Bond Pricing Calculator

A bond's fair price is the present value of every future cash flow you receive: each coupon payment plus the face value at maturity, discounted back to today at the yield to maturity. The formula is Price = sum of C divided by (1 + r) to the power of t for every period, plus F divided by (1 + r) to the power of n, where C is the coupon per period, r is the per-period yield, F is face value, and n is the total number of periods. A 10-year, $1,000 face bond with a 5% coupon paid semiannually at a 6% yield is worth about $925.61, because each $25 coupon and the $1,000 redemption are discounted at 3% per six-month period over 20 periods.

Premium when the coupon rate sits above the market yield, discount when the coupon rate sits below it, and at par when they match. The intuition is simple: if a bond pays a 6% coupon while comparable bonds yield 4%, investors will bid the price above face value to capture the extra income, so it becomes a premium bond. The reverse holds for a 4% coupon in a 6% world; nobody pays par for below-market income, so the price falls to a discount that boosts the total return back up to the prevailing yield.

Because the coupon and face value of an existing bond are fixed at issue. The only thing that can change to reflect a new market yield is the price. When yields rise, the same fixed cash flows get discounted at a steeper rate, which shrinks their present value, so the price falls. When yields drop, those same cash flows are discounted more lightly and the price rises. This inverse relationship is mechanical: it falls out of the discount factor 1 divided by (1 + r) to the power of t, where a higher r in the denominator always produces a smaller present value.

Modified duration is the bond's percentage price sensitivity to a 1 percentage point change in yield. A bond with a modified duration of 7 falls about 7% in price if yields rise by 1 percentage point, and rises about 7% if yields fall by the same amount. Two patterns drive duration: longer maturity and lower coupon both lengthen duration, which means a 30-year zero-coupon bond is far more rate-sensitive than a 2-year coupon bond. That is why long bonds get hit hardest when the Fed hikes, and why pension funds and insurers match long-dated liabilities with long-duration bonds.

Current yield is the annual coupon divided by today's price. It is a quick snapshot of cash income relative to what you pay, but it ignores any capital gain or loss at maturity. Yield to maturity (YTM) is the full internal rate of return on the bond if you hold it to maturity, blending coupon income with the gain or loss from buying below or above par. A bond bought at $925 paying $50 a year has a current yield of 5.4%, but its YTM is higher because you also get a $75 capital gain when it redeems at $1,000. Current yield is useful for cash flow planning; YTM is the right number for comparing two bonds on total return.