Fixed Income Risk & Return Flashcards
Sources of Returns from investing in a fixed-rate bond
- Coupon & principal payments
- Interest earned on coupon payments that are reinvested over the investor’s holding period for the bond
- Any capital gain or loss if the bond is sold before maturity
Assumptions
- All of the promised coupon and principal payments on time
- Interest rate earned on reinvested coupon payments is the same as the prevailing yield to maturity
Key results given assumptions
- If bond is held to maturity, annualized rate of return = YTM of the bond when purchased, if the YTM of the bond (and hence the reinvestment rate) does not change over the life of the bond.
- If a bond is sold before maturity, rate of return = YTM at purchase if the YTM has not changed since purchase.
- If the YTM increases (decreases) after the bond is purchased but before the f irst coupon date, if bond is held to maturity, realized return that is higher (lower) than the original YTM of the bond when purchased.
- If the YTM increases (decreases) after the bond is purchased but before the first coupon date, if bond is held for a short (long) period, realized return that is lower than the original YTM of the bond when purchased.
Investment Horizon
We call the time that the bond will be held the investor’s investment horizon, which may be shorter than the bond’s maturity.
Horizon Yield
A bond investor’s horizon yield is the compound annual return earned from the bond over the investment horizon.
It is calculated by comparing the purchase price of the bond to the end value derived from holding the bond, which includes coupons, interest earned on reinvested coupons, and the sale price (or principal payment amount, if the bond is held to maturity).
If Reinvestment rate = YTM, Realised return = ?
realised return = YTM
If Reinvestment rate > YTM, realised return = ?
Realised Return > YTM
If Reinvestment rate < YTM, realised return = ?
Realised Return < YTM
If YTM increases before first coupon, realised return (increases/decreases) if held till maturity
increases (higher than original YTM as reinvestment rate is higher)
If YTM decreases before first coupon, realised return (increases/decreases) if held till maturity
decreases (lower than original YTM as reinvestment rate is lower)
IF YTM is constant, what is the realised return if bond is held to maturity/sold before maturity
Realised Return = YTM
(Assuming Reinvestment rate = YTM)
Carrying Value
At dates between the purchase and the maturity, the value of a bond at the same YTM as when it was purchased is its carrying value, and it re flects the amortization of the discount or premium since the bond was purchased.
Short investment horizon: Price Risk >/< Reinvestment risk
Price Risk > Reinvestment risk
Long investment horizon: Price Risk >/< Reinvestment risk
Price Risk < Reinvestment risk
Price Risk
the uncertainty about a bond’s price due to uncertainty about the prevailing market YTM at a time of sale
Reinvestment Risk
uncertainty about the total of coupon payments and reinvestment income on those payments due to the uncertainty about future reinvestment rates.
Macaulay Duration
Point at which RR = YTM even though YTM has changed ie. Reinvestment risk -= Price Risk
Weighted average of the PV of future cash flows with the weights being the PV of future cash flows.
Represents the average time in which investment is recovered (close to maturity as majority CF is received then)
= sum of w*x / sum of w
Where w = PV of Future CF
x = time (years)
Duration Gap
= Macaulay’s Duartion - Investment Horizon
If YTM increases, -ve DG for profits
If YTM decreases, +ve DG for profits
Macaulays Duration for ZCB
= Maturity
Flat Structure
Constant Spot Rates
Assuming a flat structure, High Coupon Bonds = (Higher/Lower) Mac Duration
Lower - more weightage to coupons, less weightage to FV
Assuming a flat structure, Low Coupon Bonds = (Higher/Lower) Mac Duration
Higher - less weightage to coupons, more weightage to FV
If Maturity Increases, Mac Duration….
Increases
If YTM Increases, Mac Duration….
