Exam FM · Bond Yield Measures · Tutorial 430 of 998
How Maturity Affects Yield Under Different Bond Terms
Use the shorter-maturity yield to determine whether extending a bond’s term raises, lowers, or leaves its yield unchanged.
Intermediate
11 min read
What You'll Learn
Set up the yield equation for bonds with level coupons and a single redemption payment.
Compare yields for adjacent maturities while holding price, coupon, and redemption fixed.
Derive the maturity test from the change in value at the shorter bond’s yield.
Interpret the test using the coupon-to-redemption ratio when redemption is positive.
Recognize the special case of zero redemption without dividing by redemption.
Check maturity comparisons with fully calculated bond cash flows.
Why Maturity Can Change Yield in Either Direction
Suppose two bonds have the same price and coupon but different maturities. It is tempting to assume that the longer bond must always have a higher yield, because its cash flows extend further into the future. That conclusion is not generally valid. The direction depends on how the coupon and redemption payments compare with the yield on the shorter bond.
The comparison below holds price, coupon amount, and redemption value fixed, and changes only the number of coupon periods. It concerns the yield implied by the promised cash flows under the stated model; it does not guarantee an investor’s realized return if payments are missed or coupons are reinvested at a different rate.
Definition: Consider a bond with price \(P>0\), level coupon \(C\geq 0\) paid at the end of each period, redemption value \(R\geq 0\) paid with the final coupon at the end of period \(n\), and yield \(i>-1\) per coupon period. Its yield equation is
$$
P=C a_{\overline{n}|}+Rv^n
=C\sum_{k=1}^{n}v^k+Rv^n,\qquad v=\frac{1}{1+i}.
$$
Assume at least one promised payment is positive. If coupons are annual, \(i\) is an annual effective yield; otherwise it is the yield per coupon period. A yield quoted on a different basis must be converted before comparing it with a per-period coupon rate.
For fixed positive cash flows, the present value decreases strictly as \(i\) increases: each discount factor \(v^k=(1+i)^{-k}\) decreases. Thus, for a fixed price and a specified set of these cash flows, there is at most one yield with \(i>-1\). The maturity comparison uses this fact: calculate what the longer bond would be worth at the shorter bond’s yield, then see whether that price is above or below the fixed price.
The Adjacent-Maturity Test
Let the shorter bond have \(n\) coupon periods and yield \(i_n\), so that
$$
P=C\sum_{k=1}^{n}v^k+Rv^n,\qquad v=\frac{1}{1+i_n}.
$$
Now add one more coupon period, with the same \(C\), the same \(R\), and the same price \(P\). At the old yield \(i_n\), the added coupon has present value \(Cv^{n+1}\). The redemption payment is also delayed by one period, changing its present value by \(Rv^{n+1}-Rv^n\). Therefore the longer bond’s value at the shorter bond’s yield differs from \(P\) by
$$
Cv^{n+1}+R(v^{n+1}-v^n)
=v^{n+1}\bigl(C-Ri_n\bigr).
$$
The factor \(v^{n+1}\) is positive. The sign of the value change is therefore the sign of \(C-Ri_n\). If the longer bond is worth more than \(P\) at the shorter yield, its yield must be higher to bring its value back down to \(P\). If it is worth less, its yield must be lower. If the value is unchanged, the same yield prices both bonds.
Theorem: With \(P\), \(C\), and \(R\) fixed, compare maturities \(n\) and \(n+1\), and let \(i_n\) be the yield for maturity \(n\). Under the bond cash-flow model above:
If \(C-Ri_n>0\), the yield for maturity \(n+1\) is greater than \(i_n\).
If \(C-Ri_n<0\), the yield for maturity \(n+1\) is less than \(i_n\).
If \(C-Ri_n=0\), the yield is unchanged.
When \(R>0\), define the coupon rate per period as \(C/R\). The test can then be read as a comparison with that rate: extending maturity raises yield if \(i_n<C/R\), lowers yield if \(i_n>C/R\), and leaves yield unchanged if \(i_n=C/R\). This is a comparison with the shorter bond’s yield, not a rule that longer maturities always move yields in one direction.
Worked Comparisons
Worked Example: Extending a Bond Raises Its Yield
A hypothetical bond has annual coupons of \(8\), redemption value \(100\), and price \(P\). Compare a two-year version with a one-year version, holding those three amounts fixed. Suppose the two-year bond’s yield is \(5\%\) per year. Find \(P\), calculate the one-year yield, and determine the direction of the maturity effect.
