VRM 10. Interest Rates

Relationship Between Two Discrete Frequencies

Suppose rate $R_1$ is compounded $m_1$ times per annum, Let the equivalent rate compounded $m_1$ times per annum be $R_1$ . The $R_1$ rate means that an amount A compounds to:

$A\left(1+\frac{R_1}{m_1}\right)^{m_1}$

at the end of one year.

The $R_1$ rate means that an amount A compounds to:

$A\left(1+\frac{R_2}{m_2}\right)^{m_2}$

at the end of one year.

Rate $R_1$ compounded $m_1$ times per year is therefore equivalent to rate $R_1$ compounded $m_1$ times per year when:

$A\left(1+\frac{R_1}{m_1}\right)^{m_1}=A\left(1+\frac{R_2}{m_2}\right)^{m_2}$

$\Rightarrow R_2=\left[\left(1+\frac{R_1}{m_1}\right)^{\frac{m_1}{m_2}}-1\right]m_2$

EXAMPLE –

Suppose the rate is 5% with semi-annual compounding, and the equivalent rate with quarterly compounding needs to be calculated. In this case $m_1 = 2, \; m_2 = 4, \; and \; R_1 = 0.05$

The equivalent rate with quarterly compounding is therefore __________________

Spot Rates

The spot rate is the interest rate earned when cash is received at just one future time. It is also referred to as the zero-coupon interest rate, or just the “zero.”

Suppose USD 100 is invested today and is repaid with USD 120 in three years with no intermediate payments. The three-year spot rate is the rate that equates USD 120 in three years with USD 100 today. If the rate R is measured with annual compounding, then

$100(1+R)^3=120\Rightarrow R=6.27\%$

With semi-annual compounding for example, the rate is given by:

$100\left(1+\frac{r(t)}{2}\right)^{2t}$

When the discount factor d(t) is applied to this, it should bring it back to 100. Hence:

$100\left(1+\frac{r(t)}{2}\right)^{2t}d(t)=100\Rightarrow d(t)=\left(1+\frac{r(t)}{2}\right)^{-2t}$

Par Rates

The value of the coupon rate for which the value of the bond will be equal to its face value is referred to as the par rate.

Let A(T) be the value of an instrument that pays USD 1 on every payment date (this is referred to as an annuity):

A(T) = d(0.5) + d(1.0) + d(1.5) + ⋯ + d(T)

It can be easily shown that for a semi-annual paying bond, the par rate is given by

$p=\frac{2\times100\times(1-d(T))}{A(t)}$

Suppose the spot interest rates, expressed with semi-annual compounding, for 0.5, 1.0, 1.5, and 2.0 years be 4%, 5%, 5.5%, and 6% (respectively).

The two-year par rate (%) is

Thus, a two-year bond paying a coupon semi-annually at a rate of __________________ per year is worth par.

Valuing Bond Using Par Rates

The par rate in conjunction with the annuity factors A(T) to provide a way of valuing bonds with other coupons. Consider a bond with maturity T, coupon c, and a face value of 100. The value of the bond (V) is

$V=\frac{c}{2}A(T)+100d(T)$

It is known that:

$\frac{p}{2}A(T)+100d(T)=100$

Substituting for d(T)

$V=100+\frac{c-p}{2}A(T)$

In the earlier example, the par rate is _________________. When the coupon is ___________, this formula gives the value of the bond as:

Forward Rates

Forward rates are the future spot rates implied by today’s spot rates. For example, suppose the offered one-year rate is 3% and the offered two-year rate is 4% (both with annual compounding). Suppose F is the forward rate for the second year. The forward rate is such that USD 100, if invested at 3% for the first year and at a rate of F for the second year, gives the same outcome as 4% for two years. This means that:
When rates are expressed with semi-annual compounding (as is frequently the case in fixed-income markets), an extension of this analysis shows that the forward rate per six months for a six-month period starting at time T is $\frac{\left(1+\frac{R_2}{2}\right)^{T+0.5}}{\left(1+\frac{R_1}{2}\right)^T}-1$
where $R_1 \text{and} R_2$ are the spot rates for maturities T and T + 0.5 (respectively) with semi-annual compounding. Thus, the annualized forward rate expressed with semi-annual compounding is twice this.

When rates are expressed with continuous compounding, the forward rate for the period between time $R_1 \text{and} R_2$ is

$F=\frac{R_2T_2-R_1T_1}{T_2-T_1}$

where

$R_1$ is the spot rate for maturity $T_1\text{and}R_2$ is the spot rate for maturity $R_1$ . This formula is approximately true when other compounding frequencies are used for the rates.

