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Yield and Yield Spread Measures for Fixed-Rate Bonds

Fixed Income

Periodicity and Annualized Yields

Learning Outcome Statement:

calculate annual yield on a bond for varying compounding periods in a year

Summary:

This LOS focuses on calculating the annual yield of bonds with different compounding periods within a year. It emphasizes the importance of standardizing yield measures to facilitate comparison across different bonds. The periodicity, or the number of compounding periods per year, plays a crucial role in determining the annualized yield. The content also covers the conversion of yields between different periodicities using a specific formula.

Key Concepts:

Periodicity

Periodicity refers to the number of compounding periods in a year and typically matches the frequency of coupon payments. It is crucial for calculating the annualized yield of a bond.

Effective Annual Rate (EAR)

EAR is the interest rate for a period of one year taking compounding into account. It varies with the number of compounding periods per year and is calculated using the future value of reinvested coupons.

Semiannual Bond Basis Yield

This is the yield calculated on a semiannual basis, commonly used in the United States and the United Kingdom. It is calculated by multiplying the yield per semiannual period by two.

Periodicity Conversions

This involves converting the annual percentage rate (APR) from one periodicity to another. This is essential when comparing bonds with different compounding frequencies.

Formulas:

Future Value Calculation

=FV(rate,nper,pmt,[pv],[type])=FV(\text{rate}, \text{nper}, \text{pmt}, [\text{pv}], [\text{type}])

Calculates the future value of a series of cash flows at a given rate and number of periods.

Variables:
raterate:
periodic reinvestment rate
npernper:
number of periods in a year
pmtpmt:
rate per period
pvpv:
present value (optional)
typetype:
indicates when payments occur (optional)
Units: currency

Periodicity Conversion Formula

(1+APRmm)m=(1+APRnn)n(1 + \frac{APR_m}{m})^m = (1 + \frac{APR_n}{n})^n

Used to convert an APR from one periodicity to another, facilitating comparisons between different compounding intervals.

Variables:
APRmAPR_m:
annual percentage rate for m periods per year
mm:
number of periods per year for the initial APR
APRnAPR_n:
annual percentage rate for n periods per year
nn:
number of periods per year for the converted APR
Units: percentage

Other Yield Measures, Conventions, and Accounting for Embedded Options

Learning Outcome Statement:

compare, calculate, and interpret yield and yield spread measures for fixed-rate bonds

Summary:

This LOS covers various yield measures and conventions, including the calculation of yield-to-maturity for different compounding frequencies and the impact of embedded options in bonds. It explains how to convert yields between different periodicities and how to account for callable bonds, providing a comprehensive understanding of how yields are affected by these factors.

Key Concepts:

Periodicity Conversion

Periodicity conversion involves adjusting the yield calculation to account for different compounding frequencies. A general rule is that more frequent compounding at a lower annual rate corresponds to less frequent compounding at a higher annual rate.

Yield-to-Maturity (YTM)

YTM is the internal rate of return on a bond, assuming that the bond will be held to maturity and all coupons will be reinvested at the same rate. YTM calculations must be adjusted for the frequency of coupon payments and the day count convention.

Yield Measures for Callable Bonds

For bonds with embedded options like callable bonds, traditional YTM measures are inadequate. Instead, yield-to-call measures are used, which consider the possibility that the bond may be called before maturity.

Government Equivalent Yield

This yield measure restates a corporate bond yield calculated with a 30/360 day count convention to an actual/actual day count convention used for government bonds, facilitating comparison between different types of bonds.

Yield-to-Worst (YTW)

YTW is the lowest yield possible on a bond without defaulting, considering all potential call dates and the maturity date. It provides a conservative measure of the bond's yield.

Formulas:

Periodicity Conversion Formula

(1+APRmm)m=(1+APRnn)n(1 + \frac{APR_m}{m})^m = (1 + \frac{APR_n}{n})^n

This formula is used to convert the yield from one compounding frequency to another.

Variables:
APRmAPR_m:
Annual percentage rate for compounding frequency m
mm:
Number of compounding periods per year for the initial compounding frequency
APRnAPR_n:
Annual percentage rate for compounding frequency n
nn:
Number of compounding periods per year for the target compounding frequency
Units: dimensionless

Yield-to-Call Formula

PV=∑i=1NPMT(1+r)i+Call price+PMT(1+r)NPV = \sum_{i=1}^{N} \frac{PMT}{(1 + r)^i} + \frac{Call\ price + PMT}{(1 + r)^N}

This formula calculates the present value of a callable bond's cash flows, considering the possibility of the bond being called at a specific date.

Variables:
PVPV:
Present value or price of the bond
PMTPMT:
Coupon payment per period
rr:
Yield per period or market discount rate
NN:
Number of periods to the call date
CallpriceCall price:
Price at which the bond can be called
Units: currency

Yield Spread Measures for Fixed-Rate Bonds and Matrix Pricing

Learning Outcome Statement:

compare, calculate, and interpret yield and yield spread measures for fixed-rate bonds

Summary:

This LOS focuses on understanding the yield and yield spread measures for fixed-rate bonds, including the calculation of various yield measures such as yield-to-worst, option-adjusted yield, and G-spread. It also covers the interpretation of these measures in the context of bond valuation and investment decision-making.

Key Concepts:

Yield-to-Worst (YTW)

Yield-to-worst is the lowest yield possible on a bond without the issuer actually defaulting. It considers all potential yields from callable bonds to their maturity or call dates and selects the minimum yield as the YTW.

Option-Adjusted Spread (OAS)

Option-adjusted spread is the spread calculated by taking the Z-spread and adjusting it for the value of embedded options within the bond, typically used for callable or convertible bonds.

G-spread

G-spread refers to the yield spread of a corporate bond over a government bond with a similar maturity. It reflects the additional yield an investor earns over a risk-free government bond, compensating for the higher risk of the corporate bond.

Z-spread

Z-spread, or zero-volatility spread, is the constant spread that needs to be added to each spot rate of the yield curve to make the present value of the bond's cash flows equal to its market price. It provides a measure of the credit risk and liquidity premium of the bond.

Formulas:

Z-spread calculation

PV=∑t=1NC(1+zt+Z)t+FV(1+zN+Z)NPV = \sum_{t=1}^{N} \frac{C}{(1 + z_t + Z)^t} + \frac{FV}{(1 + z_N + Z)^N}

This formula calculates the present value of the bond's cash flows, adjusted by the Z-spread over the spot rates, to match the bond's market price.

Variables:
PVPV:
Present value of the bond's cash flows
CC:
Coupon payment
ztz_t:
Spot rate at time t
ZZ:
Z-spread
FVFV:
Face value of the bond
NN:
Number of periods until maturity
Units: currency units

Option-Adjusted Spread (OAS)

OAS=Z−Option value in basis points per yearOAS = Z - \text{Option value in basis points per year}

This formula adjusts the Z-spread by subtracting the value of the embedded options, providing a measure that reflects the yield spread without the influence of the options.

Variables:
OASOAS:
Option-adjusted spread
ZZ:
Z-spread
OptionvalueOption value:
Value of the embedded option expressed in basis points per year
Units: basis points
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