Decreases (as lower sensitivity at higher rates)
If Coupons are high, Mac Duration….
is lower
Modified Duration
= Macaulays Duartion/ (1+YTM)
For semi: /2 as usual
Modif ied duration provides an estimate for the percentage change in a bond’s price given a 1% change in YTM
Approx % change in bond price =
-ModDur x Change in YTM
Always an Inverse Relation
ModDur is almost = MacDur
ModDur cannot be calc for PB/CB
Approximated Modified Duartion
= V2 - V1 / (2V0 x change in YTM)
V2 = Higher Price (due to -ve change in ytm)
V1 = lower price (due to +ve change in YTM)
Money Duration
AKA Dollar Duration
Expressed in currency
= Annual ModDur x Full Price of Bond Position
Price Value per Basis Point (PVBP)
Change in Full price of Bond when YTM changes by 1 Basis point
Step 1 Calculate YTM
Step 2 Calculate Prices for +1 BP (V1) and -1 BP (V2)
Step 3 = (V2 - V1)/2 = PVBP (or V0-V1)
Full Value = PVBP * Bond Position/Par Value
Bonds with longer maturity will have (higher/lower) interest rate
higher (in usual instances)
Bonds with higher coupons will (increase/decrease) interest rate risk
decrease - less sensitive to changes in yield as duration will be shorter
FRNS - interests increase –> coupon increase ie price risk is
low
An increase in Bonds YTM will (increase/decrease) interest rate risk
decrease
Macaulay Duration for Perpetuity
=(1 + YTM)/YTM
Exception for MacDauration
Deep Discount Bonds
As maturity increases, Duration increases, but at very high maturity, duration starts declining
Effective Duration
For Callable/Putable Bonds
Analyzing interest rate risk for bonds with embedded options is based on shifts in the benchmark curve (e.g., government par rates), rather than changes in the bond’s own yield. This measure of price sensitivity is referred to as effective duration ie % change in price per 1% change in benchmark yield
= (V2 - V1)/2xV0xchange in benchmark
EffDur and Credit Spread
Effective duration separates the effects of changes in benchmark yields from changes in the spread for credit and liquidity risk. Modi ied duration makes no distinction between changes in the benchmark yield and changes in the spread.
Effective duration re flects only the sensitivity of the bond’s value to changes in the benchmark yield curve and assumes all else (including spreads) remains the same.
Effective Convexity
EffDur does not take into account that the yield curve’s convexity - Eff Con incorporates this
Value derived from EffCon is much closer to the actual Value rather than the Value derived from EffDur not taking into account EffCon
Convexity of Bonds
While the convexity of any option-free bond is positive, a callable bond can exhibit negative convexity. This is because at low yields, the call option becomes more valuable, and the call price puts an effective limit on increases in bond value,
Value using ED
Change in BP = ED x Change in YTM
This gives percentage
If change in y=+, change in BP=–, vv
Value using ED & EC
Change in BP = EDChange in YTM + 1/2EC*(Change in YTM in %)^2
In this only ED*Change in YTM is positive or negative according to the inverse relationship between YTM and Price
EC Part remains positive for both price is always understated whether YTM increases or decreases
Negative Convexity
While the convexity of any option-free bond is positive, a callable bond can exhibit negative convexity. This is because at low yields, the call option becomes more valuable, and the call price puts an effective limit on increases in bond value,
Putable Bond Yield Curve
- Positive Convexity
- At higher yields, the put becomes more valuable, so that the value of the putable bond decreases less than that of an option-free bond as yield increases. This means the duration of a putable bond is less than that of an equivalent option-free bond at high yields.
Key Rate Duration
Accounts for nonparallel shifts in benchmark yield.
AKA partial duration, is defined as the sensitivity of the value of a bond or portfolio to changes in the benchmark yield for a specific maturity, holding other yields constant.
The sum of a bond’s key rate durations equals its effective duration.
KRD = ModDur x weight in portfolio
Shaping Risk
Key rate duration is particularly useful for measuring shaping risk, de fined as the effect of a nonparallel shift in the yield curve on a bond portfolio. We can use the key rate duration for each maturity to compute the effect on the portfolio of a yield change at that maturity. The effect on the overall portfolio is the sum of these individual effects.
Analytical v Empirical Durations
Analytical = ModDur, MacDur, EffDur, KRD
Empirical = actual observed historical relationship between benchmark yield changes and bond price changes. Used when assumption of constant credit spread is not justified.