When forward rates for successive periods are compounded, spot rates are obtained. For example, suppose all rates are expressed with semi-annual compounding, and that $F_1, F_2, \ldots F_n$ are forward rates in n successive six-month periods. Then:

$\left(1+\frac{R}{2}\right)^n=\left(1+\frac{F_1}{2}\right)\left(1+\frac{F_2}{2}\right)\dots\left(1+\frac{F_n}{2}\right)$

Thus, if a large financial institution can borrow or lend at spot rates, it can lock in the forward rate.

Properties Of Spot, Forward, And Par Rates

Key properties of the rates discussed till now are as follows:
- If the term structure is flat (with all spot rates the same), all par rates and all forward rates equal the spot rate.
- If the term structure is upward-sloping, the par rate for a certain maturity is below the spot rate for that maturity.
- If the term structure is downward-sloping, the par rate for a certain maturity is above the spot rate for that maturity.
- If the term structure is upward-sloping, forward rates for a period starting at time T are greater than the spot rate for maturity T.
- If the term structure is downward-sloping, forward rates for a period starting at time T are less than the spot rate for maturity T.

The situation for an upward-sloping term structure case is illustrated in this Table.

Maturity (yrs)	Spot	6-Month Fwd	Par
0.5	2.01	4.04	2.01
1	3.02	4.86	3.01
1.5	3.63	5.27	3.62
2	4.04	5.83	4.01
2.5	4.40	5.94	4.36
3	4.65	6.24	4.60
3.5	4.88	6.36	4.82
4	5.06	6.54	4.99
4.5	5.23	6.67	5.14
5	5.37		5.28

The situation for a downward-sloping term structure case is illustrated in this Table.

Maturity (yrs)	Spot	6-Month Fwd	Par
0.5	5.06	4.24	5.06
1	4.65	3.73	4.66
1.5	4.35	3.73	4.36
2	4.19	3.17	4.21
2.5	3.99	3.07	4.01
3	3.84	3.12	3.86
3.5	3.73	3.08	3.76
4	3.65	3.10	3.68
4.5	3.59	3.08	3.62
5	3.54		3.57

Flattening And Steepening Term Structures

A flattening term structure occurs
- When long- and short-maturity rates both move down, but long-maturity rates move down by more than short-maturity rates (known as a bull flattener); or
- When long- and short-maturity rates both move up, but short-maturity rates move up by more than long-maturity rates (known as a bear flattener).

A steepening term structure occurs
- When long- and short-maturity rates both move down but short-maturity rates move down by more than long-maturity rates; or
- When long- and short-maturity rates both move up, but short-maturity rates move up by less than long-maturity rates.
Note that a flattening term structure is not necessarily one that becomes flatter and a steepening term structure is not necessarily one that becomes steeper. If a term structure is already upward-sloping, then a steepening will cause it to be more upward-sloping and a flattening will cause it to be less upward-sloping. But if it is downward-sloping, the reverse is true. The steepening/flattening language refers to the relative rate movements and does not depend on the initial slope of the yield curve.
Suppose a trader thinks the current upward-sloping term structure will steepen so that 20-year rates increase faster than ten-year rates. The trader can short 20-year bonds and buy ten-year bonds. If the trader is right, the 20-year bonds will decline in value relative to the ten-year bonds and the trader will make money. Similarly, a trader who thinks that the term structure will flatten should buy 20-year bonds and short ten-year bonds.

Frequently Asked Questions

What is the relationship between different compounding frequencies for interest rates ?

The relationship between different compounding frequencies can be calculated using specific formulas to determine equivalent interest rates.

What are spot rates ?

Spot rates are the interest rates earned on cash received at a single future time, often referred to as zero-coupon rates.

How do par rates differ from spot rates ?

Par rates are the coupon rates at which a bond's price equals its face value, whereas spot rates apply to cash received at a single future time.

How do you calculate a bond's value using par rates ?

Bond value can be calculated using par rates along with annuity factors to determine its present value.

What are forward rates ?

Forward rates are future interest rates implied by current spot rates, representing expected rates for future periods.

How do you determine the equivalent interest rate with different compounding frequencies ?

Equivalent rates can be determined by adjusting for different compounding periods using a specific formula.

What happens when the term structure is upward-sloping ?

In an upward-sloping term structure, forward rates are typically higher than the corresponding spot rates for future periods.

What is a flattening term structure ?

A flattening term structure occurs when long-term rates decrease more than short-term rates, reducing the slope of the yield curve.

What is a steepening term structure ?

A steepening term structure occurs when long-term rates increase more than short-term rates, increasing the slope of the yield curve.

How can traders profit from changes in the term structure ?

Traders can profit by taking positions that benefit from expected changes, such as steepening or flattening of the yield curve, by buying or shorting bonds of different maturities.

