale price is affected
(Price risk)
YTM UP → Sale price DOWN → Realized Return DOWN
YTM DOWN → Sale price UP → Realized Return UP
Change in YTM is written as bp: 1 bp = 0.0001 = 0.01%
Duration
Macaulay Duration
is the weighted average term to maturity of the cash flows from a bond.
is the length of time taken by the investor to recover his invested money in the bond through coupons and principal repayment.
Calculation
general formula
1) Find the number of period and the CF in each period
2) find the PV of each \(CF_{i}\), and the weight (%) of each \(PV(CF_{i})\) in the total PV(CF)
Macaulay= \(\sum [Period \times Weight(\%)]\)
closed-form formula
Macaulay Duration = \( [ \frac{1+r}{r}-\frac{1+r+\left [ N \times (c-r) \right ]}{c \times [(1+r)^N-1]+r} ]-\frac{t}{T}\)
r: yield to maturity (YTM)
c: coupon rate
N: No. period to maturity
t: No.days from the last coupon payment to the settlement date.
T: No. days in the coupon period
t/T: the fraction of the coupon period that has gone by since the last coupon payment
Modified Duration
is the approximate % of price change for a 1% change in yield
assumes that the price/yield relationship is a straight line.(when it is actually convex)
Annual Modified Duration = Annual Macaulay Duration / (1+YTM)
Approximate Modified Duration: \(\frac{Price(YTM.down)-Price(YTM.up)}{2\times Initial.Price\times \bigtriangleup YTM}\)
\( \bigtriangleup YTM\): is the change of YTM when it either increases or decreases
YTM down = Original YTM - \( \bigtriangleup YTM\)
YTM up = Original YTM + \( \bigtriangleup YTM\)
Effective Duration
Must be used for bonds with embedded call / put options
Cash flows depend on interest rate levels and paths
Not necessarily better for small changes in yield
based on
1) Parallel shift in benchmark yield curve
2) Pricing model for bonds with options
Effective Duration = \(\frac{Price(Curve.Decrease)-Price(Curve.Increase)}{2\times Initial.Price\times Decimal.Change.In.Curve}\)
Effects of embedded options
Put option: reduce effective duration when interest rises
Call option: reduce the effective duration when interest falls
Key Rate Duration
Key rate duration (or Partial duration) measures price sensitivity to a change in the benchmark yield for a specific maturity
Key rate durations maybe used to estimate effect on bond price of a steeper or flatter yield curve
Macaulay, Modified, and Effective duration measures price sensitivity to a parallel shift in yield curve
Factors affecting duration
Longer maturity → Higher duration
Higher coupon rate → Lower duration
Higher YTM → Lower duration
Hiểu Duration như điểm cân bằng cán cân
Portfolio Duration Measures
Method 1: weighted average number of periods until portfolio cash flows are due to be received
Theoretically correct but rarely used
Based on cash flow yield, IRR of cash flow
Cannot be used if bonds have embedded options
Method 2: Weighted average of bonds' durations
May be used with effective durations
Assumes parallel shifts in yield curve
Money Duration
Money duration per 100 of par value = Annual Modified Duration x Full Price per 100 of par value
Money duration = Annual Modified Duration x Full Price
Provides an estimate of change in bond's full price per 1% change in YTM
Change in bond's full price = \(\bigtriangleup YTM\) x Money Duration
Price Value of a Basis Point (PVBP)
Is the change in full price for a 1bp change in YTM
1) Calculate Price when YTM down 1bp (A), and when YTM up 1bp (B)
PVBP = \( \frac{(A)-(B)}{2}\)
for semiannual bond, Calculate Price when YTM down 0.5 bp (A), and when YTM up 0.5 bp (B)
Yield Duration vs Curve Duration
Yield duration statistics measure the sensitivity of a bond's full price to the bond's own yield-to-maturity (the Macaulay duration, modified duration, money duration )
Curve duration statistics measure the sensitivity of a bond's full price to the benchmark yield curve (effective duration)
Convexity Adjustment
Duration-based estimates of bond prices are below the actual prices for option-free bonds
Price based on duration are underestimates of actual prices
For only a small changes in yield, then duration-based estimate is usable. But for a larger changes in yield, the difference becomes larger
Approximate Convexity:
assumes expected cash flows do not change when yield changes
Approximate Convexity
= \(\frac{V_{-}+V_{+}-2V_0}{(\Delta YTM)^2\times V_0} \)
Effective Convexity:
takes into account changes in cash flows due to embedded options, while approximate convexity does not
Approximate Effective Convexity
= \(\frac{V_{-}+V_{+}-2V_0}{(\Delta Curve)^2\times V_0} \)
Bondholders prefer more convexity, other things equal
Callable bond (which issuer can buy back) have a ceiling on the bond price, even if the straight bond value can soar (as YTM decreases)
Callable bond can have negative convexity with lower duration
Putable bond (buyer can sell) have a floor on the bond price, even if the straight bond value can dive (as YTM decreases)
Putable bond can have lower duration
When there is a change in YTM
1) we calculate the duration effect
2) we calculate the adjustment for convexity (that would show the effect on actual price)
Term structure of volatility
Lower duration → less sensitivity of price change due to yield change
Higher duration → more sensitivity of price changes due to yield change
If short-term rates are driven by monetary policy and long-term rates driven by expected inflation and expected growth, short-term YTM may be more volatile than long-term YTM; term structure of yield volatility slopes downward.
Bond with lower duration may have higher volatility of YTM
Yield curve shifts are not necessarily parallel
Duration and Investment Horizon
Macaulay Duration = investment horizon at which price risk and reinvestment risk just of offset
On the investment horizon
Moving left from Macaulay Duration Point: Price risk dominates
Moving right from Macaulay Duration Point: Reinvestment risk dominates
Duration Gap = Macaulay Duration - Investment Horizon
\(>0\) : YTM UP will cause Return DOWN
\(<0\) : YTM DOWN will cause Return DOWN
Credit Spreads and Liquidity
Benchmark yield: composed of real risk-free rate and expected inflation
Spread to benchmark: include premiums for credit risk and illiquidity
Estimate price effect of change in spread using duration and convexity (unit: %): \(-duration(\Delta Spread)+\frac{1}{2}convexity(\Delta Spread)^2\)
Order effect
First-order effect: Duration estimates the change in bond's price along the straight line that is tangent to this curved line
Second-order effect: Convexity adjusts the percentage price change estimate provided by modified duration , to better approximate the relationship between a bond's price and it YTM (convex)