Fixed income

Weight on the exam = 12%

Number of formulas = 23

Full price
Full price is the price of a bond including accrued interest.

P_{f} - full price

P_{c} - clean price

AccInt - accrued interest

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Duration
Duration is the change in the value of a fixed income security that will result from a 1% change in interest rates.

D - duration

ΔP_{b} - percentage change in bond price

ΔY - yield change in percent

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Yield on a risky bond
This equation demonstrates the relation between yield on a risky bond and default free bonds

Y_{rb} - yield on a risky bond

Y_{df} - yield on a default free bond

DRP - default risk premium (credit spread)

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Value of a callable bond
Callable bond is a bond that can be redeemed by the issuer prior to its maturity. Usually a premium is paid to the bond owner when the bond is called.

V_{cbond} - value of a callable bond

V_{option free} - value of an option-free bond

V_{call} - value of the call

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TIPS coupon payment
TIPS (Treasury Inflation Protected Security) is a treasury security that is indexed to inflation in order to protect investors from the negative effects of inflation.

C_{TIPS} - TIPS coupon payment

Par_{adj} - inflation-adjusted par value

C_{rate} - stated coupon rate

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Absolute yield spread
Absolute yield spread measures the difference in spread between two bonds in terms of basis points.

AYS - absolute yield spread

Y_{higher} - yield on the higher-yield bond

Y_{lower} - yield on the lower-yield bond

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Relative yield spread
Relative yield spread measures the yield spread relative to the reference bond.

RYS - relative yield spread

AYS - absolute yield spread

Y_{bm} - yield on the benchmark bond

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Yield ratio
Yield ratio is the ratio of the yields between the two bonds.

YR - yield ratio

Y_{subj} - subject bonds yield

Y_{bm} - yield on the benchmark bond

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After-tax yield
After-tax yield is the amount that an investor will actually get from that investment after all of the taxes have been taken out.

Y_{at} - after-tax yield

Y_{tax} - taxable yield

t_{m} - marginal tax rate

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Taxable-equivalent yield
Taxable-equivalent yield is the pretax yield that a taxable bond needs to possess for its yield to be equal to that of a tax-free municipal bond.

Y_{tax eq} - taxable-equivalent yield

Y_{tax free} - tax-free yield

t_{m} - marginal tax rate

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Zero-coupon bond value
Zero-coupon bond value is a debt security that doesn't pay interest (a coupon) but is traded at a discount, rendering profit at maturity when the bond is redeemed for its full face value.

V_{zero} - zero-coupon bond value

V_{m} - maturity value

i - semiannual discouint rate

n - number of years

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Bond price
Bond is a debt investment in which an investor loans money to an entity (corporate or governmental) that borrows the funds for a defined period of time at a fixed interest rate.

P_{bond} - bond price

C_{1} - semiannual coupon payment 1

C_{2} - coupon in year 2

C_{2N} - coupon in year 2

P_{bond} - bond price

YTM - yield to maturity

Par - bond's par value

.
N - number of years to maturity

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Current yield
Current yield only considers the coupon interest and no other sources for an investors return.

Y_{cur} - current yield

C_{an} - annual cash coupon payment

P_{bond} - bond price

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Bond equivalent yield
Bond equivalent yield is used for a calculation for restating semi-annual, quarterly, or monthly discount-bond or note yields into an annual yield.

BEY - bond equivalent yield

CFY_{m} - monthly cash flow yield

YTM_{ap} - annual-pay yield to maturity

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EAY
Effective annual yield converts holding period yield (HPY) to a compound annual yield based on a 365-day year to make the yield comparable with other investments.

EAY - effective annual yield

HPY - holding period yield

t - days to maturity

365 - days in a year

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Nominal spread
Nominal spread is simply the difference in basis points between the Treasury and non-treasury security.

S_{nom} - nominal spread

YTM_{bond} - bond's yield to maturity

YTM_{treasury} - treasury's yield to maturity

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Z-spread
Z-spread is the constant spread that will make the price of a security equal to the present value of its cash flows when added to the yield at each point on the spot rate Treasury curve where a cash flow is received . In other words, each cash flow is discounted at the appropriate Treasury spot rate plus the Z-spread.

Z_{spread} - Z-spread

OAS- option adjusted spread

Cost_{opt} - option cost in percent

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Spot rate from forward rates
The equation shows how to determine spot rate from forward rates.

S_{3} - 3-year spot rate

_{1} f_{0} - current 1-year forward rate

_{1} f_{1} - 1-year forward rate one year from now

_{1} f_{2} - 1-year forward rate two years from now

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Forward rate from spot rates
The equation shows how to determine forward rate from spot rates.

_{1} f_{2} - 1-year forward rate two years from now

S_{3} - 3-year spot rate

S_{2} - 2-year spot rate

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Effective duration
Duration is the approximate percentage change in price for a 100 basis point change in rates. Effective duration takes into account the way in which changes in yield will affect the expected cash flows.

D_{e} - effective duration

P_{-} - bond price when yields fall

P_{+} - bond price when yields rise

P_{0} - initial price

ΔY - change in yield in decimal form

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Percentage change in bond price
This equation simply shows how to calculate percentage change in bond price after the change in yield using effective duration.

%P_{bond} - percentage change in bond price

D_{e} - effective duration

ΔY_{%} - change in yield in percent

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Portfolio duration
Portfolio duration equals the weighted average of the portfolio's bond's durations.

D_{port} - portfolio duration

w_{1} - weight of bond 1 (market value of bond 1 devided by the market value of the portfolio)

D_{1} - duration of bond 1

w_{2} - weight of bond 2

D_{2} - duration of bond 2

w_{N} - weight of bond N

D_{N} - duration of bond N

N - number of bonds in the portfolio

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Percentage change in price
Calculation of percentage change taking into account duration and convexity effects.

%P - percentage change in price

Effect_{dur} - duration effect

Effect_{conv} - convexity effect

D - duration

Δy - change in yield

Conv - convexity

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Price value of basis point
Price value of basis point is a measure used to describe how a basis point change in yield affects the price of a bond.

P_{bp} - price value of basis point

D - duration

V_{bond} - bond value

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