A bond's price equals the present value of all its future cash flows — coupon payments and face value — discounted at the market interest rate: Price = Σ[C/(1+r)^t] + F/(1+r)^T. Bond prices and interest rates move inversely: when rates rise, existing bond prices fall, because future fixed payments are discounted more heavily. Bonds trade at par (price = face value) when the coupon rate equals the market rate, at a premium when the coupon exceeds market rates, and at a discount otherwise. This inverse relationship is not a market anomaly but a mathematical necessity of present-value discounting.
Price a 5-year, 5% coupon bond at market rates of 3%, 5%, and 7% to observe the price-rate inverse relationship. Verify that coupon rate equal to market rate always gives a price of par. Use a spreadsheet to handle multi-period discounting for precision.
You already know how to calculate the present value of a future cash flow: divide by (1 + r)^t, where r is the discount rate and t is the number of periods. A bond is simply a bundle of future cash flows — a series of coupon payments (typically annual or semiannual) and the return of face value at maturity. Pricing a bond means computing the present value of all those cash flows, discounted at the current market interest rate. That is the entire concept; the rest is application.
The formula makes this explicit: Price = C/(1+r) + C/(1+r)² + ... + C/(1+r)^T + F/(1+r)^T, where C is the coupon payment, r is the market rate, T is time to maturity, and F is face value. The coupon payments form an annuity (hence the soft prerequisite), and the face value is a single lump sum. In a spreadsheet, you sum these terms explicitly; on an exam, you use the annuity formula to collapse the coupon stream.
The inverse price-rate relationship follows directly from the present-value logic. Fix the cash flows — they are contractual and do not change. Now increase the discount rate r. Every term in the sum gets smaller (dividing by a larger number), so the total price falls. This is not a market phenomenon or a coincidence; it is a mathematical necessity. The relationship is also nonlinear: prices fall faster when rates rise starting from low levels than from high levels, a property formalized as *convexity* in more advanced fixed-income analysis.
Whether a bond trades at par, a premium, or a discount tells you immediately how the coupon rate compares to current market rates. When coupon rate = market rate, price = par. When coupon rate > market rate, price > par (premium — investors pay extra for above-market coupon income). When coupon rate < market rate, price < par (discount — buyers demand compensation for below-market income). This three-way classification is worth internalizing because it lets you reason about bond prices qualitatively without any calculation.
One distinction that trips up many students: the coupon rate is a fixed number stamped on the bond contract. The discount rate (or yield to maturity) is the market's current required return, and it changes daily. When you price a bond, you use the market rate as r — not the coupon rate. The coupon rate only determines the size of the coupon cash flows. Mixing these up leads to pricing a bond at par no matter what the market does, which misses the entire point of bond valuation.