Two Values After a Yield Change
A bond can have two different values at the same date. Its carrying book value follows the schedule established when it was purchased, using the original yield. Its market value reflects the yield currently required by the market for the bond’s remaining cash flows. If market yields change after purchase, the market value changes, but the original-yield book-value schedule does not change merely because of that market movement.
This distinction matters whenever a problem asks for a bond’s value after purchase. The date alone does not determine which value is wanted: the relevant yield and the meaning of “value” matter too. The calculations below use coupon dates so that each period has one consistent yield period.
For the standard calculations in this tutorial, assume the bond has fixed, known coupon and redemption payments, no default, no embedded options, and no transaction costs or taxes. Assume the holding is carried using the original effective yield, with no separate impairment or accounting adjustment. These assumptions isolate the effect of a market-yield change. In a real sale, the sale proceeds are market-based; a different accounting rule can also affect reported carrying amounts.
At coupon date \(k\), there are \(n-k\) coupon payments remaining. Using \(j\), the carrying value is
The market value at the same date, using the current market yield \(h\), is
The final term in each expression includes both the last coupon and redemption: when \(r=n-k\), the cash flow is \(C+R\). If \(h=j\), the formulas give the same value. If \(h\ne j\), the values generally differ.
Why the Carrying Schedule Stays Intact
The coupon-date recurrence makes the distinction especially clear. Starting with the original carrying value \(B_0\), interest for one period is \(jB_{k-1}\). The coupon \(C\) is then paid, leaving the next carrying value:
This recurrence uses the original yield \(j\), not the current market yield. It records the effective interest and coupon payment associated with the original purchase basis. A later market quote does not alter the purchase price or the yield that generated this schedule.
By contrast, market value is a new present-value calculation at the yield investors currently require. For fixed positive future payments, a lower market yield raises their present value and a higher market yield lowers it. That change affects \(M_k\), not \(B_k\) under the stated carrying convention.
Worked Calculations
Worked Example: A Market Yield Falls After the First Coupon
A three-year bond has annual coupons of $60 and redemption value $1,000. It was purchased to yield 8% effective annually. Find its original purchase price and its carrying book value immediately after the first coupon. At that date, the market yield for the remaining payments falls to 5%. Find the market value then, and compare the two date-one values.
Method. First calculate the purchase price at the original yield, then apply the original-yield recurrence. Reprice the remaining cash flows separately at the new market yield. A common mistake is to substitute 5% into the book-value recurrence.
The purchase price is the present value of the three coupons and redemption payment at 8%:
Immediately after the first coupon, the carrying value is
There are two remaining payments from date one: $60 in one year and $1,060 in two years. At the original yield, their present value confirms the carrying value:
At the new market yield of 5%, the market value is instead
Thus the carrying book value is $964.334706 and the market value is $1,018.594104, or $964.33 and $1,018.59 to the nearest cent. The market value is higher because the market yield fell below the original yield. The calculation checks: discounting the remaining cash flows at 8% gives \(B_1\), while discounting those same cash flows at 5% gives \(M_1\).
Worked Example: A Market Yield Rises After a Premium Purchase
A two-year bond pays annual coupons of $80 and redeems for $1,000. Its original annual effective yield is 5%. Immediately after the first coupon, the market yield rises to 10%. Find the purchase price, the date-one carrying book value, and the date-one market value.
Method. Use 5% for the original price and carrying schedule, then use 10% only to reprice the one remaining payment. The trap is treating the market-yield increase as a change in the bond’s historical carrying schedule.
The original purchase price is
After one period, the original-yield recurrence gives
This is also the present value at 5% of the only remaining payment, $1,080:
At the new 10% market yield, the remaining payment has market value
To the nearest cent, the carrying value is $1,028.57 and the market value is $981.82. The market value is lower because the required yield rose. The book-value check is that the purchase price was above redemption value, so at the original 5% yield the premium is being amortized; the date-one carrying value is lower than the initial price.