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Relationship Between Two Discrete Frequencies

Suppose rate $R_1$ is compounded $m_1$ times per annum, Let the equivalent rate compounded $m_1$ times per annum be $R_1$ . The $R_1$ rate means that an amount A compounds to:

$A\left(1+\frac{R_1}{m_1}\right)^{m_1}$

at the end of one year.

The $R_1$ rate means that an amount A compounds to:

$A\left(1+\frac{R_2}{m_2}\right)^{m_2}$

at the end of one year.

Rate $R_1$ compounded $m_1$ times per year is therefore equivalent to rate $R_1$ compounded $m_1$ times per year when:

$A\left(1+\frac{R_1}{m_1}\right)^{m_1}=A\left(1+\frac{R_2}{m_2}\right)^{m_2}$

$\Rightarrow R_2=\left[\left(1+\frac{R_1}{m_1}\right)^{\frac{m_1}{m_2}}-1\right]m_2$

EXAMPLE –

Suppose the rate is 5% with semi-annual compounding, and the equivalent rate with quarterly compounding needs to be calculated. In this case $m_1 = 2, \; m_2 = 4, \; and \; R_1 = 0.05$

The equivalent rate with quarterly compounding is therefore __________________

Spot Rates

The spot rate is the interest rate earned when cash is received at just one future time. It is also referred to as the zero-coupon interest rate, or just the “zero.”

$100(1+R)^3=120\Rightarrow R=6.27\%$

With semi-annual compounding for example, the rate is given by:

$100\left(1+\frac{r(t)}{2}\right)^{2t}$

When the discount factor d(t) is applied to this, it should bring it back to 100. Hence:

$100\left(1+\frac{r(t)}{2}\right)^{2t}d(t)=100\Rightarrow d(t)=\left(1+\frac{r(t)}{2}\right)^{-2t}$

Par Rates

The value of the coupon rate for which the value of the bond will be equal to its face value is referred to as the par rate.

Let A(T) be the value of an instrument that pays USD 1 on every payment date (this is referred to as an annuity):

A(T) = d(0.5) + d(1.0) + d(1.5) + ⋯ + d(T)

It can be easily shown that for a semi-annual paying bond, the par rate is given by

$p=\frac{2\times100\times(1-d(T))}{A(t)}$

Suppose the spot interest rates, expressed with semi-annual compounding, for 0.5, 1.0, 1.5, and 2.0 years be 4%, 5%, 5.5%, and 6% (respectively).

The two-year par rate (%) is

Thus, a two-year bond paying a coupon semi-annually at a rate of __________________ per year is worth par.

Valuing Bond Using Par Rates

The par rate in conjunction with the annuity factors A(T) to provide a way of valuing bonds with other coupons. Consider a bond with maturity T, coupon c, and a face value of 100. The value of the bond (V) is

$V=\frac{c}{2}A(T)+100d(T)$

It is known that:

$\frac{p}{2}A(T)+100d(T)=100$

Substituting for d(T)

$V=100+\frac{c-p}{2}A(T)$

In the earlier example, the par rate is _________________. When the coupon is ___________, this formula gives the value of the bond as:

Forward Rates

Forward rates are the future spot rates implied by today’s spot rates. For example, suppose the offered one-year rate is 3% and the offered two-year rate is 4% (both with annual compounding). Suppose F is the forward rate for the second year. The forward rate is such that USD 100, if invested at 3% for the first year and at a rate of F for the second year, gives the same outcome as 4% for two years. This means that:
When rates are expressed with semi-annual compounding (as is frequently the case in fixed-income markets), an extension of this analysis shows that the forward rate per six months for a six-month period starting at time T is $\frac{\left(1+\frac{R_2}{2}\right)^{T+0.5}}{\left(1+\frac{R_1}{2}\right)^T}-1$
where $R_1 \text{and} R_2$ are the spot rates for maturities T and T + 0.5 (respectively) with semi-annual compounding. Thus, the annualized forward rate expressed with semi-annual compounding is twice this.

When rates are expressed with continuous compounding, the forward rate for the period between time $R_1 \text{and} R_2$ is

$F=\frac{R_2T_2-R_1T_1}{T_2-T_1}$

where

$R_1$ is the spot rate for maturity $T_1\text{and}R_2$ is the spot rate for maturity $R_1$ . This formula is approximately true when other compounding frequencies are used for the rates.

When forward rates for successive periods are compounded, spot rates are obtained. For example, suppose all rates are expressed with semi-annual compounding, and that $F_1, F_2, \ldots F_n$ are forward rates in n successive six-month periods. Then:

$\left(1+\frac{R}{2}\right)^n=\left(1+\frac{F_1}{2}\right)\left(1+\frac{F_2}{2}\right)\dots\left(1+\frac{F_n}{2}\right)$

Thus, if a large financial institution can borrow or lend at spot rates, it can lock in the forward rate.

