Why Bond Prices Move Inversely With Yields

Why the Direction Matters

A bond’s promised payments may stay exactly the same while its market price changes. The link between price and required yield follows from present-value arithmetic: a higher yield discounts each future payment more heavily. Understanding this direction is useful when interpreting quoted prices and comparing a bond’s value under different yield assumptions.

The result is about a fixed set of future cash flows. It does not say that every observed bond price change must have been caused only by a change in yield. For example, the bond’s promised payments or perceived ability to make them could change as well.

Definition: For a bond with fixed cash flows $C_t$ payable at the ends of periods $t=1,\ldots,n$, its price $P(y)$ at a yield $y$ per period is the present value of those cash flows: $$ P(y)=\sum_{t=1}^{n}\frac{C_t}{(1+y)^t} =\sum_{t=1}^{n}C_t v_y^t, \qquad v_y=\frac{1}{1+y}. $$ Here $C_t$ is the cash payment at period $t$, and $y>-1$. For an ordinary bond, the cash flows are nonnegative, with at least one positive payment.

The yield and the payment period must use a consistent basis. With annual payments and an effective annual yield, $y$ is the effective rate per year. With semiannual payments and a nominal annual yield convertible semiannually, use the nominal rate divided by two as the yield per half-year. In either case, the exponent counts the number of matching payment periods.

The Present-Value Reason

When $y$ rises, $1+y$ rises, so its reciprocal $v_y=1/(1+y)$ falls. Every future payment is multiplied by a power of this smaller discount factor. Since the powers and the cash flows are nonnegative, no future payment’s present value increases.

For a bond with positive payments at one or more future dates, at least one present-value term decreases strictly. Thus the total price decreases strictly. The following derivative gives a compact proof of the same result.

Result: If the cash flows $C_t$ are fixed and nonnegative, and at least one $C_t$ at a future date is positive, then $P(y)$ is strictly decreasing for $y>-1$.

Differentiate the finite sum term by term. Each term’s derivative is negative when its cash flow is positive and its payment date is in the future.

\frac{dP}{dy} = \sum_{t=1}^{n} -tC_t(1+y)^{-t-1}.

For $y>-1$, $(1+y)^{-t-1}$ is positive. Also, $t>0$ and $C_t\geq 0$. Therefore every term in the derivative is nonpositive, and any term for which $C_t>0$ is negative. Because at least one future payment is positive, the sum is strictly negative:

\frac{dP}{dy}<0.

This proof establishes the direction of the relationship; it does not require the bond to trade at face value. A bond can be priced above, at, or below face value at a particular yield. The result says that, with its cash flows held fixed, a higher yield gives a lower present value than a lower yield.

Worked Examples

Worked Example: A Zero-Coupon Bond at Two Yields

A hypothetical zero-coupon bond pays $1,000 exactly three years from today. Assume no default and use effective annual yields with annual periods. Compare its price at a 4% yield and a 5% yield.

Method. There is one cash flow, $C_3=\$1{,}000$, so its price is the payment discounted for three years. The discount-factor powers are $1.04^3=1.124864$ and $1.05^3=1.157625$.

At 4%:

P(0.04)=\frac{\$1{,}000}{1.04^3} =\frac{\$1{,}000}{1.124864} \approx \$889.00.

At 5%:

P(0.05)=\frac{\$1{,}000}{1.05^3} =\frac{\$1{,}000}{1.157625} \approx \$863.84.

Check. The only payment is unchanged, while its discount factor falls from $1/1.124864$ to $1/1.157625$. The calculated price falls by about $\$25.16$, from $\$889.00$ to $\$863.84$, as the yield rises. Prices are rounded to the nearest cent only at the final step.

Common mistake avoided: Do not discount this single payment for one year just because the bond has one payment. Its payment date is three years away, so the exponent is three.

Worked Example: A Coupon Bond Priced Above and at Face Value

A hypothetical bond pays an annual coupon of $60 for three years and returns $1,000 of face value with its final coupon. Assume the promised payments are fixed and use effective annual yields. Find its price at 5% and 6%.

Method. The cash flows are $60 at the ends of years one and two, and $1,060 at the end of year three. At 5%, the discount factors are $1/1.05=0.952381$, $1/1.05^2=0.907029$, and $1/1.05^3=0.863838$, to six decimal places.

The 5% price is:

\begin{aligned} P(0.05) &=\frac{60}{1.05}+\frac{60}{1.05^2}+\frac{1{,}060}{1.05^3}\\ &=(60)(0.952381)+(60)(0.907029)+(1{,}060)(0.863838)\\ &\approx 57.1429+54.4218+915.6679\\ &\approx \$1{,}027.23. \end{aligned}

At 6%, the discount factors are $1/1.06=0.943396$, $1/1.06^2=0.889996$, and $1/1.06^3=0.839619$, to six decimal places:

\begin{aligned} P(0.06) &=\frac{60}{1.06}+\frac{60}{1.06^2}+\frac{1{,}060}{1.06^3}\\ &\approx 56.6038+53.3998+889.9964\\ &\approx \$1{,}000.00. \end{aligned}

Check. The price decreases from $\$1{,}027.23$ to $\$1{,}000.00$ as the yield increases. At 6%, the annual coupon of $60 equals 6% of the $1,000 face value, and the computed price is face value. That check is consistent with the present-value calculation.

