Fixed Income Valuation Flashcards
bond price is inversely related…
to interest rates
Convexity effect
Given the same coupon rate & same maturity, bond prices increase more when rates drop than they drop in price when rates rise
Coupon effect
Given the same maturity, lower coupon bond is more price sensitive than a higher coupon bond when rates change
Maturity effect
For the same coupon rate: longer term bond is more price sensitive than a shorter term bond (few exceptions, but mostly true)
Spot rates
Rates that correspond to a bond’s cash flow dates. Rather than using the same discount rate, each cash flow is discounted by its associated spot rate (YTM on zero coupon bonds maturing at the date of each cash flow)
When pricing a bond is between coupon dates, its price has 2 parts
- The flat price (PVflat)
- accrued interest
PVflat is also known as dirty price
PVflat + accrued interest = PVfull (clean price)
Dealers quote flat price of bonds
Up to this point, we’ve been pricing bonds assuming that we’re pricing them out on the date a coupon is paid. In reality, they’ll be plenty of times you’ll have to price the bond in between coupon payment dates. That’s the point of clean/dirty price and accrued interest.
accrued interest (fixed income)
the proportional share of the next coupon payment belonging to the seller
Accrued interest = t/T x coupon payment
T = # of days between payment
t = # of days since the last payment to the trade settlement date
E.g. let’s say i’m holding onto a bond after coupon payment date and then i decide to sell it. I might think “if i had just waited a little bit longer until the next coupon payment date, i’d get the coupon payment”. It doesn’t work that way. The time period that I held the bond for after payment date - whoever is buying the bond from me has to pay me the interest for that time period. In short, I get interest right up to the date that my bond settles -> that’s an example of accrued interest
2 conventions to count days
- actual/actual (most common for govt bonds)
- 30/360 (typically corporate bonds)
The value of a bond between coupon dates formula
PVfull = PV x (1 + r)t/T
The ‘PV’ is not PVflat
Matrix Pricing
You need matrix pricing for fixed rate bonds w/out an active market or haven’t been issued yet -> there’s no market price to calculate
- you basically estimate PV and coupon rate based on prices of more frequently traded comparable bonds (i.e. similar tenor, coupons, credit quality etc.)
Yield measures for fixed rate bonds
For fixed rate bonds, yield measures typically are annualized and compounded (if maturity > 1 year)
semi-annual bond basis yield or semi-annual bond equivalent yield
semi-annual bond basis yield = (yield/semi-annual period) x 2
Most bonds in the United States make semiannual coupon payments (periodicity of two), and yields (YTMs) are quoted on a semiannual bond basis, which is simply two times the semiannual discount rate.
annualizing = bond equivalent yield
compounding = EAY
Adjusting periodicity
Current yield
Current yield is basically income yield
Current yield = Annual cash coupon payment/bond price [PVflat]
The current yield does not account for gains or losses as the bond’s price moves toward its par value over time.
Simple yield
Simple yield not only takes into account coupon payments, but also amortization of discount/premium
Simple yield = [(PMT/yr) + straight line amortization of gain/loss]/PVflat
yield-to-worst
For a callable bond, an investor’s yield will depend on whether and when the bond is called. The yield-to-call can be calculated for each possible call date and price.
The lowest of yield-to-maturity and the various yields-to-call is termed the yield-to-worst
quoted margin (floating rates note valuation)
The quoted margin is simply the yield spread over the reference rate.
Remember, the coupon rate on floating rate notes = reference rate ± margin.
That aforementioned margin is the the quoted margin. Quoted margin reflects credit quality at that time
The values of floating rate notes are more stable than those of fixed-rate debt of similar maturity because the coupon interest rates are reset periodically based on a reference rate.
required margin
required margin is the spread required by investors to reflect changes in credit quality. The required margin or discount margin is the number of basis points above or below the reference rate that would cause the note’s price to return to par value at each reset date.
