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Michael Brenndoerfer

Time Value of Money and Interest Rates › Worked Example: Bond Valuation

Michael BrenndoerferUpdated February 6, 20266 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 1Time Value of Money and Interest Rates › Worked Example: Bond Valuation

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Let us apply our time value framework to a plain fixed-coupon bond with a stated maturity and principal repaid at maturity. Its cash flows consist of periodic coupons and that final principal repayment. Valuing this bond is a direct application of present value concepts; other bond structures can have different cash flows.

Consider a 5-year bond with:

Face value: $1,000
Coupon rate: 4% (annual payments)
Market yield: 5%

The bond's price equals the present value of all future cash flows:

P = \sum_{t=1}^{5} \frac{40}{(1.05)^t} + \frac{1000}{(1.05)^5}

where:

$P$ : bond price (present value of all cash flows)
$40$ : annual coupon payment ($1,000 × 4%)
$1.05$ : one plus the yield (1 + 0.05)
$1000$ : face value returned at maturity
The first term $\sum_{t=1}^{5} \frac{40}{(1.05)^t}$ : present value of the coupon stream (an annuity)
The second term $\frac{1000}{(1.05)^5}$ : present value of the principal repayment

The structure of this formula reflects the two components of a bond's value. The coupon stream is an annuity: five equal payments of $40, one at the end of each year. We can either discount each payment separately or use the annuity formula. The principal repayment is a single lump sum occurring at maturity, which we discount using the basic present value formula. The bond price is simply the sum of these two present values.

In[31]:

Code

def annuity_pv(payment, r, periods):
    """Present value of ordinary annuity."""
    if r == 0:
        return payment * periods
    annuity_factor = -np.expm1(-periods * np.log1p(r)) / r
    return payment * annuity_factor


def bond_price(face_value, coupon_rate, yield_rate, years):
    """
    Calculate bond price given face value, coupon rate, yield, and maturity.
    Assumes annual coupon payments.
    """
    coupon = face_value * coupon_rate

    # PV of coupons (annuity)
    pv_coupons = annuity_pv(coupon, yield_rate, years)

    # PV of face value
    pv_face = face_value / (1 + yield_rate) ** years

    return pv_coupons + pv_face


# Bond parameters
face_value = 1000
coupon_rate = 0.04
yield_rate = 0.05
years = 5

price = bond_price(face_value, coupon_rate, yield_rate, years)
coupon = face_value * coupon_rate

Out[32]:

Console

Bond Valuation
========================================
Face value:    $1,000
Coupon rate:   4.0% ($40/year)
Market yield:  5.0%
Maturity:      5 years

Bond price:    $956.71
Discount:      $43.29

The bond trades at a discount ($956.71 vs. $1,000 face) because its 4% coupon is below the 5% market yield. Investors require compensation for accepting below-market coupons, which comes in the form of a lower price.

Let's visualize how the bond's cash flows contribute to its price:

Out[33]:

Visualization

Bar chart showing nominal bond cash flows over 5 years with larger final payment. — Nominal cash flows of a 5-year bond with 4% coupon. The final year includes both the coupon payment and principal repayment.

Bar chart showing four successively smaller present values for the equal $40 coupons, followed by a much larger year-5 present value because the final cash flow includes principal. — Present value of bond cash flows discounted at 5% yield. Equal coupon payments have lower present values at later dates; the principal-bearing year-5 payment still contributes the most.

The final cash flow of $1,040 (coupon plus principal) accounts for the bulk of the bond's value, but its contribution is substantially reduced when discounted back five years.

Out[34]:

Visualization

Downward sloping curve showing bond price decreasing as yield increases, crossing par at 4% yield. — Inverse relationship between bond price and yield for a 5-year, 4% coupon bond. When yield equals the coupon rate (4%), the bond trades at par ($1,000). Below par when yields exceed coupons; above par when yields fall below coupons.