Learning Module 8: Yield and Yield Spread Measures for Floating-Rate Instruments
Fixed Income
Floating-Rate Notes (FRN) Pricing Model
\[ PV \;=\; \frac{\frac{(MRR + QM)\times FV}{m}}{\left(1+\frac{MRR + DM}{m}\right)^{1}} \;+\; \frac{\frac{(MRR + QM)\times FV}{m}}{\left(1+\frac{MRR + DM}{m}\right)^{2}} \;+\; \ldots \;+\; \frac{\frac{(MRR + QM)\times FV}{m} + FV}{\left(1+\frac{MRR + DM}{m}\right)^{N}} \tag{1} \]
Where:
- \(PV\) = present value, or the price of the floating-rate note
- \(MRR\) = the market reference rate, stated as an annual percentage rate (it is sometimes known generically as Index)
- \(QM\) = the quoted margin, stated as an annual percentage rate
- \(FV\) = the future value paid at maturity, or the par value of the bond
- \(m\) = the periodicity of the floating-rate note, the number of payment periodsper year
- \(DM\) = the discount margin = required margin stated as an annual percentage rate
- \(N\) = the number of evenly spaced periods to maturity
- Notice that in Equation 1, because we are using annual rates for MRR, QM, and DM, we must divide by m periods in the year
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### Floating-Rate Notes (FRN) Pricing Model
$$
PV \;=\; \frac{\frac{(MRR + QM)\times FV}{m}}{\left(1+\frac{MRR + DM}{m}\right)^{1}}
\;+\;
\frac{\frac{(MRR + QM)\times FV}{m}}{\left(1+\frac{MRR + DM}{m}\right)^{2}}
\;+\; \ldots \;+\;
\frac{\frac{(MRR + QM)\times FV}{m} + FV}{\left(1+\frac{MRR + DM}{m}\right)^{N}}
\tag{1}
$$
Where:
- $PV$ = present value, or the price of the floating-rate note
- $MRR$ = the market reference rate, stated as an annual percentage rate (it
is sometimes known generically as Index)
- $QM$ = the quoted margin, stated as an annual percentage rate
- $FV$ = the future value paid at maturity, or the par value of the bond
- $m$ = the periodicity of the floating-rate note, the number of payment
periodsper year
- $DM$ = the discount margin = required margin stated as an annual percentage
rate
- $N$ = the number of evenly spaced periods to maturity
- Notice that in Equation 1, because we are using annual rates for MRR, QM,
and DM, we must divide by m periods in the year Pricing formula for money market instruments quoted on a discount rate basis.
\[ PV = FV \times \left(1 - \frac{Days}{Year} \times DR \right) \tag{2} \]
Where:
- \(PV\) = present value, or the price of the money market instrument
- \(FV\) = the future value paid at maturity, or the face value of the money market instrument
- \(Days\) = the number of days between settlement and maturity
- \(Year\) = the number of days in the year
- \(DR\) = the discount rate, stated as an annual percentage rate
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### Pricing formula for money market instruments quoted on a discount rate basis.
$$
PV = FV \times \left(1 - \frac{Days}{Year} \times DR \right) \tag{2}
$$
Where:
- $PV$ = present value, or the price of the money market instrument
- $FV$ = the future value paid at maturity, or the face value of the money
market instrument
- $Days$ = the number of days between settlement and maturity
- $Year$ = the number of days in the year
- $DR$ = the discount rate, stated as an annual percentage rate Transforming Equation 2 algebraically to isolate the DR term
\[ DR = \frac{Year}{Days} \times \frac{(FV - PV)}{FV} \tag{3} \]
- The unique characteristics of a money market discount rate can be examined with Equation 3, which transforms Equation 2 algebraically to isolate the \(DR\) term
- The first term, Year/Days, is the periodicity of the annual rate
- The second term reveals the odd character of a money market discount rate
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### Transforming Equation 2 algebraically to isolate the DR term
$$
DR = \frac{Year}{Days} \times \frac{(FV - PV)}{FV} \tag{3}
$$
- The unique characteristics of a money market discount rate can be examined
with Equation 3, which transforms Equation 2 algebraically to isolate the
$DR$ term
- The first term, Year/Days, is the periodicity of the annual rate
- The second term reveals the odd character of a money market discount rate Pricing formula for money market instruments quoted on an add-on rate basis
\[ PV = \frac{FV}{\left(1 + \frac{Days}{Year} \times AOR \right)} \tag{4} \]
Where:
- \(PV\) = present value, the principal amount, or the price of the money market instrument
- \(FV\) = the future value, or the redemption amount paid at maturity including interest
- \(Days\) = the number of days between settlement and maturity
- \(Year\) = the number of days in the year
- \(AOR\) = the add-on rate, stated as an annual percentage rate
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### Pricing formula for money market instruments quoted on an add-on rate basis
$$
PV = \frac{FV}{\left(1 + \frac{Days}{Year} \times AOR \right)} \tag{4}
$$
Where:
- $PV$ = present value, the principal amount, or the price of the money
market instrument
- $FV$ = the future value, or the redemption amount paid at maturity including
interest
- $Days$ = the number of days between settlement and maturity
- $Year$ = the number of days in the year
- $AOR$ = the add-on rate, stated as an annual percentage rate