Derive present value instead of memorizing that bond prices fall when yields rise.
By the end: Price a fixed future payment and distinguish market losses from default.
If 100 invested for one year earns 5%, it becomes 105. With annual compounding, the second year earns interest on 105, producing 110.25. To find what a promised future amount is worth today, reverse that multiplication. Discounting translates future cash flows into an equivalent current amount at a stated required return.
Our bond pays 100 in two years and no interim coupons. A coupon is a periodic interest payment, so this is a zero-coupon bond. At a required annual yield of 5%, its price is about 90.70. Buying for 90.70 and receiving 100 two years later earns 5% annually if the promise is honored. The yield is not a cash payment received each year.
Begin with the direction that requires only multiplication. A one-year investment of 80 at 5% earns 4 because five hundredths of 80 is 4. It ends with 84. If the full 84 remains invested for a second year at 5%, the second year's interest is 4.20. The ending value is 88.20. Compounding means that the second interest calculation includes the first year's interest. It does not mean that the percentage itself rises. The annual rate stays 5%; the amount to which it applies grows.
Discounting asks the reverse question. How much today grows into a payment of 100 after two years at 5% per year? Work backward from the payment date. Divide 100 by 1.05 to find the equivalent amount one year before maturity, about 95.2381. Divide that result by 1.05 again to get about 90.7029 today. Keep extra digits during the calculation and round the final currency amount to 90.70. If you round every intermediate step too early, the final pennies can differ even though the economic method is the same.
The multiplication check makes the dates explicit. Starting with about 90.7029 today, one year of 5% growth produces about 95.2381. A second year produces 100. The bond promises only the final 100, however. The intermediate 95.2381 is a comparison value along the assumed constant-yield path, not a coupon paid to the owner. A zero-coupon holder has no interest cash receipt available to spend or reinvest before maturity. Its return comes from paying less today than the amount ultimately received.
Face value names the amount specified for repayment at maturity in this example. Price names the amount exchanged for the claim today. Yield names the annualized return implied by price and promised payments. They can have different numerical values without anything being missing. Paying 90.70 for a face payment of 100 does not mean the issuer immediately owes an extra cash interest payment of 5. The purchase price already accounts for the timing. Payment certainty is an assumption here, so this calculation does not price the chance that the promise will be broken.
Ownership also has accounting counterparts. In an additional paper example, one household buys this existing bond from another for 90.70 using deposits at the same bank. The buyer loses a deposit asset of 90.70 and gains a bond at that transaction value. The seller loses the bond and gains a deposit of 90.70. The bank moves deposit balances between customers without changing total deposits. The bond issuer receives nothing from this resale and its face obligation remains 100. A purchase from an issuer at original issuance would involve a different counterparty and a financing transaction.
If comparable new investments offer 10%, the old promise to pay 100 in two years must cost less to compete. Its price becomes about 82.64. The promised payment did not change. The discount rate did. Conversely, a lower required yield raises the price. Longer-dated fixed payments generally respond more to a given yield change because the discount factor applies for more years.
A coupon bond is a sequence of payments, each discounted from its own date. Its coupon rate, current yield, and yield to maturity are different measures. Current yield divides the annual coupon by today's price and ignores the gain or loss when principal is repaid. Yield to maturity accounts for all promised payments under its assumptions. The zero-coupon case avoids that extra complexity while preserving the price mechanism.
A clean repricing comparison holds the valuation date fixed. Imagine two possible market conditions at the same instant, with the same bond still exactly two years from payment. At 2%, divide 100 by 1.02 to get about 98.0392 one year backward, then divide again to get about 96.1169 today. At 8%, divide 100 by 1.08 to get about 92.5926, then divide again to get about 85.7339. The higher required return corresponds to the lower current price. Nothing about these alternative prices requires the issuer to change the promised 100.
The price difference is about 10.38 currency units. Relative to the 2% price, the decline is about 10.80%, found by dividing 10.3830 by 96.1169 and multiplying by 100. The yield increase is six percentage points, not a six percent price loss. Price and yield have different units and a nonlinear relationship. Doubling a required yield does not halve the price. Each year's discount factor changes, and those factors multiply. This is why a precise price calculation is more informative than applying a remembered percentage rule.
