Credit Spreads and Risk-Adjusted Borrowing Costs

Why Borrowers Pay Different Rates

A change in a central bank’s policy rate can affect market interest rates, but it does not determine every borrower’s rate by itself. A lender also considers the possibility that the borrower will not repay as promised. As that perceived risk changes, the borrower’s rate may move even when a comparable benchmark rate does not.

The difference between a borrower’s rate and a suitable benchmark is the credit spread. It helps separate the benchmark component of a borrowing rate from the additional rate associated with the borrower and the loan. The comparison is useful only when the two rates are measured on a comparable basis. These distinctions are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A credit spread is the borrower’s quoted interest rate minus the yield on a selected benchmark with comparable currency, maturity, and rate convention. It is usually reported in percentage points or basis points. One basis point (bp) is 0.01 percentage point, so 100 bp equals 1 percentage point.

Choosing and Interpreting a Benchmark

A spread comparison is informative only if it does not accidentally compare unlike rates. For example, comparing a five-year corporate borrowing rate with a one-year benchmark rate mixes differences in credit risk with differences in maturity. Compare rates in the same currency and with the same compounding or quotation convention, and use a benchmark of similar maturity. Differences in seniority, security, tax treatment, or liquidity may also affect rates, so they matter when interpreting the result.

A government yield or a market reference rate may be used as a benchmark, depending on the purpose of the comparison. Neither should automatically be treated as a perfectly risk-free rate in every context. The benchmark itself may reflect market conditions and risks. In this tutorial, the benchmark is the specified reference rate for the comparison; the calculated spread is relative to that choice.

A spread is a difference between rates, not a percentage increase in the benchmark. If a borrower’s effective annual rate is 5.10% and the comparable benchmark is 3.75%, the spread is 1.35 percentage points, or 135 bp. It is not 1.35% of 3.75%.

Worked Example: Calculating a Spread From Comparable Quotes

A business is offered a three-year loan at an effective annual rate of 5.10%. A market report gives a three-year benchmark yield of 3.75% in the same currency, also as an effective annual rate. Calculate the spread and explain what the comparison establishes.

Method. Subtract the comparable benchmark rate from the borrower’s rate:

\text{Spread}=5.10\%-3.75\%=1.35\text{ percentage points}.

Since 1 percentage point equals 100 bp, the spread is $1.35\times100=135$ bp. The loan rate is therefore 135 bp above this benchmark. The calculation measures the rate difference; by itself, it does not identify how much of that difference is due to expected defaults rather than other factors.

From a Spread to a Borrowing Cost

For a fixed principal and term, a higher annual rate generally raises the amount the borrower must repay. Under annual compounding with one payment at the end of the term, the maturity amount is the principal multiplied by one plus the annual rate, raised to the number of years. This comparison assumes the rates stay fixed for the entire term and that no interim payments or fees apply.

Worked Example: Comparing Three-Year Repayment Amounts

A borrower receives $50,000 and must repay the loan in one amount after three years. The benchmark rate is 4.1% effective annually, while the borrower’s effective annual rate is 6.2%. Assume annual compounding, fixed rates for all three years, and no fees or interim payments. Find the credit spread and the difference in maturity amounts.

Step 1: Find the spread. The spread is $6.2\%-4.1\%=2.1$ percentage points, or $210$ bp.

Step 2: Calculate repayment at the borrower’s rate. The annual accumulation factor is $1+0.062=1.062$. Its square is $1.062^2=1.127844$, and its cube is $1.127844\times1.062=1.197770328$. Therefore:

\$50{,}000(1.062)^3 =\$50{,}000(1.197770328) =\$59{,}888.5164 \approx \$59{,}888.52.

Step 3: Calculate the benchmark comparison amount. The annual accumulation factor is $1+0.041=1.041$. Its square is $1.041^2=1.083681$, and its cube is $1.083681\times1.041=1.128111921$. Thus:

\$50{,}000(1.041)^3 =\$50{,}000(1.128111921) =\$56{,}405.59605 \approx \$56{,}405.60.

