Debt Maturity and Refinancing Risk

Why Debt Maturity Matters

A change in market interest rates does not necessarily change the government's interest bill at once. The timing depends on when existing debt matures and must be refinanced, and on whether any debt has a floating rate that resets before maturity. A government with long-dated fixed-rate bonds can be insulated from a rate increase for a time; one with near-term maturities or frequent rate resets may face higher costs sooner.

Maturity structure also affects refinancing risk: the possibility that a government must replace a large amount of maturing debt when borrowing conditions are unfavorable, or cannot refinance it on acceptable terms. A longer maturity can reduce the frequency and concentration of refinancing, but it may come with a higher borrowing rate. The balance depends on the maturity schedule and financing conditions; neither short nor long maturity is always cheapest or safest.

The analysis here focuses on the pass-through from market rates to cash interest costs and on rollover exposure. It does not treat a change in the market value of outstanding bonds as an immediate change in the government's coupon payments. The treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Maturity, Refinancing, and Interest-Cost Pass-Through

Definition: A debt instrument's maturity is the date its principal is due for repayment. Refinancing occurs when a borrower issues new debt to obtain funds to repay maturing principal. Interest-cost pass-through is the process by which a change in market borrowing rates affects the interest paid on the debt stock. For fixed-rate debt, the change generally affects cash interest costs when debt is refinanced; for floating-rate debt, it affects costs when the rate resets.

A maturity schedule describes how much principal comes due at different dates. For example, if one quarter of a fixed-rate debt stock matures and is refinanced each year, approximately one quarter of the stock can take on the new rate at each annual refinancing, assuming the total principal remains constant. This is different from saying that one quarter of the interest bill changes: the rate change applies to principal, and its dollar effect depends on the rate difference.

A simple model makes the mechanism precise. Suppose a government has a constant principal stock $D$, all debt initially pays a fixed annual rate $r_0$, and the market rate for refinancing becomes a constant $r_1$. Assume a fraction $q$ of the stock is refinanced at the end of each year, at par, and the new borrowing remains at $r_1$. Assume no new borrowing, principal repayment from other sources, floating-rate debt, or change in the refinancing fraction. These are simplifying assumptions, not features of all public debt.

r_t=(1-q)r_{t-1}+qr_1

Here $r_t$ is the average annual rate on the debt stock after $t$ refinancing steps. The $(1-q)$ share retains its previous rate, while the $q$ share takes the new rate. The annualized interest cost on the constant stock at that point is $D r_t$. This is a run-rate calculation; actual payments during a year depend on the dates when the bonds mature and interest is paid.

Why the Pass-Through Formula Holds

Subtract the average rate from the new market rate. The recurrence gives:

r_1-r_t=(1-q)(r_1-r_{t-1})

Thus, at each refinancing step, the fraction of the remaining gap between the old average rate and the new rate that is still left is multiplied by $1-q$. Starting from $r_0$, repeated application gives:

$$ r_1-r_t=(1-q)^t(r_1-r_0) $$

Equivalently, the share of the original rate gap that has passed through is $1-(1-q)^t$, and:

r_t=r_0+\left[1-(1-q)^t\right](r_1-r_0)

This formula applies to the stated constant-fraction model. If the fraction refinanced changes over time, the remaining gap is multiplied by $1-q_j$ in period $j$, so after $t$ steps it is $(r_1-r_0)\prod_{j=1}^{t}(1-q_j)$. If bonds mature in distinct, irregular amounts, calculate the effect from their actual schedule rather than assuming a constant $q$.

Worked Example: Gradual Pass-Through After a Rate Increase

A hypothetical government has $D=\$100$ billion of fixed-rate debt, all paying an annual rate of 2.00%. A market-rate increase means that refinanced debt will pay 5.00%. Assume 25% of the constant principal stock is refinanced at the end of each year, at par, and no debt is floating-rate or added. Find the repriced share, the remaining original rate gap, and the annualized interest cost after four refinancing steps.

Method. Use the constant-fraction refinancing model with $q=0.25$, $r_0=0.0200$, $r_1=0.0500$, and $t=4$. The original rate gap is $0.0500-0.0200=0.0300$, or 3.00 percentage points.

The share of the original gap remaining after four steps is:

$$ (1-q)^t=(1-0.25)^4=0.75^4=0.31640625 $$

Therefore, the share that has repriced is $1-0.31640625=0.68359375$, or 68.359375%. The repriced principal is $\$100\text{ billion}\times0.68359375=\$68.359375$ billion. The remaining principal carrying the old rate is $\$100\text{ billion}\times0.31640625=\$31.640625$ billion.

