- Calculate and interpret the impact of different compounding frequencies on a bond’s value.
- Define spot rate and compute spot rates given discount factors.
- Interpret the forward rate, and compute forward rates given spot rates.
- Define par rate and describe the equation for the par rate of a bond.
- Interpret the relationship between spot, forward, and par rates.
- Assess the impact of maturity on the price of a bond and the returns generated by bonds.
- Define the “flattening” and “steepening” of rate curves and describe a trade to reflect expectations that a curve will flatten or steepen.
- Describe a swap transaction and explain how a swap market defines par rates.

- Video Lecture
- |
- PDFs
- |
- List of chapters

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

at the end of one year.

The rate means that an amount A compounds to:

at the end of one year.

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

EXAMPLE –

Suppose the rate is 5% with semi-annual compounding, and the equivalent rate with quarterly compounding needs to be calculated. In this case

The equivalent rate with quarterly compounding is therefore __________________

- 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

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

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

- 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

- 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.

- 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

It is known that:

Substituting for d(T)

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

- 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
- where 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 is

where

is the spot rate for maturity is the spot rate for maturity . 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 are forward rates in n successive six-month periods. Then:

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

- 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 |

- 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.