Passage of time gives a different comparison. Suppose a buyer pays about 90.7029 when two years remain and the required yield stays at 5%. One year later the bond is only one year from payment, so its price is about 95.2381. It has increased by about 4.5352, which is 5% of the original unrounded price. The buyer could sell then and realize that one-year return under our assumptions. This increase comes from moving nearer to the payment date while yield stays constant. It should not be confused with a yield-driven gain on a fixed date.
Now change both conditions explicitly. The buyer still starts at about 90.7029, but after one year the remaining one-year yield is 8%. The bond's price then is about 92.5926. The gain over the purchase price is about 1.8897, or 2.0833%. The price is lower than it would have been with an unchanged 5% yield, yet higher than the original purchase price because a year has passed. A statement that yields rose is therefore insufficient to calculate an investor's holding-period return. We also need purchase price, sale date, remaining maturity, and any payments received.
Maturity sensitivity can be explored on paper by comparing equal face payments at different dates. At 5%, 100 due in one year costs about 95.24, while 100 due in two years costs about 90.70. At 6%, the respective prices are about 94.34 and 89.00. The one-year price falls about 0.94% and the two-year price about 1.88%. Both fall, but the longer wait exposes the current value to another year's changed discount factor. This comparison isolates timing by holding payment size, default assumptions, and the valuation date fixed.
F is the payment at maturity, y the annual yield as a decimal, and n the years until payment. No coupons or default are assumed.
| Same two-year promise | Yield per year | Price today |
|---|---|---|
| Receive 100 at maturity | 0% | 100.00 |
| Receive 100 at maturity | 5% | 90.70 |
| Receive 100 at maturity | 10% | 82.64 |
The model above always prices a payment of 100 exactly two years away. No discount selects 0%, Moderate yield selects 5%, and Higher yield selects 10%. The slider selects whole percentage rates between 0% and 10%. These choices replace the required yield immediately. They do not advance a clock, accumulate interest, change the face payment, or simulate a trading strategy. The two bars share a currency scale, but their labels identify different dates. Equal lengths at 0% mean the discount factor is one, not that the future has arrived.
Use the 3% slider value as a calculation you can check without relying on a preset. First calculate one plus three hundredths, giving 1.03. Multiply 1.03 by itself to obtain the two-year growth factor of 1.0609. Divide 100 by 1.0609 to obtain about 94.26. The displayed price should agree after rounding. Move to 6% and repeat. The two-year growth factor is 1.1236, so the price is about 89.00. The difference is about 5.26 units. These are alternative valuations today, not a sequence of realized gains or losses.
There is no maturity slider or coupon input. To explore coupons, use this additional paper bond instead. It promises 5 after one year and 105 after two years. At a 5% required annual yield, the first payment is worth about 4.7619 today, found by dividing 5 by 1.05. The second payment is worth about 95.2381 today, found by dividing 105 by 1.1025. Add the present values to get 100. Each cash flow must be translated from its own date before addition. Adding 5 to a discounted final payment would mix dates.
For the same coupon bond at a 10% required yield, the first payment is worth about 4.5455 and the second about 86.7769. Together they are worth about 91.3223 today. The coupon rate is still 5%, because the annual coupon of 5 is five hundredths of face value 100. Current yield is about 5.48%, found by dividing the annual coupon of 5 by the current price of 91.3223. It is not the required yield of 10%. It excludes the additional gain between purchase price and principal repayment, as well as the dates of those payments.
Yield to maturity is the rate that makes all promised discounted payments add up to today's price under the stated convention. It provides a consistent pricing comparison. For a coupon bond, achieving that same compound return on total wealth through maturity also requires an assumption about reinvesting coupons. The zero-coupon model has no intervening coupons, so no coupon reinvestment choice arises. Neither yield measure guarantees the result if the issuer defaults, the investor pays fees, or the bond is sold before maturity at an unknown price. Those would require assumptions beyond the displayed calculation.
