THE MODIFIED DURATION IS CALCULATED IF THE MACAULAY DURATION IS KNOWN. - an estimate of the % price change for a bond given a change in its YTM. - its just a simple adjustment to the Macaulay duration modified duration = macaulay duration / (1 + r) r = yield per period; ie YTM

Understanding FI Risk and Return Flashcards by Wes L

For a hold to maturity bond investor and rates go up after the purchase but before the first coupon payment, calculate the investor’s realized rate of return. Orginal I= 12%, increased to 15%
n=5, 10% coupon, PV= 92.79, interest rate goes up to 15%

*the bondholder will reinvest the coupon payments at 15% for the 5 yr period
n=5, PV=0, I/Y=15, PMT=10
FV= 67.42 (the amount in excess of the coupons, $17.42 (67.42 - 50), is the “interest-on-interest” gain from
compounding)

the total return at maturity is then 167.42 (67.42+100)

thus, the realized rate of return is 12.53%
n=5, FV=167.42, PV= -92.79, pmt=0
CPT I/Y = 12.53

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If an investor buys a 5-yr, 10% coupon bond at 92.79, where interest rates move from 12%-15%, and he sells after 3-yrs, what is the effect?

the 3-yr coupons = 34.73 (n=3, I/Y=15, pv=0, pmt=10, FV= 34.73)
**to calculate the sale price after 3 yrs, THERE ARE ONLY 2 YRS REMAINING ON THE BOND and FV =100
- n=2, fv=100, I/Y=15, pmt= 10,
solve for PV= 91.87 (the sale price!)
total return = 126.60 (91.87 + 34.73)
so, the 3-yr “horizon yield” is: n=3, pv= 92.79 (the orginal PV amt), FV= 126.60 (the total return)
- cpt I/y = 10.91%

Compared to holding this bond to maturity, selling the bond after 3 years results in a lower return (10.91% v 12%)
Although the coupons were invested at a higher rate, the capital loss was greater than the gain from reinvesting coupons.

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Interest rate risk affects bondholders in two different ways:

coupon reinvestment risk
market price risk

*these are offsetting risks; different time horizons will experience different exposures

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When does coupon reinvestment risk matter?

when an investor has a long-term horizon

- a buy-and-hold investor has only coupon reinvestment risk

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When does market price risk matter?

when the investor has a short-term horizon relative to the time to maturity of the bond

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Duration

the duration of a bond measures the sensitivity of the bond’s full price (including AI) to changes in interest rates
- indicates the % change in the price of a bond for a 1% change in interest rates.

the higher the duration, the more sensitive the bond is to change in interest rates
duration is expressed in years
- the time it will take a bond to repay its value

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The two categories of duration are:

yield duration

2. curve duration

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Yield duration is …

yield duration is the sensitivity of the bond price with respect to the bond’s own yield to maturity

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Curve duration is…

curve duration is the sensitivity of the bond price with respect to a benchmark yield curve
- such as a govt yield curve, the spot curve, or the forward curve

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Types of duration under yield duration:

YIELD DURATION

macaulay
modified
money
PVBP

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Types of duration under curve duration:

CURVE DURATION

- effective

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Macaulay duration definition

indicates the investment horizon for which coupon reinvestment risk and market price risk offset each other

is a weighted average of the time to receipt of the bond’s promised payments
the weights are the time period’s share of the full price that corresponds to each of the bond’s promised future coupon payments
- ie; calculate the present value of each CF, multiply by its weight and sum the total

ex. a 10-yr, 8% annual bond

Per  CF      PV       Weight (PV/price)  Period * Weight
1        8      find        
2       8
3       8
4       8
5       8
6       8
7       8
8       8
9       8
10    108
Total:       price               1                         7.0029 (example
             (sum pv)                                          (sum)

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Modified duration

THE MODIFIED DURATION IS CALCULATED IF THE MACAULAY DURATION IS KNOWN.

an estimate of the % price change for a bond given a change in its YTM.
its just a simple adjustment to the Macaulay duration

modified duration = macaulay duration / (1 + r)

r = yield per period; ie YTM

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A % price change for a bond given a change in its YTM =

%PVfull = -AnnualModifiedDuration * deltaYTM

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A 2-yr annual $100 bond has a Macaulay duration of 1.87 years. The YTM is 5%. Calculate the modified duration.

modified duration = macaulay duration / (1 + YTM)

= 1.87 / 1.05 = 1.78 years

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Approximate Modified Duration

if the yield is A and increases by 1% (A+1%), then the price of the bond will decrease by B% (the output of the equation )
used to estimate the value of modified duration; if Macaulay D isn’t known
this method estimates the slope of the line tangent to the price-yield curve
- y-axis: price
- x-axis: YTM
Appx Modified Duration = (PV-) - (PV+) / ( 2 * ▲Yield * PVo)

ie: = (price of bond when yield is decreased) - (price of bond when yield is increased) / ( 2 * delta yield * initial bond price)
- once the appx modified duration is solved, the appx Macaulay duration can be calculated

Approximate Macaulay Duration

can be solved for once the appx modified duration is known
it is the weighted avg time to receipt to receipt of interest and principal payments

appx Macaulay Duration = Appx modified duration * ( 1 + r )

r = periord YTM

A 12% annual bond with 10-yr to maturity currently trading at par. Assume a 10bps change in ytm; what is the bond’s approximate modified duration?

What is the formula?
What steps are needed to calculate?

