Fixed Income

Question 1

Excess Spread

(define and calculate)

Answer 1

Spread in excess of the fair spread for suffering credit losses; an unannualized excess return over a specific holding period (NOTE: may need to deannualize to match hold period)

important to analyze source of mgr’s excess return

goal of any mgr is to earn excess return over fair mkt returns by finding mispriced securities

E [ExcessSpread] ≈ Spread0 − (EffSpreadDur × ΔSpread) − (POD × LGD)

***NOTE: the first term (Spread0) might need to be adjusted for periods in the year (eg 6 month hold would be x 0.5), AS DOES THE PODxLGD TERM

Question 2

CDS Spread

(define and calculate)

Answer 2

standardized fixed coupon adjusted by an up-front pmt/receipt to freflect FV of the protection

essentially a premium paid to party who has credit risk of the bond

CDS Price ≈ 1 + (Fixed coupon − CDS spread) × EffSpreadDurCDS

Question 3

Inflation-linked Bonds

Answer 3

Principal expands in line with an inflation index

exhibit lower return volatility than conventional bonds and equities

returns depend on volatility of REAL, not NOMINAL interest rates

–> volatility of real rates typically lower than that of nominal rates, which impact returns of conentional bonds and equities

Question 4

Calculate Bond Portfolio Returns

Answer 4

given key rate durations and a change in yield curve, use the change in yield curve for the specified rate duration

Sum of Durations for each tenor times the change in yield

Question 5

Modified Duration

Answer 5

% change in bond price for a 1% change in yield

Macaulay / (1+ periodic discount rate)

NOTE: make sure to divide by coupon frequency for periodic discount rate

if MD is 7, value of bond expected to decrease by 7% with a 1% increase in yield

Question 6

Effective Duration

Answer 6

used for embedded options

HINT: think E for effective and E for embedded

if no options, effdur will be same as Modified Duration

Question 7

Money Duration

Answer 7

(Modified Duration) x (change in yield) x (mkt value)

–> Multiply by 1/100² to get BPV (price value of a basis point)

related to dollar sensitivity

Question 8

Spread Duration

Answer 8

measures a credit-risky bond’s sensitivity to changes in credit spreads

Key point: NOT looking at underlying benchmark rates, just the spread movement

Higher effective spread duration = higher sensitivity to changes in spread

Question 9

Empirical Duration

Answer 9

takes step back and looks at how prices move in actual terms (“empirical,” duh)

how muc hdoes bond price ACTUALLY move from shift in rates

just looks at regression of historical price and int rate changes

Question 10

Return Decomposition

FS1, v important

Answer 10

1. Yield income: coupon / price
2. Rolldown Return: return bond will have if the yield curve does not change
   
   1. think of upward sloping curve (maturity horizontal yield vertical axis)
   2. Rolling Yield = #1 + #2 (note difference, it’s NOT rolldown return)
   3. once you have 1&2 you can think about what is attributable to changes in benchmark rates and and spreads (ie changes in the yield curve), steps 3 and 4:
3. Manager predicted change based on duration, convexity, and benchmark rates
   
   1. first component of this formula is the duration change, second is the “second order” change, due to shape:
   2. see photo
4. mgr predicted price changed based on D, C, and yield SPREADS
5. projected foreign currency G/L
   
   1. if foreign currency strengthens you benefit
6. LESS expected credit losses

Question 11

Structural Risk

Answer 11

the risk of non-parallel shifts in yield curve

causes assets to act differently from liabilities –> might not be able to meet liabilities when they come due

non-parallel shifts = diff rates move by diff amts

Convexity = bad from an immunization point of view –> dispersion –> higher dispersion = higher exposure to non-parallel yield shifts = higher structural risk

Question 12

Immunizing Multiple Liabilities

Answer 12

PVA >= PVL

BPVa = BPV_L (so assets move the same amount as liabilities with changes in yield curve)

convexity of assets slightly exceeds Convexity of liabilities –> look for the minimum convexity you can get because you DO need dispersion of asset cash flows to be slightly wider than that of liabilities because asset cash flow needs to happen before Liability cash flow

Question 13

BPV

Answer 13

MD x V x 0.0001

Question 14

Duration Gap

Answer 14

BPV_A - BPV_L

if <0, need to increase duration of assets

can do so by going long futures

of futures needed = desired change in BPV / BPV_{futures (of one futures contract)}