Worked Example: A Yield Change at a Later Coupon Date
A three-year bond pays an annual coupon of $40 and redeems for $1,000. Its original yield is 5% effective annually. Immediately after the second coupon, the market yield rises to 8%. Calculate the carrying book value and market value at that date.
Method. Carry the original schedule through both coupon dates, then value the sole remaining payment at the new market yield. This separates the timing of the market quote from the yield used to carry the investment.
At date one, the carrying value is the original-yield present value of the date-two coupon and the date-three coupon plus redemption:
At date two, after the second $40 coupon has been paid, the recurrence gives
The remaining $1,040 payment discounted at the original 5% yield confirms this result:
At an 8% market yield, that same payment has market value
To the nearest cent, \(B_2=\)$990.48 and \(M_2=\)$962.96. The values differ because they discount the remaining cash flow at different yields. In particular, the new market yield does not replace 5% in the recurrence that produced \(B_2\).
Interpreting the Difference
At a coupon date, the formulas themselves provide a reliable diagnostic. If the problem asks for book value under the original-yield carrying convention, use \(j\) throughout the carrying schedule. If it asks for the current market value, discount the remaining promised cash flows at \(h\). Do not average the yields or apply the new yield only to part of the remaining term.
| Quantity | Yield used | What changes after a market-yield shift? |
|---|---|---|
| Carrying book value \(B_k\) | Original yield \(j\) | Nothing changes from the market-yield shift alone |
| Market value \(M_k\) | Current market yield \(h\) | It changes as the required yield changes |
When \(h=j\), the two values coincide because they discount the same remaining cash flows at the same rate. For positive promised payments, if \(h<j\), then each market discount factor is larger than its original-yield counterpart, so \(M_k>B_k\). If \(h>j\), each market discount factor is smaller, so \(M_k<B_k\). These comparisons concern market value versus book value at the same date and assume the stated fixed cash flows.
If the bond is sold, the transaction price is generally related to market value, not the old carrying amount. The difference between proceeds and carrying amount may be relevant to a realized gain or loss, but it does not mean the past book-value schedule should be rewritten. The distinction is useful in bond analysis because it prevents a market-price movement from being confused with the amortization of a premium or accretion of a discount.
Common Mistakes
- Resetting the book value to market value. A changed yield changes the market price calculation; under the stated original-yield carrying convention, it does not reset the carrying schedule.
- Using the original yield for market value. That returns the carrying value, not the value at the new market yield.
- Discounting coupons already paid. At a coupon date immediately after payment, exclude that coupon and value only payments still due.
- Mixing annual and coupon-period yields. A yield used in each recurrence or discount period must match the coupon-period timing. If coupons are not annual, use the corresponding yield per coupon period.
- Ignoring the timing convention. “Immediately before” and “immediately after” a coupon can refer to different cash flows and values. The examples explicitly use values immediately after the coupon is paid.
Check Your Understanding
For each problem, distinguish the original-yield carrying value from the value at the stated current market yield. Unless stated otherwise, values are immediately after the coupon at an annual coupon date, and yields are effective annual rates.
- A two-year bond has annual coupons of $70, redemption value $1,000, and an original yield of 6%. Immediately after its first coupon, the market yield is 4%. Find the carrying value and market value at that date. (Answer: Carrying value $1,009.43; market value $1,028.85.)
- A two-year bond pays an annual coupon of $60 and redeems for $1,000. Its original yield is 4%; after the first coupon, the market yield is 5%. Find both date-one values and state which is larger. (Answer: Carrying value $1,019.23; market value $1,009.52; carrying value is larger.)
- A three-year bond has annual coupons of $50, redemption value $1,000, and an original yield of 8%. Immediately after the first coupon, the market yield falls to 6%. Calculate the date-one carrying value and market value. (Answer: Carrying value $946.50; market value $981.67.)
- A four-year bond pays annual coupons of $30 and redeems for $1,000. Its original yield is 5%. Immediately after the second coupon, the market yield is 4%. Calculate both date-two values and the difference between them. (Answer: Carrying value $962.81; market value $981.14; market value exceeds carrying value by $18.33.)