IFT Notes for Level I CFA^{®} Program

Before discussing yield measures for money market instruments, it is important to understand the concept of periodicity. **Periodicity** is the number of compounding periods in a year, or number of coupon payments made in a year.

The **stated annual rate** for a bond will depend on the periodicity we are assuming. The stated annual rate is also called the annual percentage rate or APR.

**Instructor’s Note**

A quarterly coupon paying bond has a periodicity of four, while a semi-annual bond has a periodicity of two, and a monthly-pay bond with a given annual yield would have a periodicity of twelve.

“Compounding more frequently within the year results in a lower (more negative) yield-to-maturity.”

Consider a 5-year, zero-coupon bond priced at 80 per 100 par value. What is the stated annual rate for periodicity = 4, periodicity = 2, and periodicity = 1?

__When periodicity = 4__: compounding happens four times a year. N = 20; (5 years x 4 = 20). PMT = 0 as it is a zero-coupon bond. PV = -80; FV = 100; CPT I/Y = 1.12. This is the rate for each quarter. The stated annual rate is 1.12 x 4 = 4.487%.

__When periodicity = 2__: N = 10; PV = -80; PMT = 0; FV= 100; CPT I/Y = 2.2565. The stated annual rate is 2.25 x 2 = 4.51%.

__When periodicity = 1__: N = 5; PV = -80; PMT = 0; FV= 100; CPT I/Y = 4.56%. With a periodicity of 1, the stated annual rate is the same as the effective annual rate.

The formula for conversion based on periodicity is

**Example**

A 4-year, 3.75% semi-annual coupon payment government bond is priced at 97.5. Calculate the annual yield to maturity stated on a semi-annual bond basis and convert the annual yield to:

- An annual rate comparable to bonds that make quarterly coupon payments.
- An annual rate comparable to bonds that make annual coupon payments.

**Solution to 1:**

The stated annual yield to maturity on a semiannual bond basis can be calculated using a financial calculator: N = 8; PMT = 1.875; FV = 100; PV = -97.5; CPT I/Y. I/Y = 2.2195%. Hence, the stated annual yield to maturity = 2.2195% x 2 = 4.439%.

=

= 4.415%

The annual rate of 4.439% for compounding semiannually compares with 4.415% for compounding quarterly.

**Solution to 2:**

= 4.488%

The annual percentage rate of 4.439% for compounding semiannually compares with an effective annual rate of 4.488%.

The **effective annual rate** (EAR) is the yield on an investment in one year taking into account the effects of compounding. This rate has a periodicity of one as there is only one compounding period per year. EAR is used to compare the rate of return on investments with different frequency of compounding (periodicities).

**Semiannual bond equivalent yield: **Yield per semi-annual period times two. If the yield per semi-annual period is 2%, then the semi-annual bond equivalent yield is 4%.

**Street convention:**It is the yield to maturity using a 30/360 day convention assuming payments are made on scheduled dates, even if the payment date fell on a weekend or a holiday.**True yield:**Yield to maturity calculated using an actual calendar of weekends and holidays. For instance, assume the coupon date falls on 15 March 2015, which is a Sunday. Street convention assumes the payment is made on that date, whereas true yield assumes the payment is made on 16 March if it is a business day. The coupon payment is discounted back from 16 March instead of 15 March.**Government equivalent yield:**Yield to maturity calculated using the actual day/count convention used for U.S. Treasuries.**Current yield:**Sum of the coupon payments received over the year divided by the flat price. It is also called the income or interest yield. Example: A 5-year, 8% semiannual coupon payment bond is priced at $960. Its current yield is 80/960 = 0.0833 = 8.33%. Current yield is not an accurate measure of the rate of return as it ignores the frequency of coupon payments, reinvestment income, and capital gain/loss on a bond.

**Yield-to-call:**Calculates the rate of return on a callable bond if it is bought at market price and held until the call date. The difference between YTM and the yield-to-call is that YTM assumes the bond is held to maturity. Calculation of yield-to-call is the same as YTM where N = number of periods to call date and FV= call price.**Yield-to-first call (YTFC):**It is the internal rate of return if the bond was bought at market price and held until the first call date.**Yield-to-second call:**Similarly, the yield on a callable bond if it was bought at market price and held to the second call date is called yield-to-second call.**Yield-to-worst:**Yield is calculated for every scenario. The lowest yield is called the yield-to-worst.

**Example **

An analyst observes the following statistics for two bonds:

Bond A | Bond B | |

Annual Coupon Rate | 6.00% | 10.00% |

Coupon Payment Frequency | Semi-annually | Quarterly |

Years to Maturity | 4 years | 4 years |

Price (per 100 par value) | 95 | 110 |

Current Yield | ? | ? |

Yield to Maturity | ? | ? |

- Calculate both yield measures for the two bonds.
- How much additional compensation, in terms of yield to maturity, does a buyer of Bond A receive for bearing additional risk compared with Bond B

**Solution to 1:
**The current yield for Bond A is 6/95 = 6.316% and the yield to maturity for Bond A is 7.469%.

The current yield for Bond B is 10/110 = 9.091% and the yield to maturity for Bond B is 7.106%.

Compare the yields for the same periodicity to answer this question. 7.106% for a periodicity of four converts to 7.169% for a periodicity of two. The additional compensation for the greater risk in Bond A is 30 bps (0.07469 – 0.07169).

**Example**

Consider a bond which is selling for $100 and has the following cash flows:

Calculate the yield-to-first call and yield-to-second call.

**Solution:**

The yield-to-first call can be calculated as follows: PV= -100; N = 5; PMT = 10; FV = 102; CPT I/Y; I/Y = 10.325%. YTFC = 10.325%.

The yield-to-second call in our example will be 10.09%: PV= -100; N = 8; PMT = 10; FV = 101; CPT I/Y; I/Y = 10.09%.

**Example**

A bond with 4 years remaining until maturity is currently trading for 101.75 per 100 of par value. The bond offers a 5% coupon rate with interest paid semiannually. The bond is first callable in 2 years and is callable after that date on coupon dates according to the following schedule:

End of Year | Call Price |

2 | 102.50 |

3 | 101.50 |

4 | 100.00 |

- What is the bond’s annual yield-to-first-call?
- What is the bond’s yield-to-worst?

**Solution to 1:**

The yield-to-first-call can be calculated with the following key strokes:

PV = -101.75, FV = 102.5, N = 4, PMT = 2.5, CPT I/Y = 2.6342.

To arrive at the annualized yield-to-first-call, the semiannual rate must be multiplied by two. (2.6342 × 2 = 5.2684)

**Solution to 2:**

The yield-to-worst is 4.52%. The bond’s yield to worst is the lowest of the sequence of yields-to-call and the yield to maturity. Yield-to-first-call = 5.27%, Yield-to-second-call = 4.84%, and Yield to maturity = 4.52%.