Pricing a Bond with PV

A bond's price is the sum of all its future cash flows, each discounted back to today at the yield rate you require.

Key takeaways

• Bond pricing applies present value (PV) logic: future dollars are worth less than today's dollars
• The discount rate used is the yield-to-maturity (YTM) — the return you expect
• Higher required yield means lower bond price; lower required yield means higher price
• The coupon, face value, and time-to-maturity all feed into the calculation
• Understanding the mechanics lets you spot mispriced bonds and check calculator results

The core idea: discounting cash flows

A bond is a stream of cash flows: coupon payments every six months (or annually, depending on the bond), plus the return of principal at maturity. To find what that stream is worth today, you discount each payment back using the yield rate as your discount rate.

The present value formula for a single cash flow is:

PV = Cash flow / (1 + discount rate)^time

For a bond, you apply this to every coupon payment and the final principal repayment, then sum them all.

Consider a concrete example: a bond with a 4% annual coupon, $1,000 face value, maturing in 5 years, and you require a 5% yield. The bond pays $40 each year for five years, then returns the $1,000 principal. Discount each:

• Year 1 coupon: $40 / 1.05^1 = $38.10
• Year 2 coupon: $40 / 1.05^2 = $36.28
• Year 3 coupon: $40 / 1.05^3 = $34.55
• Year 4 coupon: $40 / 1.05^4 = $32.91
• Year 5 coupon: $40 / 1.05^5 = $31.38
• Year 5 principal: $1,000 / 1.05^5 = $783.53

Sum: $38.10 + $36.28 + $34.55 + $32.91 + $31.38 + $783.53 = $956.75

This bond should trade around $956.75 if the market yield is 5%. If it's trading at $960, it's slightly overpriced. If it's at $950, it's a bargain.

Why discount rate matters

The discount rate is your required yield — what you need to earn to compensate for time and risk. If interest rates rise and new bonds offer 5.5%, you will demand a higher yield from your existing 4% coupon bond. To achieve that higher return, the price must fall so that the coupon yield becomes part of a larger total return.

Conversely, if rates fall to 3.5%, your 4% bond becomes more attractive. Its price rises because fewer people need such a high return to compensate.

This inverse relationship — price down, yield up; price up, yield down — is fundamental to all bond trading.

Semi-annual payments and fractional years

Most U.S. corporate and Treasury bonds pay coupons semi-annually. When pricing such bonds, you typically express the semi-annual yield as half of the annual yield, and count time in half-year periods.

Example: A bond with a 4% coupon (2% semi-annual), 5 years to maturity (10 half-year periods), and 5% annual yield (2.5% semi-annual):

• Period 1 coupon: $20 / 1.025^1 = $19.51
• Period 2 coupon: $20 / 1.025^2 = $19.03
• ... (periods 3–9 similarly discounted)
• Period 10 coupon: $20 / 1.025^10 = $15.68
• Period 10 principal: $1,000 / 1.025^10 = $781.20

The sum of these ten terms gives the bond's fair price. In practice, you use a spreadsheet or financial calculator to avoid manual arithmetic.

The bond pricing formula

If you encounter a situation without a calculator, the formal expression is:

Price = Σ [C / (1 + y)^t] + FV / (1 + y)^n

Where:

• C = coupon payment per period
• y = yield per period (as a decimal)
• t = period number (1, 2, 3, ... n)
• FV = face value
• n = total number of periods

This looks daunting but is just a systematic way to capture what we did above: each coupon is divided by the appropriate power of (1 + yield), and the principal is also discounted.

Pulled to par and accrued interest

A bond trading at its face value ($1,000) is said to be trading at par. The price you receive on a secondary market purchase often includes accrued interest — the coupon earned since the last payment date, owed to you by the seller.

For PV calculations, you usually work with the clean price (excluding accrued interest). The actual settlement involves adding back the accrued interest. This distinction matters when you're timing a purchase or calculating total cash outlay.

How investors use this in practice

Professional fixed-income traders run PV calculations continuously. If they observe a bond trading at a price that seems inconsistent with current yields, they investigate: Is there a liquidity premium? Credit deterioration? Or a genuine misprice worth trading on?

Individual investors rarely do this by hand anymore. Brokerage platforms and financial websites display bond prices and yields together. But understanding the mechanics helps you interpret what's happening and why a bond price moved when the Fed changed rates.

Checking your work with real data

Suppose you're looking at a Treasury bond quote: face value $1,000, 3% coupon, 2.5 years to maturity, market price $1,018. Can you back out the yield?

Using the PV formula with the price as the unknown and solving for yield is called computing the yield-to-maturity (YTM). A spreadsheet's YIELD function handles this. If you plug in the bond details, you might find YTM = 2.8%, which means the bond is overpriced if you require a higher yield, or underpriced if 2.8% meets your threshold.

Limitations of the PV model

The PV approach assumes:

1. All coupons are reinvested at the same yield rate. In reality, the first coupon payment lands months away and might be reinvested in a different-yielding bond.
2. The bond is held to maturity. If you sell before maturity, your return depends on the price at sale, not the original yield.
3. No default risk is priced in. For corporate and municipal bonds, credit quality matters. The yield reflects some default probability; the formula doesn't explicitly model it.
4. The discount rate is constant. Long bonds, in particular, are sensitive to changes in rates during the holding period.

These limits don't break the PV model — they're just constraints to keep in mind when you use it.

Summary: the mechanics in action

Start with the coupon and maturity. Choose a required yield. Discount every cash flow at that rate. Sum the present values. That's the bond's price.

If the market price differs, either the market has a different view of the bond's riskiness, or you have a trading opportunity. Mastering this exercise builds intuition for how duration works, why rates matter, and what drives bond returns.

Related concepts

• ./06-yield-to-maturity-ytm.md
• ./01-face-value-explained.md
• ./02-coupon-rate-fixed-vs-floating.md

Decision flow

Next

With the PV framework in hand, you're ready to see how spreadsheets automate bond pricing — and how to use Excel and Google Sheets functions to calculate price, yield, and duration in seconds.