Method. First value the two-year cash flows at the stated yield. Then use the one-year yield equation and apply the adjacent-maturity test. This avoids assuming that a longer term automatically raises or lowers yield.
At \(i=0.05\), the discount factor is \(v=1/1.05=20/21\). The two-year annuity factor is
$$
a_{\overline{2}|}=v+v^2=\frac{20}{21}+\frac{400}{441}
=\frac{420+400}{441}=\frac{820}{441}.
$$
Thus the price is
$$
P=8\left(\frac{820}{441}\right)+100\left(\frac{400}{441}\right)
=\frac{6{,}560+40{,}000}{441}
=\frac{46{,}560}{441}
=\frac{15{,}520}{147}
\approx 105.5782.
$$
For the one-year bond, the coupon and redemption are both paid at the end of the year, so
$$
P=\frac{8+100}{1+i_1}=\frac{108}{1+i_1}.
$$
Therefore
$$
1+i_1=\frac{108}{15{,}520/147}
=\frac{15{,}876}{15{,}520}
=\frac{3{,}969}{3{,}880},
\qquad
i_1=\frac{89}{3{,}880}\approx 0.0229381.
$$
The one-year yield is approximately \(2.2938\%\). Since \(C-Ri_1=8-100(0.0229381)\approx 5.7062>0\), extending maturity raises the yield. The two-year yield, \(5\%\), is indeed higher than the one-year yield.
Check. Substituting the exact one-year yield gives \(108/(1+i_1)=15{,}520/147\), the price calculated from the two-year cash flows. The comparison uses the same price and contractual coupon and redemption amounts for both terms.
Worked Example: Extending a Bond Lowers Its Yield
A hypothetical bond pays an annual coupon of \(8\), redeems at \(100\), and has a two-year yield of \(10\%\). The price is fixed at the value implied by that two-year yield. Find the one-year yield and compare it with the two-year yield.
Method. Calculate the common price from the two-year cash flows, solve the one-year equation, and use the sign test. The relevant comparison is between the shorter yield and \(C/R=8\%\).
At \(10\%\), \(v=10/11\), so
$$
a_{\overline{2}|}=\frac{10}{11}+\frac{100}{121}
=\frac{110+100}{121}=\frac{210}{121}.
$$
The price is
$$
P=8\left(\frac{210}{121}\right)+100\left(\frac{100}{121}\right)
=\frac{1{,}680+10{,}000}{121}
=\frac{11{,}680}{121}
\approx 96.5289.
$$
For the one-year bond,
$$
1+i_1=\frac{108}{P}
=\frac{108(121)}{11{,}680}
=\frac{13{,}068}{11{,}680},
\qquad
i_1=\frac{1{,}388}{11{,}680}
=\frac{347}{2{,}920}
\approx 0.1188356.
$$
The one-year yield is approximately \(11.8836\%\), which exceeds the two-year yield of \(10\%\). The test confirms the direction:
$$
C-Ri_1=8-100(0.1188356)\approx -3.8836<0.
$$
Hence extending the term lowers the yield.
Check. The one-year cash flow is \(108\), and \(108/(1+347/2{,}920)=11{,}680/121\), the same price as for the two-year bond. The longer yield is lower, as the negative sign predicts.
Worked Example: A Bond at the Coupon Rate Has No Maturity Effect
A hypothetical bond has annual coupon \(C=8\), redemption value \(R=100\), and price \(P=100\). Show that its yield is \(8\%\) for any positive integer maturity \(n\).
Method. Substitute \(i=C/R=0.08\) into the yield equation and simplify the annuity and redemption terms. This proves the result for every maturity rather than checking only two terms.
At \(i=0.08\), \(v=1/1.08\), and \(i=0.08\) satisfies \(C=Ri\). The standard finite-annuity identity is \(i a_{\overline{n}|}=1-v^n\), so
$$
8a_{\overline{n}|}+100v^n
=100(0.08)a_{\overline{n}|}+100v^n
=100(1-v^n)+100v^n
=100.
$$
Thus the price is \(100\) at an \(8\%\) yield for every \(n\). Since the cash flows are positive and price decreases strictly with yield, this is the unique yield for each maturity. Equivalently, \(C-Ri=8-100(0.08)=0\), so adding a period leaves the yield unchanged.