BPV of one futures contract = BPV_CTD / CV_CTD

conversion factor maps the CTD to a notional bond underlying the futures contract

similar with swaps:

Size(NP) = change in BPV / BPVswap

Question 15

Butterfly Spread and Formula

Answer 15

butterfly trade = leveraged way to cpature value when curvature changes

take long and offsetting short position in the bullet and offsetting barbell

short funds the long so no investor capital is required

long and short duration CANCEL each other for 0 net duration

profit primarily from change in curvature

• (ST yield) + (2 x medium-term yield) - LT yield

Question 16

Forward Rate Bias

Answer 16

defined as an observed divergence from IRP, where active investors seek to profit by borrowing in lower-yielding currency and investing in a higher-yielding curency

lower-yielding currencies trade at a forward PREMIUM

fully hedged foreing currency FI invmts will tend to yield the domestic RfR

Question 17

G Spread

Answer 17

the bond’s YTM minus an interpolated YTM of the two adjacent maturity OTR gov bonds

estimates a gov benchmark YTM that exactly matches the maturity of the bond

YIELD SPREAD = just the corp bond’s yield minus the nearest dated gove bond

to get to G-spread need to interpolate:

(M - S) / (L-S) –> easier way, M = target in the middle –> gives the weight of the LONGER dated benchmark bond

target = (1-weight) x shorter + (weight x longer)

if gov bond yield curve is flat, it will equal the Yield Spread

(example p.103)

Question 18

I-spread

Answer 18

bond’s YTM minus maturity interpolated swap fixed rate

advantage: based on a tradeable derivative that can be used to hedge inflation or measure carry returns

Calculate: if only given yield, first add the adjacent two swap spreads to their gov bond (benchmark) yields

then interpolate the same way as G-spread:

(M-S) / (L-S)

example p. 105

Question 19

ASW

Answer 19

Asset Swap Spread

bond’s fixed coupon minus the maturity interpolated swap fixed rate

ie it’s the same as the G-spread but using the COUPON instead of the bond’s YTM to find the spread

NOTE: can do the same thing if given an i-spread…just figure out i spread first and swap out coupon for the swap rate (instead of swapping for YTM for g-spread)

Question 20

Yield Spread

Answer 20

simple difference between bond’s all-in YTM and the closest maturity current OTR gov bond

if gov bond yield curve is flat, it will be equal to the g-spread

Question 21

Structural vs. RF Credit Models

Answer 21

Structural:

use mkt-based variables to estimate issuer’s asset value and asset value volatility

define likelihood of default as probability of asset value falling below that of liabilities

zero net assets = default threshold

so “distance to default” (DtD) is the measure

Reduced-Form:

look for relations between macro conditions and borrowers

estimate POD or spreads:

POD ≈ Credit Spread / LGD

Question 22

Altman’s Z-Score

Answer 22

used in RF model

uses key financial ratios weighted by coefficients - dont need to know the model, just need to be able to plug in

INTERPRETATION:

> 3.0 = low chance of default

1.8 - 3.0 = some risk of default

< 1.8 = likely to default

Question 23

VaR Methods

Answer 23

Parametric

assumes returns normal

uses return and stdev to calculate loss at given percentile of distribution

5% = 1.65

1% = 2.33

Advantage: simple

disadvantage: doesnt work well for non-normal distributions (eg DOESNT work with options)

Historical Simulation

applies hist movements in key risk factors (rates spreads etc) to existing portfolio to generate return dist

advantage: uses actual mkt data, CAN handle portfolios with non-normal

Disadvantage: heavily dependent on historical sample, and history might not be guide for the future

Monte Carlo Analysis

generates return distr through random simulations from user defined model

advantage: flexilibty to incorporate non-normal distr
disadvantage: high model risk

Question 24

Calculate VaR

Answer 24

1) scale up yield volatility by multiplying the square root of the time (in days) –> recall that variance scales up with time, so StDev (the sqrt of variance) scales up by multiplying by the SqRt of time)
2) multiply that by the yield to get to ONE standard deviation
3) multiply by # of StDevs (eg 2.33) –> gives you VaR
4) Multiply that by negative MD to get % change in value
5) multiply that by portfolio value to get $ change in value

Example in photo

Question 25

Drawbacks of VaR and How to Fix them (ie cVaR etc)

Answer 25

VaR as a tail risk measure has the following drawbacks:

1. Tail events tend to be more frequent and more severe than VaR forecasts.
2. It fails to capture changes in correlation and liquidity during times of market stress.
3. It fails to quantify the expected loss during a tail event.