Common mistake avoided: The final cash flow is $1,060, not $1,000: it includes both the year-three coupon and repayment of face value. Discounting only the face value at year three would omit a promised payment.

Worked Example: Matching a Semiannual Yield to the Payment Period

A hypothetical bond has $1,000 face value, pays a coupon of $40 every six months, and matures in three years. It therefore makes six coupon payments, with $1,040 paid at the sixth date. Assume the promised payments are fixed. Compare nominal annual yields of 10% and 12%, each convertible semiannually.

Method. Convert each quoted yield to its six-month rate. The rates per coupon period are $0.10/2=0.05$ and $0.12/2=0.06$. The discounting exponents run from one through six, because there are six half-year periods.

At a 5% rate per half-year, the six discount factors are approximately $0.952381,\ 0.907029,\ 0.863838,\ 0.822702,\ 0.783526,\ 0.746215$. The first five payments are $40 and the final payment is $1,040:

\begin{aligned} P(0.05) &=40(0.952381+0.907029+0.863838+0.822702+0.783526)\\ &\quad+1{,}040(0.746215)\\ &\approx \$949.24. \end{aligned}

Equivalently, the final term can be separated into a $40 coupon and $1,000 of face value. At a 6% rate per half-year, the discount factors are approximately $0.943396,\ 0.889996,\ 0.839619,\ 0.792094,\ 0.747258,\ 0.704961$:

\begin{aligned} P(0.06) &=40(0.943396+0.889996+0.839619+0.792094+0.747258)\\ &\quad+1{,}040(0.704961)\\ &\approx \$901.65. \end{aligned}

Check. The price falls from $\$949.24$ to $\$901.65$ when the quoted nominal annual yield rises from 10% to 12%. The calculation uses rates per half-year because the payments are semiannual. Prices are rounded to the nearest cent only at the final step.

Common mistake avoided: Using 10% or 12% directly as the rate for each half-year would not match the stated nominal annual yields. The semiannual rates are one-half of the quoted nominal annual rates.

What the Result Does—and Does Not—Say

The inverse relationship depends on holding the bond’s future cash flows fixed. It compares two present-value calculations for the same payments, at two different required yields. It does not require the market price to equal face value at either yield, nor does it guarantee that the price observed in a market moved for only one reason.

The result also relies on ordinary bond cash flows being nonnegative payments to the investor. In the derivative, a negative future cash flow would contribute a positive term. That is not the usual promised-payment pattern of a bond, and the simple decreasing-price conclusion need not hold for an arbitrary collection of positive and negative cash flows.

Keep three questions separate when applying the calculation:

Are the cash flows fixed? If a payment amount or date changes, the comparison is no longer holding the cash flows constant.
Does the rate match the period? Annual payments require a rate per year; semiannual payments require a rate per half-year.
Is the exponent the number of periods until payment? Each payment is discounted for its own time to receipt.

A frequent exam error is to state the inverse relationship as a slogan without showing what is being held constant. Another is to mix a quoted annual yield with a payment-period exponent. Writing the cash flows, period rate, and discount factors explicitly makes both errors easier to catch.

Key takeaway: For fixed, nonnegative bond cash flows paid in the future, price is the sum of cash flows multiplied by their discount factors. A higher yield lowers every discount factor, so the bond’s present value falls.

Check Your Understanding

For each problem, keep the stated cash flows fixed and match the yield period to the payment schedule.

A one-year bond pays a $40 coupon and $1,000 face value at maturity. Using effective annual yields, calculate its price at 4% and 8%. (Answer: $1,000.00 at 4%; $962.96 at 8%.)
A zero-coupon bond pays $1,200 in two years. Using effective annual yields, calculate its price at 5% and 10%. (Answer: $1,088.44 at 5%; $991.74 at 10%.)
A two-year bond pays annual coupons of $50 and returns $1,000 face value with the second coupon. Find its price at effective annual yields of 5% and 7%. (Answer: $1,000.00 at 5%; approximately $963.84 at 7%.)
A four-period bond pays $30 each half-year and returns $1,000 with its fourth payment. Compare nominal annual yields of 6% and 8%, both convertible semiannually. (Answer: $1,000.00 at 6%; approximately $963.70 at 8%.)
A bond has fixed positive payments at the ends of years one and two. Explain, using the derivative of its present-value expression, why its price decreases strictly for yields greater than $-1$. (Answer: $P'(y)=-C_1(1+y)^{-2}-2C_2(1+y)^{-3}<0$ when $C_1,C_2>0$ and $y>-1$.)

What You'll Learn