Check. The coupon and redemption amounts, the price, and the yield equation all use annual periods. The result is not a claim that bonds of different prices or coupon amounts must have the same yield.
Worked Example: Zero Redemption Requires the Direct Sign Test
A hypothetical instrument pays a coupon of \(25\) at the end of each year and has redemption value \(R=0\). Its price is \(45\). Compare its yield for one coupon period and two coupon periods.
Method. Solve each yield equation, then compare the results. Because \(R=0\), the ratio \(C/R\) is undefined; use \(C-Ri_n\) directly instead.
For one year, \(45=25/(1+i_1)\), giving
$$
i_1=\frac{25}{45}-1=-\frac{4}{9}\approx -0.444444,
$$
or approximately \(-44.4444\%\). For two years, the equation is
$$
45=25(v+v^2),\qquad v=\frac{1}{1+i_2}.
$$
Dividing by \(25\) gives \(v+v^2=9/5\), or \(5v^2+5v-9=0\). The positive discount-factor root is
$$
v=\frac{-5+\sqrt{205}}{10}\approx 0.9317821.
$$
Equivalently, multiplying the price equation by \((1+i_2)^2\) gives
$$
9i_2^2+13i_2-1=0.
$$
The root in the required domain \(i_2>-1\) is
$$
i_2=\frac{-13+\sqrt{205}}{18}\approx 0.0732123,
$$
or approximately \(7.3212\%\). The other quadratic root is less than \(-1\), so it cannot represent a yield with a positive discount factor.
Check. The selected discount factor satisfies \(v+v^2=1.8\), so the two-year price is \(25(1.8)=45\). Also, \(C-Ri_1=25>0\), so the longer yield must be higher: \(7.3212\%>-44.4444\%\). The test works without dividing by \(R\).
Common Mistakes and Practical Use
The adjacent-maturity result matters because yield comparisons can be sensitive to which bond feature is held fixed. The theorem compares bonds at the same price with the same coupon amount and redemption value. If price also changes, or if coupon amount or redemption changes, the theorem alone does not determine which yield is higher.
Assuming longer maturity always raises yield. Example 2 shows the reverse: when the shorter yield is above \(C/R\), an added period lowers yield.
Using the coupon rate without matching the period. If coupons are quarterly, compare the quarterly yield with the coupon amount divided by redemption per quarter. Convert rates before comparing them.
Using \(C/R\) when \(R=0\). That ratio is undefined. The direct expression \(C-Ri_n\) remains valid and equals \(C\) in this case.
Ignoring the sign of a negative yield. Yields can be negative while remaining above \(-1\). The test still applies to that yield; do not replace the algebra with an intuition based only on positive rates.
Changing more than maturity. The test does not compare bonds if their prices, coupon amounts, or redemption values differ. Re-establish the cash-flow equation before drawing a conclusion.
A reliable approach is to identify the shorter bond’s yield first, compute \(C-Ri_n\), and then state the direction for the longer bond. If the sign is zero, verify that the same yield prices both maturity cash flows. This keeps the conclusion tied to the actual cash-flow change rather than to a broad intuition about term.
Check Your Understanding
For each comparison, hold price, coupon amount, and redemption value fixed unless the problem says otherwise. Use the shorter bond’s yield to determine the direction.
A bond has annual coupon \(12\), redemption \(100\), and price \(100\). Compare one-year and two-year maturities. What is the one-year yield, and does the yield change when maturity is extended? (Answer: \(12\%\); it is unchanged.)
A bond has annual coupon \(6\), redemption \(100\), and price \(104\). Compare one-year and two-year maturities. Find the one-year yield and state whether the longer yield is higher or lower. (Answer: One-year yield \(=106/104-1\approx1.9231\%\); the two-year yield is higher.)
A zero-coupon bond has redemption \(100\) and price \(81\). Compare one-year and two-year maturities. Find both yields and state the direction. (Answer: One-year yield \(\approx23.4568\%\); two-year yield \(=1/9\approx11.1111\%\); the longer yield is lower.)
A bond has annual coupon \(9\), redemption \(100\), and price \(108\). Compare one-year and two-year maturities. Calculate the one-year yield, then use \(C-Ri_1\) to determine the direction without solving the two-year yield. (Answer: One-year yield \(=1/108\approx0.9259\%\); \(C-Ri_1>0\), so the two-year yield is higher.)