Appx Modified Duration = (PV-) - (PV+) / ( 2 * ▲Yield * PVo)

need to know current bond price and current ytm to start
n=10, pmt=12, fv=100, pv=100; cpt I/Y = 12 (and BC its trading at par)

Find the price if ytm increased by 10bps
Find the price if ytm decreased by 10 bps
10bps = .1% = .001
solve

100.57 - 99.44 / (2 * .001 * 100)

= 5.65 years

The modified duration of a bond is 6.54. The approximate percentage change in price using duration only for a yield decrease of 120bps is ….

which formula?
what does 120bps convert to?
what to watch for

%PVfull = -modified duration * delta ytm
120bps = 1.2%

%PVfull = -6.54 * -1.20

make sure to remember the (-) before modified duration and if YTM decreases, its a (-) as well.

The effective duration is the _____ appropriate measure of interest risk of a bond with an embedded call option

the MOST

bonds with embedded options and mortgage-backed securities do not have a well-defined YTM

Effective Duration

falls under the curve duration category
it is a measure of the sensitivity of the bond’s price to a change in a benchmark yield curve (instead of its own YTM)
the best measure of interest rate sensitivity for bonds with embedded options and mortgage-backed securities, or securities without well-defined YTM (pension fund liabilities)
- because these securities may be prepaid before the maturity date

Effective duration = = (PV-) - (PV+) / ( 2 * ▲CURVE * PVo)

*note its a similar formula to appx modified duration, but its the delta curve (not delta yield)

As the coupon rate increases, the Macaulay duration ____

decreases

- a higher coupon rate has a lower duration

As the time to maturity increases, Macaulay duration ___

increases

- a longer time to maturity has higher duration

A higher YTM results in a ____ duration

lower duration

A callable bond characteristics

- the difference between a callable bond and a non-callable bond is the value of the embedded call option - when interest rates are low, the price of a callable bond will always be lower than a non-callable - which is why callable bonds have negative convexity when rates fall - has lower effective duration than a normal bond when interest rates are low - there is a risk that the bond will be called

Why would an investor buy a putable bond?

- to protect against falling prices as rates rise - the value of the put option increases as rates rise, which limits the sensitivity of the bonds price to changes in the benchmark rates

Convert Macaulay duration to Modified duration

modified duration = Macaulay / ( 1 + r) r needs to be adjusted for periodicity (semi-annual or quarterly)

Money duration is

- is a measure of the price change in units of the currency in which the bond is denominated, given a change in annual yield to maturity Money duration = Annual modified duration * PVfull *if given Macaulay duration, it must be converted to annual modified duration = modified duration = Macaulay / (1 + r) ▲PVfull = -money duration * ▲yield

Price value of a basis point (PVBP)

- is related to money duratino - PVBP estimates the change in the full price given a 1bp change in the yield to maturity - 1bp = .01% = .0001 PVBP = (PV-) - (PV+) / 2 where PV- and PV+ are full prices calculated by decreasing and increasing the YTM by 1bp

What is the quick way to calculate the PVBP?

take the money duration and * by 1bp ie money duration = $200,000 200,000 * .0001 = $20

A $100, 5-yr bond pays 10% semi-annually. The YTM is 10% and priced at par. The modified duration of the bond is 3.81. Calculate the PVBP.

First, calculate the money duration money duration = $100 * 3.81 = $381 Then PVBP PVBP = $381 * .0001 = $0.0381

Convexity v duration

- duration assumes a linear relationship between the changes in a bond's price and changes in YTM - in reality, bond prices do not move along a straight line, but exhibit a convex relationship - convexity is added to duration for a more exact measure of a bond's price for a change in YTM

Convexity formulas

Changed in the price of a bull bond %PVfull = [ ( -AnnModDur * ▲Yield) ] + [ .5 * AnnConvexity * (▲yield^2) ] first set of [ ] is duration, second set of [ ] is the convexity adjustment Approximate convexity = [ (PV-) + (PV+) ] - [ 2 * PVo ] / [ ( ▲Yield^2) * PVo) ] *not PV- and PV+ are added with this equaiton The change in PVfull price in units of currency, given a change in YTM ▲PVfull = [ - (money dur * ▲yield ) ] + [ .5 * MoneyConvex * (▲yield^2) ]

The relationship between various bond parameters with convexity is _____ as with duration Convexity for: - the lower the coupon rate, - the lower the yield to maturity, - the longer the time to maturity, - the greater the dispersion of cash flow or cash payments spread over time,

The relationship with convexity and duration is the SAME - the lower the coupon rate, THE GREATER THE CONVEXITY - the lower the yield to maturity, THE GREATER THE CONVEXITY - the longer the time to maturity, THE GREATER THE CONVEXITY - the greater the dispersion of cash flow or cash payments spread over time, THE GREATER THE CONVEXITY

Effective Convexity

Effective: again deals with bonds with unpredictable CF - uses the benchmark yield curve, not ytm = [ (PV-) + (PV+) ] - [ 2 * PVo ] / [ ( ▲CURVE^2) * PVo) ]

Convexity is good for bond investors * when rates go down, the bond which IS MORE convex will appreciate more * when rates go up, the bond which IS MORE convex will depreciate less

Short-term investors are concerned with the impact on the _____ given a sudden change in YTM

- Flat price | * typically, short-term bonds have greater yield volatility than long-term

What is the duration gap of a bond

- defined as the Macaulay duration - investment horizon If 1. Mac dur < investment horizon; DG is negative - coupon reinvestment risk dominates 2. Mac dur > investment horizon; DG is positive - market price risk dominates 3. Mac dur = investment horizon; DG is 0 - coupon reinvestment risk offsets market price risk

High-yield junk bonds will have a ______ empirical duration than analytical duration

- have a lower empirical duration | * credit spreads and benchmark govt yields are negatively correlated