Measuring Bond Price Changes in Dollars and Percentages

Why Measure a Bond Price Change Two Ways?

A yield change affects the present value of a bond’s promised cash flows. Reporting the resulting price change in dollars answers how much value one bond gains or loses. Reporting it as a percentage of the starting price answers how large that change is relative to the bond’s initial value. These measures are related, but they are not interchangeable: the same percentage move can represent very different dollar amounts for bonds of different sizes.

This tutorial focuses on repricing cash flows at two yields and comparing the resulting prices. It uses a fixed-cash-flow valuation: the payment amounts and payment dates do not change between the two calculations. For clarity, examples use annual effective yields and value the bond immediately after a coupon date, so accrued interest is zero. If a bond uses another payment frequency, the yield and periods must instead be expressed on a consistent basis.

Definition: Let $P_0$ be a bond’s price at the initial yield and $P_1$ its price at the new yield, using the same cash flows and valuation date. The signed dollar price change is $\Delta P=P_1-P_0$. The signed percentage price change is $\frac{P_1-P_0}{P_0}\times100\%$. A negative result means the price fell; a positive result means it rose.

The percentage calculation uses the initial price as its denominator because the question is how much the price changed relative to where it started. A yield shift should also be identified precisely. For example, a yield increase from 5% to 6% is a rise of 1 percentage point, or 0.01 when yields are written as decimals. It is not a 1% relative increase in the original yield.

Reprice First, Then Compare

For annual cash flows, let $C_t$ be the promised payment at the end of year $t$, and let $y$ be the annual effective yield per year. Under the usual present-value model, the price at that yield is

P(y)=\sum_{t=1}^{n}\frac{C_t}{(1+y)^t}.

This model assumes the specified cash flows are fixed, the yield is appropriate for the cash-flow periods, and the bond is valued using those promised payments. It does not incorporate a change in default expectations, taxes, transaction costs, or cash flows. Recalculate the present value at the new yield; do not apply a percentage to face value unless the question explicitly defines a different measure.

List the cash flows and dates.
Keep the payment amounts and timing the same for both valuations.

Calculate each price.
Discount every cash flow at the initial yield to get $P_0$, and at the new yield to get $P_1$.

Calculate the signed dollar change.
Subtract the initial price: $\Delta P=P_1-P_0$.

Calculate the signed percentage change.
Divide the dollar change by $P_0$, then multiply by 100%.

Why the Comparison Formula Works

The dollar change is the difference between the two values assigned to the same promised cash flows. Substituting the present-value formula at each yield gives

\Delta P = P(y_1)-P(y_0) = \sum_{t=1}^{n} C_t \left[ \frac{1}{(1+y_1)^t} - \frac{1}{(1+y_0)^t} \right].

Thus each payment contributes its value at the new yield minus its value at the initial yield; adding those differences gives the total price change. If all $C_t$ are positive and both yields exceed $-1$, raising the yield makes every discount factor smaller, so every discounted payment falls and the total price falls. This sign check is useful, but it does not replace the actual repricing needed to find the size of the change.

The two measures also respond differently to bond scale. If every cash flow is multiplied by a positive factor $k$, then both prices and the dollar change are multiplied by $k$. The percentage change does not change:

\frac{kP_1-kP_0}{kP_0}\times100\% = \frac{P_1-P_0}{P_0}\times100\%.

So a larger face amount can produce a larger dollar gain or loss even when the relative price movement is identical. Dollar change is relevant to the value of a particular position; percentage change allows comparisons relative to starting value.

Worked Example: A Zero-Coupon Bond and a One-Point Yield Increase

A hypothetical zero-coupon bond pays $1,000 at the end of year 3 and has no intervening payments. Its annual effective yield rises from 5% to 6%. Find its initial price, revised price, dollar change, and percentage change. Report prices and the dollar change to the nearest cent and the percentage change to four decimal places.

Method. Discount the single payment at each yield. The cash flow and its date remain fixed; only the yield changes.

At the initial yield,

P_0=\frac{1{,}000}{(1.05)^3} =\frac{1{,}000}{1.157625} =863.837598\ldots.

At the new yield,

P_1=\frac{1{,}000}{(1.06)^3} =\frac{1{,}000}{1.191016} =839.619283\ldots.

The signed dollar change is

\Delta P=839.619283\ldots-863.837598\ldots =-24.218315\ldots.

The signed percentage change, relative to the initial price, is

\frac{-24.218315\ldots}{863.837598\ldots}\times100\% =-2.8035729\ldots\%.

Rounded as requested, the initial price is $863.84, the revised price is $839.62, the dollar change is −$24.22, and the percentage change is −2.8036%. The negative signs agree with the fact that the yield increased.

Check. Subtracting the displayed rounded prices gives $839.62 − $863.84 = −$24.22, consistent with the unrounded calculation. The percentage uses the unrounded initial price and dollar change before rounding.

Worked Example: A Coupon Bond Initially at Par

A hypothetical two-year bond has a face value of $1,000 and pays a $100 coupon at the end of each year. It pays the $1,000 principal together with the second coupon at the end of year 2. Its annual effective yield rises from 10% to 11%. Calculate both price changes. Report money to the nearest cent and the percentage to four decimal places.

Method. Value the year-1 coupon and the year-2 coupon-plus-principal separately at each yield. At 10%, the coupon rate equals the yield, so the initial price is at par; verify that by calculating the cash-flow present values.

The initial price is

P_0 = \frac{100}{1.10} + \frac{1{,}100}{(1.10)^2} = 90.9090909\ldots+909.0909091\ldots = 1{,}000.

