Bond Valuation: PV of Coupons + Face Value

What is a bond?

A bond is a loan from an investor to an issuer (government or corporation). In return, the issuer promises:

1. Coupon payments — periodic interest, usually fixed.
2. Face value (par) — the principal, repaid at maturity.

The pricing formula

A bond's fair price is the sum of the present values of all its future cash flows: $P = \underbrace{\sum_{t=1}^{n} \dfrac{C}{(1 + y)^t}}_{\text{PV of coupons (annuity)}} + \underbrace{\dfrac{F}{(1 + y)^n}}_{\text{PV of face value}}$

where

• $C$ = periodic coupon payment,
• $F$ = face value (principal),
• $y$ = market yield per period (discount rate),
• $n$ = number of periods to maturity.

In closed form (annuity): $P = C \cdot \dfrac{1 - (1 + y)^{-n}}{y} + \dfrac{F}{(1 + y)^n}$

Step-by-step solution

Setup: $F = 1{,}000$, coupon rate $5\%$ ⟹ $C = 50$/year, $n = 10$, $y = 0.06$.

Step 1 — PV of the coupon stream (annuity). $\dfrac{1 - (1.06)^{-10}}{0.06} = \dfrac{1 - 0.5584}{0.06} = \dfrac{0.4416}{0.06} \approx 7.3601$ $PV_{\text{coupons}} = 50 \cdot 7.3601 \approx 368.00$

Step 2 — PV of the face value. $\dfrac{1{,}000}{(1.06)^{10}} = \dfrac{1{,}000}{1.7908} \approx 558.39$

Step 3 — Add: $P = 368.00 + 558.39 \approx \boxed{926.39}$

The bond is worth about $926.39 today — less than its $1,000 face value because the coupon rate (5%) is below the market yield (6%).

Bond pricing relationships

• Coupon rate = yield → bond trades at par (price = face).
• Coupon rate < yield → discount bond (price < face). Our case.
• Coupon rate > yield → premium bond (price > face).
• Higher yield → lower bond price (inverse relationship).
• Longer maturity → more rate sensitivity (higher duration).

Sanity check at different yields

• $y = 5\%$ (= coupon): price = $1,000 (par).
• $y = 6\%$: price ≈ $926.39 (our answer).
• $y = 4\%$: price ≈ $1,081.11 (premium).
• $y = 8\%$: price ≈ $798.70 (deeper discount).
• $y = 10\%$: price ≈ $692.77.

Every 1% rise in yield drops the price by roughly 7–8% on this 10-year bond.

Semiannual coupons (US convention)

Most US corporate and Treasury bonds pay semiannually. Adjustments:

• Coupon per period: $C/2$
• Periods: $2n$
• Discount rate: $y/2$

Our bond with semiannual coupons at 6% market yield: $P = 25 \cdot \dfrac{1 - (1.03)^{-20}}{0.03} + \dfrac{1000}{(1.03)^{20}} \approx 25 \cdot 14.8775 + 553.68 \approx 925.61$

Very close to the annual-compounding answer. The convention matters more as rates/maturities grow.

Yield-to-maturity (YTM)

The yield to maturity is the discount rate $y$ that makes the PV formula equal to the current market price. It's the bond's IRR — solved numerically (or with financial-calculator functions).

Duration and sensitivity

Duration measures how much a bond's price changes for a 1% yield change. Longer maturities and smaller coupons → higher duration → more rate sensitivity. Our 10-year 5% bond has duration around 7.8 years.

Common mistakes

• Using coupon rate as the discount rate. The discount rate is the market yield, not the coupon rate.
• Forgetting the face-value term. Bond = annuity + a single future lump sum.
• Mishandling semiannual vs. annual. Match period, coupon, and rate consistently.
• Ignoring accrued interest for bonds traded between coupon dates (clean vs. dirty price).

Try it in the visualization

Two stacks: coupon PVs as a series of bars shrinking to the right, and the face-value PV as one tall bar at maturity. Their sum = bond price, shown relative to the par line of $1,000.