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Fixed Income > Understanding FI Risk and Return > Flashcards

Understanding FI Risk and Return Flashcards

1
Q

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%

A

*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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2
Q

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?

A
  • 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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3
Q

Interest rate risk affects bondholders in two different ways:

A
  1. coupon reinvestment risk
  2. market price risk

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

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4
Q

When does coupon reinvestment risk matter?

A
  • when an investor has a long-term horizon

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

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5
Q

When does market price risk matter?

A
  • when the investor has a short-term horizon relative to the time to maturity of the bond
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6
Q

Duration

A

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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7
Q

The two categories of duration are:

A
  1. yield duration

2. curve duration

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8
Q

Yield duration is …

A
  • yield duration is the sensitivity of the bond price with respect to the bond’s own yield to maturity
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9
Q

Curve duration is…

A

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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10
Q

Types of duration under yield duration:

A

YIELD DURATION

  • macaulay
  • modified
  • money
  • PVBP
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11
Q

Types of duration under curve duration:

A

CURVE DURATION

- effective

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12
Q

Macaulay duration definition

A

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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13
Q

Modified duration

A

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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14
Q

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

A

%PVfull = -AnnualModifiedDuration * deltaYTM

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15
Q

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

A

modified duration = macaulay duration / (1 + YTM)

= 1.87 / 1.05 = 1.78 years

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16
Q

Approximate Modified Duration

A
  • 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

17
Q

Approximate Macaulay Duration

A
  • 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

18
Q

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?

A

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

19
Q

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

A 
%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.

20
Q

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

A

the MOST

  • bonds with embedded options and mortgage-backed securities do not have a well-defined YTM
21
Q

Effective Duration

A
  • 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)

22
Q

As the coupon rate increases, the Macaulay duration ____

A
  • decreases

- a higher coupon rate has a lower duration

23
Q

As the time to maturity increases, Macaulay duration ___

A
  • increases

- a longer time to maturity has higher duration

24
Q

A higher YTM results in a ____ duration

A
  • lower duration
25
Q

A callable bond characteristics

A
  • 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
26
Q

Why would an investor buy a putable bond?

A
  • 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
27
Q

Convert Macaulay duration to Modified duration

A

modified duration = Macaulay / ( 1 + r)

r needs to be adjusted for periodicity (semi-annual or quarterly)

28
Q

Money duration is

A
  • 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

29
Q

Price value of a basis point (PVBP)

A
  • 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

30
Q

What is the quick way to calculate the PVBP?

A

take the money duration and * by 1bp

ie
money duration = $200,000

200,000 * .0001 = $20

31
Q

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.

A

First, calculate the money duration

money duration = $100 * 3.81 = $381

Then PVBP
PVBP = $381 * .0001 = $0.0381

32
Q

Convexity v duration

A
  • 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
33
Q

Convexity formulas

A

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) ]

34
Q

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,
A

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
35
Q

Effective Convexity

A

Effective: again deals with bonds with unpredictable CF
- uses the benchmark yield curve, not ytm

= [ (PV-) + (PV+) ] - [ 2 * PVo ] / [ ( ▲CURVE^2) * PVo) ]

36
Q

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
A
37
Q

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

A
  • Flat price

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

38
Q

What is the duration gap of a bond

A
  • 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
39
Q

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

A
  • have a lower empirical duration

* credit spreads and benchmark govt yields are negatively correlated

Fixed Income (6 decks)

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