A yield curve compares yields across maturities for comparable instruments. Long yields reflect expected future short rates and a term premium, compensation for bearing longer-term interest-rate exposure and related risks. Interest-rate duration describes a bond price's sensitivity to yield changes. They need not move one-for-one with today's policy rate. Credit spreads add compensation for default risk and other differences. A higher yield is not automatically a better risk-adjusted deal.
A holder can suffer a market-value loss without the issuer missing a payment. Holding to maturity may avoid selling at the lower price, but it does not eliminate the opportunity cost, inflation risk, or a funding need before maturity. A bank promising withdrawals today cannot assume it will never have to sell long-lived assets.
Consider a separate bank holding bonds carried at 90, reserve assets of 20, deposit liabilities of 100, and equity of 10. Assets are 110, matching liabilities plus equity. Suppose the bonds' sale value falls to 84 after market yields rise. If that decline is recognized in these simplified accounts, assets become 104 and equity becomes 4. Deposits remain 100. There has been no customer withdrawal and no issuer default. A change in the value of an asset can reduce the residual belonging to shareholders without changing the amount owed to depositors.
If the bank sells the bonds at 84 to a nonbank customer of another bank, the selling bank removes its bond asset of 84 and receives reserves of 84 through settlement. The buyer gains bonds and loses deposits of 84. The buyer's bank loses reserves and deposit liabilities of 84. The sale exchanges an asset already valued at 84 for another asset of 84, so it causes no second loss under these assumptions. If the bank had instead kept the bonds recorded at 90 until sale, the sale would be the moment it recognized the 6 loss. Do not charge that same decline twice.
Liquidity and capital now answer different questions. The sale has made reserves available for payments, but it has not restored equity from 4 to 10. Keeping the bond until its promised payment date might avoid an immediate sale, but the bank must still fund any withdrawals arriving earlier. A creditor able to wait and an institution owing payment today do not face the same cash-flow problem. Accounting categories can affect the timing of reported losses; they do not give an asset a guaranteed sale price or eliminate its financing needs.
Yield curves introduce an additional comparison across dates of payment. A two-year yield and a ten-year yield concern different commitments. Expected future short rates can affect the longer yield, while a term premium compensates for bearing longer exposure and associated uncertainty. That premium can change independently of today's overnight policy rate. A credit spread is a yield difference that may reflect default risk, liquidity, and other contract differences. Calling it pure default compensation requires more evidence than observing two rates. The bond calculator holds these complications outside its single required-yield input.
When reading a reported bond loss, reconstruct the dates before interpreting the number. Establish when the asset was bought, what was paid, whether coupons have arrived, and when the current value was measured. Then distinguish the contractual payment from the sale price and purchasing power. Receiving the promised 100 in two years settles a nominal promise. It does not ensure that 100 buys the original quantity of goods. A complete account of an investor's outcome therefore needs the transaction history and the relevant price changes, while the model here isolates the narrower arithmetic of a fixed future payment.
A default-free bond pays 105 in exactly one year. Its market price is 100. What is its annual yield? What price gives a 10% yield?
At maturity, today's price multiplied by one plus the yield equals 105.
| Required yield | Today's price | One-year payment |
|---|---|---|
| 5% | 100.00 | 105 |
| 10% | 95.45 | 105 |
At price 100 the yield is 5%. At a 10% required yield, divide 105 by 1.10 to get about 95.45. The same promised 105 becomes less valuable today when competing returns rise.
A bank must sell the two-year zero-coupon bond immediately after its required yield rises from 5% to 10%. What loss does it realize per 100 of promised payment?
Compare the two prices, not the face value with the new price.
| Purchase value | Sale value | Loss |
|---|---|---|
| 90.70 | 82.64 | 8.06 |
The price falls from about 90.70 to 82.64, a loss of about 8.06. This is an interest-rate valuation loss, not a missed payment. Selling under pressure converts the market-value change into a realized loss.
These references support the mechanisms and identify institutional context. They are optional; the explanations and worked answers above stand alone. Numerical examples are teaching cases, not live financial data or personal financial advice.
Asset purchases and proposed transmission channels. Our zero-coupon present-value calculation distinguishes yield to maturity from coupon-based current yield.
Rates, expectations, asset prices, credit, and uncertain transmission lags. A channel description is not a quantitative causal estimate.