Price of Bond with irregular long first coupon period

Question

I'm trying to manually calculate the price of a bond with a long first coupon period, found a few candidate formulas on

Microsoft Excel Document: OddFPrice

mit.edu: OddFPrice

WestClinTech.com: Calculating price of bond with OFC on SQLServer

The presented formulae vary across these sources, also I can't seem to match the results given by the software when using the formula given for it (microsoft excel for instance). I suspect my current problem is an unclear understanding of the variables used in the calculations. The following is a brief description of my current mental model (using notation on excel):

Example Problem Data:

• Bond Issue Date(m/d/y): 10 / 15 / 2008
• Settlement Date: 11 / 11 / 2008
• First Coupon Date: 1 / 1 / 2010
• Maturity Date: 1 / 1 / 2022
• Coupon Rate: 7.85%
• Frequency of coupon payments: semi-annual
• Yield: 6.25
• Redemption Value per $100 face-value: $100
• Day count basis: Actual/actual

Current understanding of the terms (as described in the given equation on excel page)

• Ai: Number of days from bond issue date to settlement date
• DCi: Number of days from issue date to next imaginary coupon after issue date (not first actual coupon)
• DSC: Number of days from settlement date to next imaginary coupon after issue date (not first actual coupon)
• E: Length in days of the first coupon period
• N: Number of coupons payable between the first coupon date(exclusive) and maturity date
• NC (becomes hairy here I think): Number of imaginary coupons between the issue date and settlement date
• NLi: Length in days of the imaginary coupon period which happens to contain the settlement date
• Nq: Number of imaginary coupon periods between settlement date and first coupon date rounded down

I strongly suspect my grasp of these terms is the real problem, so I'd very much appreciate any of the following:

1. Clarification on the terms I described or confirmation if correct
2. The correct/preferred formula of the three sources
3. A detailed explanation of the problem solving process

NB: I used the term 'imaginary coupon' here in place of quasi or pseudo coupon, I do not clearly understand the meaning of those terms in these context, or whether they mean the same thing, a clarification would also be appreciated greatly.

I'm happy to provide additional details if required, thanks.

Answer 1

You should be able to do it manually very quickly in Excel, or write a simple VBA program. For example, consider a bond that pays $10 a year for five years starting 5 years from today, and then $100 ten years from today. The timeline of cash flows looks like so.

Periods 0 -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10

Cash flows 0 -- 0 -- 0 -- 0 -- 0 -- 10 -- 10 -- 10 -- 10 -- 10 -- 110

The value of the bond today is:

Value_0 = 10/(1+r_5)^5 + 10/(1+r_6)^6 + ... + 110/(1+r_10)^10

where r_5,...r_10 are the annually compounded yields. If the yields are semi-annually compounded, the value is

Value_0 = 10/(1+r_5/2)^10 + 10/(1+r_6/2)^12 + ... + 110/(1+r_10/2)^20

If the time to each cash flow is not a "round" number of periods, measure time in years and use effective annual rates. E.g., if the first cash flow comes in 1,794 days, its present value would be

10/(1+r)^(1794/365.25)

with the discount rate, r, corresponds to the effective annual rate on a 1,794 day loan (a little less than 5 years).