The average rate is the old rate plus the repriced share times the rate increase:

$$ r_4=0.0200+(0.68359375)(0.0300) =0.0200+0.0205078125 =0.0405078125 $$

The annualized interest cost is $\$100\text{ billion}\times0.0405078125=\$4.05078125$ billion, which is $\$2.05078125$ billion above the initial $\$2$ billion run rate. Rounded to the nearest $\$0.01$ billion, the annualized cost is $\$4.05$ billion. Check: the repriced and old-rate shares sum to $68.359375\%+31.640625\%=100\%$; weighting 5.00% and 2.00% by those shares gives the same 4.05078125% average rate. The common mistake is to assume the whole portfolio immediately pays 5.00%; after four steps, 31.640625% of the original rate gap remains.

Staggered and Concentrated Maturities

Two debt portfolios can have the same total principal and face the same market rate but pass through rate changes at different speeds. A staggered schedule spreads maturities across dates. A concentrated schedule, sometimes called a maturity wall, places a large share of principal due at one time. A schedule with dispersed maturities can smooth the amount needing refinancing in any single period, although it does not eliminate the possibility that rates are high when particular bonds come due.

A single average-maturity statistic cannot, by itself, show how much debt falls due in each period. A portfolio can have a moderate average maturity but still have a large near-term maturity concentration. For assessing rollover exposure, examine the amounts and dates due as well as the average. In addition, debt that matures later may still have an earlier rate reset if it is floating-rate debt.

Worked Example: A Ladder Versus a Maturity Wall

Consider two hypothetical portfolios, each with $\$40$ billion of fixed-rate debt paying 2.00%. In Portfolio A, $\$10$ billion matures at the end of each of the next four years. In Portfolio B, all $\$40$ billion matures at the end of year four. Any principal that matures is refinanced at 5.00% at par and remains outstanding; all other principal retains its 2.00% rate. Ignore issuance costs and changes in principal. Calculate the annualized interest run rate after each year's refinancing.

Method. For each date, add interest on the portion refinanced at 5.00% to interest on the portion still at 2.00%. For Portfolio A, each $\$10$ billion refinancing adds $\$10\text{ billion}\times(0.0500-0.0200)=\$0.30$ billion per year to the run rate.

Portfolio A begins with a run rate of $\$40\text{ billion}\times0.0200=\$0.80$ billion. After year one it is $\$0.80+\$0.30=\$1.10$ billion. After year two it is $\$1.10+\$0.30=\$1.40$ billion. After year three it is $\$1.40+\$0.30=\$1.70$ billion. After year four it is $\$1.70+\$0.30=\$2.00$ billion.

For Portfolio B, no principal matures in years one through three, so its annualized run rate remains $\$0.80$ billion over those years. At the end of year four, all $\$40$ billion refinances at 5.00%, giving a run rate of $\$40\text{ billion}\times0.0500=\$2.00$ billion. Check: after all principal in either portfolio has refinanced, both have the same $\$2.00$ billion annualized cost under the assumptions. The timing differs: Portfolio A faces smaller annual refinancing amounts, while Portfolio B faces a $\$40$ billion maturity concentration in year four. This example compares annualized costs after refinancing dates, not the interest paid during the maturity year.

Borrowing Cost and Rollover Risk

Shorter-term borrowing may have a lower initial rate than longer-term borrowing, but it exposes the borrower to more frequent refinancing at future rates. Longer-term fixed-rate borrowing can make costs more predictable and defer refinancing, but a borrower may pay a higher rate to lock in funding for longer. That rate difference is not guaranteed: it depends on market conditions and the terms investors require.

A clean comparison should specify its horizon and assumptions. It should also distinguish a stated annual rate from the total dollar interest over the comparison period. In a simple undiscounted calculation with constant principal, annual interest is principal multiplied by the applicable annual rate. If future refinancing rates are uncertain, the expected cost alone is not a complete measure of risk: the distribution of possible costs and the possibility of difficult market access matter too.

Worked Example: Comparing a Rate Lock With Annual Refinancing

A hypothetical government needs to finance $\$12$ billion for a three-year planning horizon. It can issue a three-year fixed-rate bond at 3.80% per year, or issue one-year debt at 3.20% for year one and refinance at 4.20% for year two and 5.20% for year three. Assume the principal remains $\$12$ billion, rates are known for this illustration, interest is not discounted, and principal at the end of the three-year horizon is excluded from both alternatives. Compare cumulative interest over the three years.

