Fixed Income Valuation

Pricing with Fractional Years (e.g., 5.2 Years)

1. The Intuition: "Time Travel"

In practice, we need to calculate price of bonds with, for instance, 5 years and 75 days to maturity. However, standard annuity formulas are in some sense rigid, i.e., the first payment is assumed to be exactly one full interval away.

We solve this issue using a two-step mental shortcut:

TODAY Year 0 NEXT COUPON Year 0.205 (75 Days) MATURITY Year 5.205 STEP 1: FWD STEP 2: BACK
  1. Step 1 (Forward): Fast-forward to the next coupon date. Calculate the value there.
  2. Step 2 (Back): Discount that future value back to "today".

2. Examples

Scenario:
• Time to Maturity: 5 Years and 75 Days.
• Coupon: 5% (Paid Semi-Annually, so $25 per payment).
• Discount Rate: 6% per annum.

Note: Because coupons are paid semi-annually, the periodic rate is \(r = 0.03\).

Case A: Zero Coupon Bond

For a Zero, we simply measure the total time in years and plug it into the PV formula.

1. Calculate Total Time (Years) $$ T_{years} = 5 + \frac{75}{365} = 5.2055 \text{ Years} $$ 2. Discount Face Value

We use the semi-annual rate (\(r=0.03\)). The exponent must double the years to count semi-annual intervals.

$$ PV = \frac{1000}{(1+r)^{2 \times T_{years}}} $$ $$ PV = \frac{1000}{(1.03)^{10.411}} = \$735.21 $$

Case B: Coupon Bond

Step 1: Value at "Next Coupon" (Future)

Imagine we are standing at the 0.205 Year mark. There are exactly 5 years remaining.

Since payments are semi-annual, 5 years means 10 payments (\(T=10\)). We use the Annuity Formula:

$$ PV_{annuity} = \frac{C}{r} \left( 1 - \frac{1}{(1+r)^T} \right) $$

Substituting \(C=25\), \(r=0.03\), and \(T=10\):

$$ P_{future} = \frac{25}{0.03} \left( 1 - \frac{1}{(1.03)^{10}} \right) + \frac{1000}{(1.03)^{10}} $$ $$ P_{future} = \$213.26 + \$744.09 = \$957.35 $$

Crucial: Don't forget the coupon ($25) we get at this moment!

Total Future Value = $957.35 + $25.00 = \$982.35$

Step 2: Discount Back to Today

Now, bring that amount back to the present. The time difference is 75 days.

$$ \text{Fractional Year } (w) = \frac{75}{365} = 0.2055 \text{ Years} $$

We discount using the semi-annual convention (\(2 \times w\) in the exponent):

$$ PV_{today} = \frac{\text{Total Future Value}}{(1+r)^{2 \times w}} $$ $$ PV_{today} = \frac{982.35}{(1.03)^{0.411}} $$
Price = $970.49
Clean vs. Dirty Price: The $970.49 calculated above is the dirty price (also full price or invoice price). The price you see on a trading screen (clean price) would subtract the accrued interest for the 107 days that have passed in the coupon period.