Fixed Income Valuation

Question 1

bond price is inversely related…

Answer 1

to interest rates

Question 2

Convexity effect

Answer 2

Given the same coupon rate & same maturity, bond prices increase more when rates drop than they drop in price when rates rise

Question 3

Coupon effect

Answer 3

Given the same maturity, lower coupon bond is more price sensitive than a higher coupon bond when rates change

Question 4

Maturity effect

Answer 4

For the same coupon rate: longer term bond is more price sensitive than a shorter term bond (few exceptions, but mostly true)

Question 5

Spot rates

Answer 5

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)

Question 6

When pricing a bond is between coupon dates, its price has 2 parts

Answer 6

1. The flat price (PV_flat)
2. accrued interest

PV_flat is also known as dirty price

PV_flat + accrued interest = PV_full (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.

Question 7

accrued interest (fixed income)

Answer 7

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

Question 8

2 conventions to count days

Answer 8

1. actual/actual (most common for govt bonds)
2. 30/360 (typically corporate bonds)

Question 9

The value of a bond between coupon dates formula

Answer 9

PV_full = PV x (1 + r)^t/T

The ‘PV’ is not PV_flat

Question 10

Matrix Pricing

Answer 10

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.)

Question 11

Yield measures for fixed rate bonds

Answer 11

For fixed rate bonds, yield measures typically are annualized and compounded (if maturity > 1 year)

Question 12

semi-annual bond basis yield or semi-annual bond equivalent yield

Answer 12

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

Question 13

Adjusting periodicity

Answer 13

Question 14

Current yield

Answer 14

Current yield is basically income yield

Current yield = Annual cash coupon payment/bond price [PV_flat]

The current yield does not account for gains or losses as the bond’s price moves toward its par value over time.

Question 15

Simple yield

Answer 15

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]/PV_flat

Question 16

yield-to-worst

Answer 16

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

Question 17

quoted margin (floating rates note valuation)

Answer 17

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.

Question 18

required margin

Answer 18

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

Question 19

IF quoted margin = discount margin (or required margin, same thing)

Answer 19

PV = 100 on reset date

Question 20

If quoted margin > discount margin (required margin)

Answer 20

PV > 100

Question 21

If quoted margin < discount margin (required margin)

Answer 21

PV < 100

Question 22

To calculate payment for floating rate note

Answer 22

PMT = [(Reference rate aka index + quoted margin) x FV]/m

m = periodicity

Question 23

To calculate I/Y for floating rate note

Answer 23

I/Y = (Index or reference rate + discount margin)/m

m = periodicity

Question 24

Yields for money market instruments

Answer 24

The yields for money market instruments are annualized but not compounded. Instead, rates of return are stated on a simple interest basis

Question 25

PV of money market securities

Answer 25

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)

Question 26

Discount rate for money market instruments

Answer 26

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

Question 27

Difference between discount rate and add on rate

Answer 27

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.

Question 28

PV when dealing w/add on rates

Answer 28

PV = FV/(1 + [days/year] x Add on rate)

Question 29

To calculate add on rate

Answer 29

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

Question 30

A yield curve shows yield by

Answer 30

maturity

Question 31

government bond spot curve (zero or strip curve)

Answer 31

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

Question 32

Key difference between the spot curve and the par curve

Answer 32

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

Question 33

on-the-run

Answer 33

In finance, on-the-run refers to most recently issued security and thereby usually the most liquid at that time

Question 34

Par curve

Answer 34

The par curve is a sequence of YTMs such that each bond is priced at par

The par rates are derived from spot rates

Question 35

Forward curve

Answer 35

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

Question 36

1y1y

Answer 36

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

Question 37

2y1y

Answer 37

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

Question 38

Forward rates given spot rates

Answer 38

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.

Question 39

G-spread

Answer 39

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

Question 40

To calculate G-spread

Answer 40

G-spread = YTM_Bond − YTM_Treasury

Question 41

i-spread (interpolated spread)

Answer 41

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

Question 42

Z-spread (Zero volatility spread)

Answer 42

The zero volatility spread is a constant spread over government spot curve

Question 43

PV using Z-spread

Answer 43

PV = PMT/(1 + S₁ + Z) + PMT/(1 + S₂ + Z)²…(PMT + FV)/(1 + S_n + Z)ⁿ