Properties Of Spot, Forward, And Par Rates

Key properties of the rates discussed till now are as follows:
- If the term structure is flat (with all spot rates the same), all par rates and all forward rates equal the spot rate.
- If the term structure is upward-sloping, the par rate for a certain maturity is below the spot rate for that maturity.
- If the term structure is downward-sloping, the par rate for a certain maturity is above the spot rate for that maturity.
- If the term structure is upward-sloping, forward rates for a period starting at time T are greater than the spot rate for maturity T.
- If the term structure is downward-sloping, forward rates for a period starting at time T are less than the spot rate for maturity T.

The situation for an upward-sloping term structure case is illustrated in this Table.

Maturity (yrs)	Spot	6-Month Fwd	Par
0.5	2.01	4.04	2.01
1	3.02	4.86	3.01
1.5	3.63	5.27	3.62
2	4.04	5.83	4.01
2.5	4.40	5.94	4.36
3	4.65	6.24	4.60
3.5	4.88	6.36	4.82
4	5.06	6.54	4.99
4.5	5.23	6.67	5.14
5	5.37		5.28

The situation for a downward-sloping term structure case is illustrated in this Table.

Maturity (yrs)	Spot	6-Month Fwd	Par
0.5	5.06	4.24	5.06
1	4.65	3.73	4.66
1.5	4.35	3.73	4.36
2	4.19	3.17	4.21
2.5	3.99	3.07	4.01
3	3.84	3.12	3.86
3.5	3.73	3.08	3.76
4	3.65	3.10	3.68
4.5	3.59	3.08	3.62
5	3.54		3.57

Flattening And Steepening Term Structures

A flattening term structure occurs
- When long- and short-maturity rates both move down, but long-maturity rates move down by more than short-maturity rates (known as a bull flattener); or
- When long- and short-maturity rates both move up, but short-maturity rates move up by more than long-maturity rates (known as a bear flattener).

A steepening term structure occurs
- When long- and short-maturity rates both move down but short-maturity rates move down by more than long-maturity rates; or
- When long- and short-maturity rates both move up, but short-maturity rates move up by less than long-maturity rates.
Note that a flattening term structure is not necessarily one that becomes flatter and a steepening term structure is not necessarily one that becomes steeper. If a term structure is already upward-sloping, then a steepening will cause it to be more upward-sloping and a flattening will cause it to be less upward-sloping. But if it is downward-sloping, the reverse is true. The steepening/flattening language refers to the relative rate movements and does not depend on the initial slope of the yield curve.
Suppose a trader thinks the current upward-sloping term structure will steepen so that 20-year rates increase faster than ten-year rates. The trader can short 20-year bonds and buy ten-year bonds. If the trader is right, the 20-year bonds will decline in value relative to the ten-year bonds and the trader will make money. Similarly, a trader who thinks that the term structure will flatten should buy 20-year bonds and short ten-year bonds.

Frequently Asked Questions

What is the relationship between different compounding frequencies for interest rates ?

The relationship between different compounding frequencies can be calculated using specific formulas to determine equivalent interest rates.

What are spot rates ?

Spot rates are the interest rates earned on cash received at a single future time, often referred to as zero-coupon rates.

How do par rates differ from spot rates ?

Par rates are the coupon rates at which a bond's price equals its face value, whereas spot rates apply to cash received at a single future time.

How do you calculate a bond's value using par rates ?

Bond value can be calculated using par rates along with annuity factors to determine its present value.

What are forward rates ?

Forward rates are future interest rates implied by current spot rates, representing expected rates for future periods.

How do you determine the equivalent interest rate with different compounding frequencies ?

Equivalent rates can be determined by adjusting for different compounding periods using a specific formula.

What happens when the term structure is upward-sloping ?

In an upward-sloping term structure, forward rates are typically higher than the corresponding spot rates for future periods.

What is a flattening term structure ?

A flattening term structure occurs when long-term rates decrease more than short-term rates, reducing the slope of the yield curve.

What is a steepening term structure ?

A steepening term structure occurs when long-term rates increase more than short-term rates, increasing the slope of the yield curve.

How can traders profit from changes in the term structure ?

Traders can profit by taking positions that benefit from expected changes, such as steepening or flattening of the yield curve, by buying or shorting bonds of different maturities.

VRM 10. Interest Rates

Relationship Between Two Discrete Frequencies

Spot Rates

Par Rates

Valuing Bond Using Par Rates

Forward Rates

Properties Of Spot, Forward, And Par Rates

Flattening And Steepening Term Structures

Frequently Asked Questions

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VRM 10. Interest Rates

Relationship Between Two Discrete Frequencies

Spot Rates

Par Rates

Valuing Bond Using Par Rates

Forward Rates

Properties Of Spot, Forward, And Par Rates

Flattening And Steepening Term Structures

Frequently Asked Questions

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