- changes usually come from changes in the issuer’s credit risk
Thus, if a floating rate note w/quoted margin of 50 bps (for example), w/no changes in credit risk, then the required margin would be 50bps also
IF quoted margin = discount margin (or required margin, same thing)
PV = 100 on reset date
If quoted margin > discount margin (required margin)
PV > 100
If quoted margin < discount margin (required margin)
PV < 100
To calculate payment for floating rate note
PMT = [(Reference rate aka index + quoted margin) x FV]/m
m = periodicity
To calculate I/Y for floating rate note
I/Y = (Index or reference rate + discount margin)/m
m = periodicity
Yields for money market instruments
The yields for money market instruments are annualized but not compounded. Instead, rates of return are stated on a simple interest basis
PV of money market securities
PV = FV x (1-[days/year] x discount rate)
This is actually pretty intuitive. Let’s say I have a money market security w/maturity of 1 year, FV = 1000, discount rate = 2% and I want to know the PV. W/out using a calculator, we know that PV would be 980
Plug into this formula: PV = 100 x (1 - 1 x 2%) = 100 x 98% = 980
If the maturity were 6 months: PV = 100 x (1 - 0.5 x 2%) = 100 x 99% = 990
We need the aforementioned formula b/c money market securities tend to have maturity of less than 1 year (usu. # days)
Discount rate for money market instruments
discount rate = (yr/days) x (FV-PV)/(FV)
(yr/days) = periodicity
(FV-PV) = interest earned
Discount rates are used for money market securities such as T-bills, commercial paper
Difference between discount rate and add on rate
Discount rate means you buy something lower than par, it matures at par. Add-on means you buy something at par, it mature at par + interest.
PV when dealing w/add on rates
PV = FV/(1 + [days/year] x Add on rate)
To calculate add on rate
Add on rate = (yrs/days) x (FV-PV)/(PV)
(yrs/days) = periodicity
(FV-PV)/(PV) = interest earned / price paid
Add on rates are typically used for money market securities such as CDs, repos
A yield curve shows yield by
maturity
government bond spot curve (zero or strip curve)
The government bond spot curve shows the YTMs for a full range of maturities for US Treasury bonds.
If the spot curve is upward sloping -> that’s normal b/c longer maturities have higher YTMs
If the spot curve is downward sloping -> that’s known as inverted yield curve
Spot curves are obviously for zero coupon bonds
Key difference between the spot curve and the par curve
The spot curve is for zero coupon bonds, so there’s no reinvestment risk. But a yield curve must reflect a group of bonds that are similar on all their characteristics. Since most bonds have coupon payments, the spot curve isn’t the appropriate the curve to use to determine what our rates are to discount each cash flow. You want to use the par curve for that
on-the-run
In finance, on-the-run refers to most recently issued security and thereby usually the most liquid at that time
Par curve
The par curve is a sequence of YTMs such that each bond is priced at par
The par rates are derived from spot rates
Forward curve
The forward curve is based on forward rates
- agreed on today, received/paid in the future
- quoted as ‘when, what’
Essentially, the forward curve is a series of 1 year forward rates plotted graphically
1y1y
1 year forward rate 1 year from now
1y1y -> add 1 and 1 = 2. This means you need the 2 year spot rate to calculate the 1y1y
2y1y
1 year forward rate 2 years from now
2y1y -> add 2 and 1 together = 3. This means you need the 3 year spot rate to calculate 2y1y
Forward rates given spot rates
Our basic relation between forward rates and spot rates (for two periods) is:
(1 + S2)2 = (1 + S1)(1 + 1y1y)
E.g. Another example of a relationship (1 + S3)3 = (1 + S1)(1 + 1y1y)(1 + 2y1y)
This again tells us that an investment has the same expected yield (borrowing has the same expected cost) whether we invest (borrow) for two periods at the 2-period spot rate, S2, or for one period at the current 1-year rate, S1, and for the next period at the forward rate, 1y1y. Given two of these rates, we can solve for the other.
G-spread
A yield spread over a government bond is also known as a G-spread.
Fixed rate bonds often use government benchmark security (usually the most recently issued, which is ‘on the run’ security)
Spread over risk-free
To calculate G-spread
G-spread = YTMBond − YTMTreasury
i-spread (interpolated spread)
An alternative to using government bond yields as benchmarks is to use rates for interest rate swaps in the same currency and with the same tenor as a bond.
Yield spreads relative to swap rates are known as interpolated spreads or I-spreads.
Higher i-spread means higher credit risk.
Spread over risky spread
Z-spread (Zero volatility spread)
The zero volatility spread is a constant spread over government spot curve
PV using Z-spread
PV = PMT/(1 + S1 + Z) + PMT/(1 + S2 + Z)2…(PMT + FV)/(1 + Sn + Z)n
