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

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

Duration

Macaulay Duration

Modified Duration

is the weighted average term to maturity of the cash flows from a bond.

is the approximate % of price change for a 1% change in yield

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

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 = $\frac{Price(Curve.Decrease)-Price(Curve.Increase)}{2\times Initial.Price\times Decimal.Change.In.Curve}$

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

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

Change in YTM is written as bp: 1 bp = 0.0001 = 0.01%

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)

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:

Effective Convexity:

Bondholders prefer more convexity, other things equal

assumes expected cash flows do not change when yield changes

Approximate Convexity= $\frac{V_{-}+V_{+}-2V_0}{(\Delta YTM)^2\times V_0} $

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} $

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: %): $-duration(\Delta Spread)+\frac{1}{2}convexity(\Delta Spread)^2$

is the length of time taken by the investor to recover his invested money in the bond through coupons and principal repayment.

Horizon yield is the Interest rate that equates Future Gain and Present Price

YTM down = Original YTM - $ \bigtriangleup YTM$

YTM up = Original YTM + $ \bigtriangleup YTM$

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)

assumes that the price/yield relationship is a straight line.(when it is actually convex)

Calculation

general formula

closed-form 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= $\sum [Period \times Weight(\%)]$

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

Effects of embedded options

Put option: reduce effective duration when interest rises

Call option: reduce the effective duration when interest falls

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)

Macaulay, Modified, and Effective duration measures price sensitivity to a parallel shift in yield curve