Learning Module 9: The Term Structure of Interest Rates: Spot, Par, and Forward Curves
Fixed Income
Calculating a Bond Price given the sequence of Spot Rates
\[ PV = \frac{PMT}{(1 + Z_1)^1} + \frac{PMT}{(1 + Z_2)^2} + \cdots + \frac{PMT + FV}{(1 + Z_N)^N} \tag{1} \]
Where:
- \(Z_1\) is the spot rate, or zero-coupon yield or zero rate, for period 1
- \(Z_2\) is the spot rate, or zero-coupon yield or zero rate, for period 2
- \(Z_N\) is the spot rate, or zero-coupon yield or zero rate, for period \(N\)
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### Calculating a Bond Price given the sequence of Spot Rates
$$
PV = \frac{PMT}{(1 + Z_1)^1} + \frac{PMT}{(1 + Z_2)^2} + \cdots +
\frac{PMT + FV}{(1 + Z_N)^N} \tag{1}
$$
Where:
- $Z_1$ is the spot rate, or zero-coupon yield or zero rate, for period 1
- $Z_2$ is the spot rate, or zero-coupon yield or zero rate, for period 2
- $Z_N$ is the spot rate, or zero-coupon yield or zero rate, for period $N$ Calculate a Par Rate by solving for PMT
\[ 100 = \frac{PMT}{(1 + z_1)^1} + \frac{PMT}{(1 + z_2)^2} + \cdots + \frac{PMT + 100}{(1 + z_N)^N} \tag{2} \]
- Equation 2 can be used to calculate a par rate by solving for \(PMT\) given a sequence of spot rates \(Z_1, Z_2, \ldots, Z_n\).
- This equation is very similar to Equation 1, except \(PV = FV = 100\).
- Recall that for a bond to trade at par, its coupon rate and yield-to-maturity must be equal. So, by solving for \(PMT\), we also solve for the yield-to-maturity for the bond to trade at par.
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### Calculate a Par Rate by solving for PMT
$$
100 = \frac{PMT}{(1 + z_1)^1} + \frac{PMT}{(1 + z_2)^2} + \cdots +
\frac{PMT + 100}{(1 + z_N)^N} \tag{2}
$$
- Equation 2 can be used to calculate a par rate by solving for $PMT$ given a
sequence of spot rates $Z_1, Z_2, \ldots, Z_n$.
- This equation is very similar to Equation 1, except $PV = FV = 100$.
- Recall that for a bond to trade at par, its coupon rate and yield-to-maturity
must be equal. So, by solving for $PMT$, we also solve for the
yield-to-maturity for the bond to trade at par. General formula for the Implied Forward Rate, \(IFR_{A,B-A}\)
\[ (1 + Z_A)^A \times (1 + IFR_{A,B-A})^{B-A} = (1 + Z_B)^B \tag{3} \]
- Equation 3 is a general formula for the implied forward rate, \(IFR_{A,B-A}\), for a security begins at \(t = A\) and matures at \(t = B\) (tenor \(B - A\))
- To solve for it, we need the spot rate, \(z_A\), and the longer-term spot rate, \(z_B\)
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### General formula for the Implied Forward Rate, $IFR_{A,B-A}$
$$
(1 + Z_A)^A \times (1 + IFR_{A,B-A})^{B-A} = (1 + Z_B)^B \tag{3}
$$
- Equation 3 is a general formula for the implied forward rate, $IFR_{A,B-A}$,
for a security begins at $t = A$ and matures at $t = B$ (tenor $B - A$)
- To solve for it, we need the spot rate, $z_A$, and the longer-term spot rate,
$z_B$