Revise Portfolio Durations
- MacDur
- EffDur = Sum of wi*xi/Sum of wi
wi = Marekt Value (MPxFV)
xi = Di = Individual durations
Effective duration is more appropriate than modified duration for estimating interest rate risk for bonds with embedded options because these bonds:
have uncertain cash flows that depend on the path of interest rate changes.
When an embedded option has significant value, relative to an equivalent option- free bond, the effective duration of a bond with an embedded option will most likely be:
lower
A bond portfolio manager who wants to estimate the sensitivity of the portfolio’s value to changes in the 5-year yield only should use a
Key Rate Duration (measure sensitivity to one specific maturity)
Restrictions on asset sales and additional borrowings by a bond issuer are best
characterized as:
Negative Covenants
A covenant that requires the issuer not to let the insurance coverage lapse on assets
pledged as collateral is an example of a(n):
Affirmative Covenant
Bermuda style embedded call option
A bond with a Bermuda style embedded call option may be called on pre-specified dates
after the first call date.
European style embedded call option
A bond with a Bermuda style A European style embedded call option specifies a single date on
which a bond may be called.
With an American style embedded call option
With an American style embedded call option, a bond may be
called any time after its first call date.
What is a surety bond?
A surety bond is issued by a third party to cover a bond and hence is an external form of credit
enhancement.
In the United States, debenture is defined as:
an unsecured bond
An investor holds $100,000 (par value) worth of TIPS currently trading at par. The coupon
rate of 4% is paid semiannually, and the annual inflation rate is 2.5%. What coupon payment
will the investor receive at the end of the first six months?
$100,000 x 1.0125 = $101,125 ie new capital
Coupon is 4%/2 on New Capital
ie $2,025.
The CFO of Premlow Insurance Co. wants to ensure that her investment portfolio aligns with
the company’s claims history and obligations as they come due. She will most likely invest in
which types of fixed-income securities?
Long Term, Investment Grade
An analyst who evaluates both fixed-income and equity indices will find that the turnover for
the former relative to the latter will be:
higher
Fixed income classifications by issuer most likely include:
Financial sector bonds
Corporate bonds are frequently classified by issuer as financial or non-financial. Floating-
rate bonds are a classification by coupon structure. Money market securities are a classification by original maturities.
Dave Kats, CFA, recommends the inclusion of a bond fund to his client. In determining the
appropriate index benchmark for the fund, Kats will look for an index that matches the
exposure of the bond fund in which specific area?
Credit quality
An analyst is likely to see a fixed-income index focused on which characteristics?
Geography, Credit Quality
A purchase of a new bond issue by a single investor is most accurately described as a(n):
private placement
Which type of issuer is most likely to issue bonds by auction?
sovereign.
Corporate bonds are
typically issued in an underwriting or private placement process while municipal bonds are
typically issued in a negotiated or underwritten process.
A bond is quoted at 96.25 bid and 96.75 ask. Based only on this information, this bond is
(liquid/illiquid)?
Illiquid
The spread between the bid and ask prices is one-half percent of par, which most likely
reflects an illiquid market for this bond. Bonds with liquid secondary markets typically
have bid-ask spreads of approximately 10 to 12 basis points.
Redding Company (Redding) has struggled financially over the last several years but is
hoping to turn things around under new leadership. Redding’s credit rating is below
investment grade, and it is looking to issue new debt to provide some much-needed capital.
Redding’s best course of option is to:
Given the situation
Redding is in, its best courses of action are either to issue callable debt (where the issuer
has the right to call in the debt issuance before maturity) or take out leveraged loans with
prepayment options. Shorter-maturity instruments are better, as the company may be
able to borrow at lower rates later when its situation improves.
Reduced rollover risk resulting from standardization is a benefit available to which type of
corporate bond issuer?
Investment Grade
A repurchase agreement is described as a “reverse repo” if:
a bond dealer is the lender.
Relative to the yields on nonsovereign bonds, sovereign bond yields may be lower because
of the:
regulatory requirements, forcing some financial institutions to hold government
debt.
the value of a long-term bond is (more/less) sensitive to interest rate changes than the value of a short-term bond.
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Long-term, low-coupon bonds are (more/less) sensitive than short-term and high-coupon bonds.
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