At the new yield, the price is

P_1 = \frac{100}{1.11} + \frac{1{,}100}{(1.11)^2} = 90.0900901\ldots+892.7846760\ldots = 982.874766\ldots.

Therefore,

\Delta P=982.874766\ldots-1{,}000 =-17.125234\ldots, $$ $$ \frac{\Delta P}{P_0}\times100\% = \frac{-17.125234\ldots}{1{,}000}\times100\% =-1.7125233\ldots\%.

The new price is $982.87, the dollar change is −$17.13, and the percentage change is −1.7125%. The price decline is less than the face value because the comparison is with the initial price, not an arbitrary reference amount.

Check. The initial present values sum to $1,000. The revised price is lower, as expected after a yield increase, and its difference from $1,000 rounds to the stated dollar loss.

Worked Example: A Premium Bond and the Correct Percentage Base

A hypothetical two-year bond has a face value of $2,000 and pays a $80 coupon annually, with the final $80 coupon and $2,000 principal paid at the end of year 2. Its annual effective yield rises from 3% to 4%. Find the dollar and percentage price changes. Report money to the nearest cent and the percentage to four decimal places.

Method. Reprice both dated payments at both yields, then divide the dollar change by the initial market price. The 3% yield is below the 4% coupon rate, so a premium initial price is reasonable; the calculation confirms its size.

At 3%,

P_0 = \frac{80}{1.03} + \frac{2{,}080}{(1.03)^2} = 77.6699029\ldots+1{,}960.599491\ldots = 2{,}038.269394\ldots.

At 4%,

P_1 = \frac{80}{1.04} + \frac{2{,}080}{(1.04)^2} = 76.9230769\ldots+1{,}923.0769231\ldots = 2{,}000.

The signed dollar change is

\Delta P=2{,}000-2{,}038.269394\ldots =-38.269394\ldots.

The percentage change is

\frac{-38.269394\ldots}{2{,}038.269394\ldots}\times100\% =-1.877543\ldots\%.

The revised price is $2,000.00, the dollar change is −$38.27, and the percentage change is −1.8775%. The percentage denominator is $2,038.269394…, the initial price. Dividing by the $2,000 face value instead would give approximately −1.9135%, which is not the percentage change relative to the bond’s starting price.

Check. The initial price exceeds face value, consistent with the coupon rate exceeding the initial yield. When the yield rises to the coupon rate, the price is exactly par; the resulting price decline has the expected negative sign.

Common Mistakes and Interpretation

A frequent error is to calculate the new price correctly but choose the wrong denominator for the percentage change. In the premium-bond example, using face value gives

\frac{2{,}000-2{,}038.269394\ldots}{2{,}000}\times100\% \approx -1.9135\%.

That calculation measures the price change as a percentage of face value, not as a percentage of the initial price. The requested change relative to the initial price is approximately −1.8775%. Always identify the reference amount named by the question before dividing.

Subtracting in the wrong order. For a change from the initial to the new price, use $P_1-P_0$. Reversing the subtraction reverses the sign.
Using the new price as the denominator. This produces a change relative to the ending value, a different measure from the change relative to the initial price.
Applying the yield shift directly to the price. A one-percentage-point yield increase does not imply a one-percent price decrease. Reprice the promised cash flows.
Mixing payment periods and yield conventions. A yield must match the timing and convention used to discount the cash flows. An annual effective yield cannot be inserted as though it were a periodic rate for another payment frequency.
Rounding too soon. Carry unrounded prices and changes through the percentage calculation, then round the reported results.

The dollar and percentage measures answer complementary questions. A $38.27 decrease may be material for one holding and small for another, depending on the number of bonds held and their starting values. Conversely, equal percentage changes across differently sized holdings need not imply equal dollar gains or losses. A useful report therefore states the initial price, new price, signed dollar change, and signed percentage change whenever the comparison warrants it.

Key takeaway: Revalue the same cash flows at the new yield, calculate $\Delta P=P_1-P_0$, and calculate the percentage change as $\Delta P/P_0$. The dollar change reflects the holding’s scale; the percentage change expresses the move relative to its initial price.

Check Your Understanding

For each problem, calculate the signed price change and interpret its sign. Round money to the nearest cent and percentages to four decimal places unless stated otherwise.

A hypothetical zero-coupon bond pays $1,200 in one year. Its annual effective yield rises from 4% to 6%. Find both prices, the dollar change, and the percentage change. (Answer: initial price $1,153.85; new price $1,132.08; change −$21.77, or −1.8868%.)
A hypothetical one-year bond pays $80 in coupons and $1,000 principal at year-end. Its annual effective yield rises from 5% to 10%. Find the initial and new prices and both changes. (Answer: initial price $1,028.57; new price $981.82; change −$46.75, or −4.5455%.)
A hypothetical two-year bond has $2,500 face value and pays a $100 annual coupon, with principal paid with the second coupon. Yield rises from 4% to 5%. Calculate both prices, the signed dollar change, and the percentage change. (Answer: initial price $2,500.00; new price $2,453.51; change −$46.49, or −1.8594%.)
A hypothetical bond’s price changes from $7,420 to $7,605 after a yield decline. Calculate the dollar and percentage changes, using the initial price as the percentage base. (Answer: increase of $185.00, or 2.4933%.)
All promised cash flows of a hypothetical bond are doubled, while the yields and payment dates remain unchanged. If its initial price was $1,500 and its new price was $1,470, state the scaled initial and new prices, the dollar change, and the percentage change. (Answer: initial price $3,000.00; new price $2,940.00; change −$60.00, or −2.0000%, unchanged in percentage terms.)

What You'll Learn