**Interest Rates: a snapshot**

The basic principle of interest rates is that the more the credit risk, the more is the interest rate that is being charged.

**Treasury Rates**

Treasury rates are the rates that the investor earns on Treasury bonds and Treasury bills. These are financial instruments used by the government to borrow in its own currency. Since, it is assumed that the government will never default, treasury rates are considered to be risk-free. This means that the investor who buys Treasury bonds or Treasury bills is certain that the interest rate and the principal payments will be made.

**LIBOR**

LIBOR is *London Interbank offered rate*. As the name suggests, it is the rate at which a bank is ready to deposit a large amount of money with other banks. (For derivatives, LIBOR is considered to be risk-free rate)

**LIBID**

LIBID is *London interbank bid rate*. This is the rate at which the bank will accept deposits from other banks. There is usually a very small spread between LIBOR and LIBID rates. (LIBOR>LIBID).

**Interest Rate calculations**

Suppose the principal amount of *P* is invested for *n* years at an interest rate of *R*. The value of the investment at termination is

If the rate is compounded *m* times a year, we have

Let’s look at the effect on an amount of $100 at an interest rate of 10% in 1 year.

Frequency of compounding | Value |

Annually (m = 1) | 110 |

Semi-annually (m = 2) | 110.25 |

Quarterly (m = 4) | 110.38 |

Monthly (m = 12) | 110.47 |

Weekly (m = 52) | 110.51 |

Daily (m = 365) | 110.52 |

**Continuous Compounding**

In this case, m tends to infinity

How do we equate continuous compounding with equivalent rate compounding *m *times per year?

Let’s say *R _{c}* is the continuous rate of interest and

*R*is the rate compounding m times per year

_{m}**Zero rates**

The n year zero-coupon interest rate is the rate of interest that is paid for an investment for n years without any periodic interest payments.

It is also referred to as the n-year spot rate, the n-year zero rate.

**Bond Pricing**

Some of the principles underlying a bond pricing are:

- Most bonds pay coupons to bond holder periodically.
- The bond’s principal (par value or face value) is paid at the end of its life.
- The theoretical price of the bond is the sum of the PV of all the cash flows that the owner will receive throughout its life.

**Example:** Treasury zero rates, measured with continuous compounding are provided below

Maturity (years) | Zero rate continuously compounded (%) |

0.5 | 5.0 |

1.0 | 5.8 |

1.5 | 6.4 |

2.0 | 6.8 |

Suppose, a 2-year treasury bond with a principal of $100 provides coupons at the rate of 8% per annum semi-annually. Calculate the price of the bond

Cash Flows

Maturity (years) | Cash Flow |

0.5 | 4 |

1.0 | 4 |

1.5 | 4 |

2.0 | 104 |

The price of the bond is

**Bond Yield**

A bond’s yield is the discount rate that, when applied to all the cash flows, gives a bond price that is equal to the market price

In the above example, if the market price of the bond is $102.09, then the equation for the bond yield would be

Solving the equation, we get y= 0.067511 or 6.7511% (using goal seek in excel)

**Par Yield**

The par yield is the coupon rate that makes the bond equal to its par value (face value or the principal value).

Using the above example, the equation for par yield would be

By solving, we get c = 6.87% with semi-annual compounding

**Treasury zero rates**

We will use the bootstrap method to determine the treasury zero rates

Bond Principal ($) | Time to maturity (years) | Annual Coupon ($) | Bond Price ($) |

100 | 0.25 | 0 | 97.5 |

100 | 0.50 | 0 | 94.9 |

100 | 1.00 | 0 | 90.0 |

100 | 1.50 | 8 | 96.0 |

100 | 2.00 | 12 | 101.6 |

As we can see, the 3-month bond provides a return of $2.5 in 3 months on an investment of $97.5.

The 3-month zero rate with quarterly compounding is

(4 × 2.5)/97.5 = 0.102564 (10.2564%) per annum

The continuously compounded rate is *10.127%*

Similarly, the 6-month rate with continuous compounding is *10.469%*

And the 1-year rate with continuous compounding is *10.536 %*

The fourth bond provides coupons at

6 -months | 4 |

1-year | 4 |

1.5- year | 104 |

Using the earlier calculations, if the one year zero-rate is denoted by *R*

We have,

Solving for R, we get R = *10. 681%* (continuously compounded)

Similarly, the 2-year zero rate is *10.808%* (continuously compounded)

Maturity (in years) | Zero-rate (continuously compounded) |

0.25 | 10.127% |

0.5 | 10.469% |

1.0 | 10.536 % |

1.5 | 10. 681% |

2.0 | 10.808% |

In practice, there are no bonds as the ones stated above, the approach is used by analysts to interpolate between the bond prices.

**Forward Rates**

Rate of interest that is applicable to a financial transaction that will take place in the future.

If R_{1} and R_{2} are the zero rates for maturities T_{1} and T_{2} and R_{F} is the forward rate for the period between T_{1} and T_{2},

Year (n) | Zero-rate for n year investment | Forward rate for n-year |

1 | 10% | |

2 | 10.5% | 11% |

3 | 10.8% | 11.4% |

4 | 11.0% | 11.6% |

5 | 11.1% | 11.5% |

Amount in 1 year = 100*e*^{0.1×1} = $110.517 (since it is the total amount to be received, the zero rate is used)

Amount in 2 years = 100*e*^{0.105×2} = $123.367

The amount between the end of year 1 and end of year 2 is given by

110.517*e*^{0.11} = $123.367 (note that the forward rate is used here)

The forward rate is therefore also the rate of interest that is implied by the zero rates for the period of time between the end of the first year and the end of the second year.