Spot Rates + Accrued Interest

Question 1

Calculate the full price of a bond

Answer 1

(1 + YTM / # of coupon periods per year)t/T

Step 1: Calculate the present value of the bond (have regard to compounding periods)
Step 2: Calculate number of days since last coupon payment (t) and number of days in coupon period (T) (to settlement date)
Step 3: Multiply the value of the bond by the discount date (attention to c periods) to the power of t / T

Example: Calculating the full price of a bond

A 5% bond makes coupon payments on June 15 and December 15 and is trading with a YTM of 4%. The bond is purchased and will settle on August 21 when there will be four coupons remaining until maturity. Calculate the full price of the bond using actual days.

Step 1: Calculate the value of the bond on the last coupon date (coupons are semiannual, so we use 4 / 2 = 2% for the periodic discount rate):
N = 4; PMT = 25; FV = 1,000; I/Y = 2; CPT → PV = –1,019.04
Step 2: Adjust for the number of days since the last coupon payment:
Days between June 15 and December 15 = 183 days.
Days between June 15 and settlement on August 21 = 67 days.
Full price = 1,019.04 × (1.02)67/183 = 1,026.46.

Question 2

Calculate the price of a bond using spot rates.

Answer 2

Coupon 1 / 1+ Spot Rate 1

+

CPN 2 / (1 + SR 2)2

+ n

Question 3

Calculate Accrued Interest

Answer 3

The accrued interest since the last payment date can be calculated as the coupon payment times the portion of the coupon period that has passed between the last coupon payment date and the settlement date of the transaction. For the bond in the previous example, the accrued interest on the settlement date of August 21 is:

$25 (67 / 183) = $9.15

The annual coupon payment is 4% × $1,000 = $40.

Using the 30/360 method, interest is accrued for 30 – 15 = 15 days in May; 30 days each in June and July; and 10 days in August, or 15 + 30 + 30 + 10 = 85 days.

Question 4

Calculate flat price / clean price / quoted price

Answer 4

The full price (invoice price) minus the accrued interest is referred to as the flat price of the bond.

flat price = full price − accrued interest

So for the bond in our example, the flat price = 1,026.46 − 9.15 = 1,017.31.

Question 5

Explain how to Interpolate a 3 year yield to maturity based on a 2 year yield to maturity of 5% and 4 year YTM of 6%.

Answer 5

To interpolate a 3-year yield to maturity (YTM) based on a 2-year YTM of 5% and a 4-year YTM of 6%, you can use a linear interpolation approach.

Here is an example with steps:

Step 1: Determine the range for interpolation

In this case, we want to interpolate the YTM for a 3-year maturity. The available data points are the 2-year YTM (5%) and the 4-year YTM (6%). So, the range for interpolation is 2 years to 4 years.

Step 2: Calculate the weightage for each YTM

In linear interpolation, we assign weightage to the available data points based on their proximity to the desired maturity. The closer the maturity, the higher the weight. In this case, the 2-year maturity is 1 year away from the 3-year maturity, and the 4-year maturity is 1 year ahead. Thus, the weightage for the 2-year YTM is 1 (1 year / 1 year), and the weightage for the 4-year YTM is 0 (1 year / 1 year).

Step 3: Calculate the interpolated YTM

The interpolated YTM is the weighted average of the available data points. The formula is as follows:

Interpolated YTM = [(Weightage of 2-year YTM) * (2-year YTM)] + [(Weightage of 4-year YTM) * (4-year YTM)]

Substituting the values, we get:

Interpolated YTM = (1 * 5%) + (0 * 6%)
= 5%

Therefore, the interpolated 3-year YTM based on a 2-year YTM of 5% and a 4-year YTM of 6% is 5%.