Step 4: Compare the amounts. Using unrounded values, the difference is $\$59{,}888.5164-\$56{,}405.59605=\$3{,}482.92035$, which rounds to $3,482.92 to the nearest cent. The benchmark-rate amount is a comparison, not a claim that the borrower could actually obtain that rate. The method avoids a common error: rounding the two maturity amounts first and then subtracting them.

Why Perceived Default Risk Can Widen a Spread

A lender who expects some loans to default may require a higher promised rate to compensate for the repayments that will not be received. The size of the rate increase depends not only on the probability of default but also on the recovery if default occurs. A simple break-even model makes this relationship precise.

Model assumptions: Consider a one-period loan with a benchmark effective annual rate $r_f$, a default probability $p$, and a recovery fraction $R$. If the loan does not default, the borrower pays the full promised amount $1+y$ per dollar lent. If default occurs, the lender receives $R(1+y)$. Assume the recovery fraction applies to the promised amount, and assume a risk-neutral lender, no fees, no liquidity or capital costs, and no risk premium beyond expected default loss.

Under these assumptions, the expected repayment per dollar lent must equal the benchmark accumulation factor $1+r_f$ for the lender to break even. If there is no default, which has probability $1-p$, repayment is $1+y$. If default occurs, with probability $p$, repayment is $R(1+y)$. Adding these probability-weighted repayments gives the expected repayment.

(1-p)(1+y)+pR(1+y) =(1+y)\bigl[1-p(1-R)\bigr].

Equating this amount to $1+r_f$ and solving for the promised rate gives:

(1+y)\bigl[1-p(1-R)\bigr]=1+r_f, \qquad y=\frac{1+r_f}{1-p(1-R)}-1.

The term $p(1-R)$ is the probability of default multiplied by the loss fraction conditional on default. It is the expected fraction of the promised amount not recovered in this simplified model. A higher $p$ or a lower $R$ increases that expected loss and, holding other assumptions fixed, increases the break-even promised rate. If $p=0$, the formula gives $y=r_f$, as it should.

This is a model-based explanation, not a general formula for inferring default probabilities from observed spreads. Actual market rates may include compensation for bearing risk, liquidity differences, taxes, embedded contract features, and other costs. Those factors violate the simplified break-even assumptions.

Worked Example: A Break-Even Rate With Default Risk

For a one-year hypothetical loan, let the benchmark effective annual rate be 4.0%, the probability of default be 2.0%, and the recovery fraction of the promised repayment be 40%. Under the risk-neutral, no-fee assumptions above, calculate the break-even promised rate and spread.

Method. The default probability is $p=0.02$, and the loss fraction if default occurs is $1-R=1-0.40=0.60$. Their product is $p(1-R)=0.02\times0.60=0.012$. Thus the probability-weighted repayment factor is $1-0.012=0.988$.

Substitute $r_f=0.04$ into the break-even formula:

y=\frac{1.04}{0.988}-1 =\frac{0.052}{0.988} =0.05263158\ldots

The promised rate is approximately 5.2632% effective annually. Its spread over the 4.0% benchmark is $5.2632\%-4.0\%=1.2632$ percentage points, or about 126.32 bp. The method avoids treating default probability alone as the loss: the 40% recovery means the loss fraction is 60%, not 100%.

Separating Benchmark Changes From Credit Repricing

A borrower’s rate can be represented as the benchmark rate plus the spread. Consequently, a change in the borrowing rate can come from a change in either component or both. A policy-rate change may influence market benchmarks, but the benchmark need not move one-for-one with a policy rate. Perceived borrower risk, lender risk appetite, or other market conditions may change the spread independently.

Worked Example: A Wider Spread Despite a Lower Benchmark

Suppose a hypothetical lender quotes a one-year benchmark rate of 4.0% and a borrower spread of 1.8 percentage points in the first period. In a later period, the benchmark is 3.5%, but the spread has risen to 3.0 percentage points. Assume a $80,000 loan with one payment after one year and no fees. Find each borrower rate and repayment, and explain the change.