Method. Multiply the constant principal by each year's rate and add the annual interest costs. This compares interest over the stated horizon only; it is not a claim about borrowing costs after year three.

With the fixed-rate bond, annual interest is $\$12\text{ billion}\times0.0380=\$0.456$ billion. Over three years, interest is $3\times\$0.456=\$1.368$ billion.

With annual refinancing, year-one interest is $\$12\text{ billion}\times0.0320=\$0.384$ billion; year-two interest is $\$12\text{ billion}\times0.0420=\$0.504$ billion; and year-three interest is $\$12\text{ billion}\times0.0520=\$0.624$ billion. Total interest is $\$0.384+\$0.504+\$0.624=\$1.512$ billion.

Under this specified rate path, annual refinancing costs $\$1.512-\$1.368=\$0.144$ billion more over the horizon. Check: the fixed rate of 3.80% is below the average of the three annual refinancing rates, $(3.20\%+4.20\%+5.20\%)/3=4.20\%$, so its cumulative interest is lower. The comparison avoids the common mistake of judging the alternatives only by the first-year rate; the 3.20% initial rate does not remain in force for all three years.

Common Mistakes and Practical Checks

Applying a new rate to all fixed-rate debt immediately. In a refinancing model, only principal that matures and is refinanced takes on the new rate. Floating-rate debt is a separate case and should be counted when it resets.
Confusing the share repriced with the share still carrying the old rate. With a constant annual refinancing fraction $q$, the remaining share after $t$ steps is $(1-q)^t$, not $1-tq$. For $q=0.25$ and $t=4$, the remaining share is $0.75^4=31.640625\%$, not 60%.
Reporting a rate change as if it were a dollar cost. A 3-percentage-point increase applied to $\$10$ billion adds $\$0.30$ billion to annual interest, not $\$3$ billion.
Treating an annualized run rate as the exact cash paid in the refinancing year. Actual cash interest depends on maturity and payment dates. State whether a calculation is a run rate after refinancing or interest paid during a particular year.
Assuming longer maturity is always cheaper, or always safer. It can reduce near-term rollover exposure while carrying a higher rate. Compare costs and refinancing dates under stated assumptions rather than treating one structure as universally preferable.

Key takeaway: For fixed-rate debt, the maturity schedule governs how quickly a market-rate change reaches interest costs. A staggered schedule can spread refinancing needs over time, while a concentrated schedule can create a large rollover requirement. Quantify both the share repriced and the dollar cost, and keep refinancing risk distinct from the interest rate paid.

Check Your Understanding

Use the stated assumptions in each problem. Answers are rounded only where indicated.

A government has $\$80$ billion of constant fixed-rate debt initially at 3.00%. It refinances 20% of the stock at the end of each year at 6.00%. After three refinancing steps, what share of the original rate gap remains, what is the average rate, and what is the annualized interest cost? (Answer: 51.20% of the gap remains; average rate 4.464%; annualized interest cost $\$3.5712$ billion.)
A $\$24$ billion fixed-rate portfolio starts at 2.00%. Equal amounts mature at the end of each of four years and are refinanced at 4.50%. Assuming constant principal, what is the annualized interest run rate after the second refinancing? (Answer: $\$0.78$ billion per year.)
A hypothetical $\$30$ billion bullet bond pays 3.50% and matures at the end of year two. It is refinanced at 5.50%, with principal remaining constant. What interest is paid in years one and two combined, and what is the annualized interest run rate after refinancing at the end of year two? (Answer: $\$2.10$ billion in years one and two combined; $\$1.65$ billion per year after refinancing.)
A government finances $\$50$ billion for three years. Annual refinancing rates are 2.80%, 4.00%, and 4.60%; an alternative fixed-rate issue locks in 3.70% for all three years. Assume constant principal, no discounting, and exclude principal repayment. Which has lower cumulative interest, and by how much? (Answer: the fixed-rate issue costs $\$5.55$ billion, versus $\$5.70$ billion for annual refinancing; the fixed-rate issue is lower by $\$0.15$ billion.)
A constant-fraction refinancing model refinances 15% of the remaining old-rate share each year. After five annual steps, what share of the original rate gap has passed through? Give the answer to four decimal places as a percentage. (Answer: $1-0.85^5=0.5562946875$, so 55.6295% has passed through.)

What You'll Learn