Fixes:

Conditional value at risk (CVaR) = expected loss given portfolio is experiencing a loss in the tail –> a measure of the avg loss in the tail –> addresses the third drawback listed previously.

Incremental VaR (IVaR) (or partial VaR) measures change in VaR from adding or removing a position in a portfolio.

Relative VaR measures VaR portfolio’s returns relative to benchmark –> can be used to assess minimum underperformance of an active manager relative to their benchmark in a given time frame with a specified probability.

Question 26

Convexity and Effective Convexity

Answer 26

convexity = second order measure of the non-linear effects of a yield change of bond price –> the rate of change of the rate of change

–> a good thing if we expect rates to change

–> longer dated bonds have higher convexity

* positive convexity = causes prices to rise by more and fall by less than duration predicts

crucial takeaway: for a given value of duration, the more spread out cash flows are, the higher convexity is

Effective Convexity = used for bonds w uncertain CFs (eg options)

Question 27

Duration Times Spread (DTS)

Answer 27

adjusts spread duration to give a higher sensitivity to spread changes when spreads themselves are higher

–> because bonds w/ higher spreads have bigger spread changes

–> eg if spread is 10%, a 1% move isnt a big deal, but would be for a bond w a spread of 1%

DTS = (EffSpreadDur) x (Spread)

Question 28

Stage of Economic Cycle and Effects on Defaults and Spreads

Answer 28

Question 29

Tracking Error for Bond Portfolios

Answer 29

Tracking error = standard deviation of active return of a portfolio

if normally distributed, portfolio return should be within +/- 1 stddev of the index’s return in 68% of time periods

so if you build a portfolio mimicking an index with tracking error of 1%, with normal distribution you should expect to achieve a return that’s within 100 bps of the return on the index 68% of the time

Question 30

Calculate Expected Change in Portfolio Price (based on expected yield and yield spread changes)

Answer 30

(-EffDur x change in yield) + (0.5 x effConvexity x yieldchange ²)

Question 31

Key Rate Duration Formula

Answer 31

KeyRateDur = - (change in port value) / (port value x change in key rate)

Question 32

Leverage to Enhance Port Return Formula

Answer 32

rp = r_l + [(V_b / V_e) x (r_l - r_b)]

Question 33

Calculate P/L of a CDS Position

Answer 33

Duration neutral heding position example from CFA Mock 1, # 17:

In order to be duration neutral, the BPV of the $1,000,000 notional value of protection sold on the 10-year index must match the BPV of the protection bought in the 3-year contract.

The BPV of the $1,000,000 notional value of protection sold on the 10-year index is equal to its effective spread duration multiplied by the notional value as follows:

BPV10-year = 9.65 × $1,000,000 = $9,650,000

To ensure that the BPV of the 3-year index is the same size, we require:

$9,650,000 = 2.85 × Notional3-year

This implies the appropriate notional value in the 3-year index = $9,650,000 / 2.85 = $3,385,965.

The absolute profit/loss on each contract is calculated as ∆CDS spread × Effective spread duration × Notional amount. Given that spreads are falling, there will be profits from selling protection in the 10-year contract and losses from buying protection in the 3-year contract.

Loss in the 3-year contract = (50 / 10,000) × 2.85 × $3,385,965 = $48,250

Profit in the 10-year contract = (30 / 10,000) × 9.65 × $1,000,000 = $28,950

Hence, net loss = –$48,250 + $28,950 = –$19,300.

Question 34

Immunizing a Single Liability

Answer 34

1) Initial MV value equals or exceeds the PV of the liability
2) Macaulay duration that matches the due date (as close as possible)
3) minimizes portfolio convexity

Question 35

Best Type of Portfolio for taking advantage of an expected FLATTENING yield curve

Answer 35

Barbell portfolio

–> the price impact of the longer-dated bond (which will go up in value bc yields going down) will be much larger than that of the shorter one (which will go down in value bc ST yields going up in flattening) because the duration of the longer dated bond(s) is larger than that of the shorter dated one(s)