First period. The borrower rate is $4.0\%+1.8\%=5.8\%$. The repayment is $\$80{,}000(1.058)=\$84{,}640$.

Later period. The borrower rate is $3.5\%+3.0\%=6.5\%$. The repayment is $\$80{,}000(1.065)=\$85{,}200$.

The benchmark fell by 0.5 percentage points, but the spread rose by 1.2 percentage points. The borrower rate therefore rose by $1.2-0.5=0.7$ percentage points, from 5.8% to 6.5%. Repayment increased by $\$85{,}200-\$84{,}640=\$560$. This comparison isolates the arithmetic; it does not establish why the spread widened.

What a Credit Spread Does and Does Not Measure

A wider spread is consistent with markets requiring more compensation for the risk or costs associated with a loan relative to the selected benchmark. It does not, without additional assumptions and information, tell us the borrower’s default probability. The simple model above requires assumptions about recovery and risk pricing; market spreads can also include premiums for uncertainty, liquidity, or other contract and market features.

A spread can also change because the benchmark is poorly matched to the loan. If the maturities differ, part of the observed difference may reflect the maturity component of rates rather than borrower credit. The earlier material on long-term yields and term premia is relevant to that distinction: compare like maturities before interpreting a difference as credit-related.

When evaluating a borrowing-cost comparison, use this sequence:

Match the rates.
Check currency, maturity, and quotation convention; note other material differences such as security or liquidity.

Calculate the spread.
Subtract the benchmark rate from the borrower rate and convert percentage points to basis points if useful.

Translate into cost only under stated terms.
Specify principal, compounding, term, payments, and fees before calculating a dollar difference.

Interpret cautiously.
Separate benchmark movements from spread movements, and do not label the entire spread as expected default loss without a supporting model.

A common exam trap is to add a spread as a percentage of the benchmark rather than as a percentage-point difference. If the benchmark is 4.0% and the spread is 1.5 percentage points, the borrower rate is $4.0\%+1.5\%=5.5\%$, not $4.0\%\times1.015=4.06\%$. Another trap is to claim that a lower policy rate must reduce every borrower’s rate. If a spread widens by more than the benchmark falls, the borrower’s total rate can rise.

Key takeaway: A credit spread is a relative rate difference, not a direct default-probability estimate. Calculate it against a comparable benchmark, then distinguish changes in the benchmark from changes in the spread before explaining borrowing costs.

Check Your Understanding

Work each problem using the stated rate convention and assumptions.

A four-year borrower rate is 5.35% effective annually, and the comparable four-year benchmark is 3.90%. Calculate the spread in percentage points and basis points. (Answer: 1.45 percentage points, or 145 bp)
A hypothetical $120,000 loan has a one-year benchmark rate of 4.2% and a borrower rate of 6.7%, both effective annual rates. With one payment after one year and no fees, find the spread and the extra repayment relative to the benchmark-rate amount. (Answer: 2.5 percentage points, or 250 bp; extra repayment $3,000)
A benchmark falls from 4.5% to 4.0%. At the same time, a borrower’s spread rises from 1.25 to 2.10 percentage points. Calculate the borrower rate in each period and its change in basis points. (Answer: 5.75% then 6.10%; increase of 35 bp)
In the one-period risk-neutral model, the benchmark is 3.0%, default probability is 1.5%, and recovery is 50% of the promised repayment. Calculate the break-even promised rate and spread. (Answer: approximately 3.7783% promised rate and 0.7783 percentage points, or 77.83 bp, spread)
A hypothetical $25,000 loan is repaid in one amount after two years. Its fixed effective annual rate is 4.4%, compared with a 3.0% benchmark; annual compounding applies and there are no fees or interim payments. Find the spread and the difference in maturity amounts. (Answer: 1.4 percentage points, or 140 bp; borrower-rate amount $27,248.40, benchmark amount $26,522.50, difference $725.90)

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