Estimating Cost From New Borrowing Rates

Why the Rate on New Borrowing Matters

A firm considering a new loan or bond issue needs an estimate of the return lenders require now. That estimate is its marginal before-tax cost of debt: the borrowing rate applicable to an additional unit of debt under the stated financing terms. It is not automatically the rate the firm pays on debt issued in the past. Existing debt may have been contracted when market conditions and the firm’s circumstances were different.

Two common sources of an estimate are a lender’s quoted rate for a proposed loan and the yield implied by the issue price and promised payments of a proposed bond. These rates can be compared only after their timing and compounding conventions are made consistent. The estimate should also match the question: a nominal quote, an effective annual rate, and a bond yield are not interchangeable labels for one number.

Definition: The before-tax marginal cost of debt is the current rate of return required by lenders on a firm’s proposed additional borrowing, before considering any tax effects. For a loan, estimate it from the lender’s rate and its compounding convention. For a bond, estimate it from the yield implied by the proceeds received and the bond’s promised payments. Unless stated otherwise, these estimates exclude fees, taxes, and other financing costs.

Interpreting a Lender’s Quoted Rate

A lender’s quote must be read with its time unit and compounding convention. A rate stated as an effective annual rate already measures the annual accumulation on a balance left invested for a year. A nominal annual rate compounded $m$ times per year is an annualized quote whose periodic rate is the nominal rate divided by $m$. Its effective annual equivalent is

i_{\text{eff}}=\left(1+\frac{j^{(m)}}{m}\right)^m-1,

where $j^{(m)}$ is the quoted nominal annual rate and $m$ is the number of compounding periods per year. The notation here identifies the quote as nominal; the resulting $i_{\text{eff}}$ is an effective annual rate. This conversion assumes the quoted periodic rate applies in each period and that interest compounds at that rate. It does not, by itself, account for fees or determine the cash flows of a loan with principal repayments.

If the quote is an effective annual rate, do not compound it again. If two offers have different conventions, first convert them to a common basis before comparing them. The contract’s stated payment schedule also matters: a rate applied to a declining loan balance does not imply that interest dollars are charged on the original principal throughout the loan term.

Worked Example: Converting a Monthly-Compounding Quote

A lender offers a firm a $2,400,000 advance at a nominal annual rate of 12% compounded monthly. Assume the full principal remains outstanding for one year, interest is compounded monthly, and there are no fees or payments during the year. Estimate the before-tax effective annual cost and the interest accrued in dollars.

Method. Convert the nominal quote to an effective annual rate, then apply that annual accumulation to the unchanged principal. This avoids treating 12% nominal as though it were already an effective annual rate.

The monthly rate is $0.12/12=0.01$, or 1%. The one-year accumulation factor is

(1+0.01)^{12}=1.12682503.

Therefore, the effective annual cost is $1.12682503-1=0.12682503$, or 12.682503%. The interest accrued is

\$2{,}400{,}000(0.12682503)=\$304{,}380.07.

The year-end balance is $\$2{,}400{,}000+\$304{,}380.07=\$2{,}704{,}380.07$. Thus, to four decimal places, the effective annual rate is 12.6825%; the dollar interest is rounded to the nearest cent. The result is a before-tax rate estimate under the stated no-payment, no-fee assumption, not a claim about a loan whose principal amortizes during the year.

Estimating Cost from a Proposed Bond’s Issue Yield

For a proposed fixed-payment bond, the issue yield is the rate that equates the proceeds received at issuance with the present value of the promised coupons and principal. Let $P_0$ be the issue price, $C$ the coupon paid once per year, $F$ the face value repaid at maturity, $n$ the number of years to maturity, and $y$ the annual yield. Under annual payments and a constant yield, the pricing equation is

P_0=\sum_{t=1}^{n}\frac{C}{(1+y)^t}+\frac{F}{(1+y)^n}.

The equation assumes the stated coupons and principal are paid in full on schedule, and that the yield is an annual rate applied consistently to the annual cash flows. The estimated before-tax cost for the proposed issue is $y$, provided the issue proceeds are the relevant amount received and there are no additional fees or issue costs in the estimate. If the bond pays more frequently than once a year, the payment and yield conventions must be aligned; do not insert annual cash flows into a formula using a periodic rate without conversion.

Justification: At the issue yield $y$, each promised payment at time $t$ is worth its payment divided by $(1+y)^t$ today. Adding those discounted values gives the price that makes an investor’s return equal to $y$. The issuer receives that issue price and promises the same future cash flows. Therefore, the yield equating their present value to the proceeds is the bond-based estimate of the issuer’s before-tax borrowing cost, under the stated assumptions.

For a conventional bond with positive coupons and a positive maturity payment, each discounted payment decreases as $y$ increases. Consequently, the total present value decreases as the yield rises. This makes yield estimation a matter of finding the rate that brings the calculated present value to the observed issue price. A rate that produces a present value above the issue price is too low; a rate that produces a present value below it is too high.

Worked Example: Yield on a Proposed Annual-Coupon Bond

A firm proposes to issue a three-year bond with face value $1,000 and annual coupons of $80. The issue price is $950.26. Estimate the before-tax issue yield to two decimal places. Assume no fees and that all three coupons and the face value will be paid as promised.

Method. Substitute the promised payments into the bond pricing equation and test a yield. A yield of 10% gives the following present value:

\begin{aligned} PV &=\frac{\$80}{1.10}+\frac{\$80}{1.10^2}+\frac{\$1{,}080}{1.10^3}\\ &=\$72.727273+\$66.115702+\$811.419985\\ &=\$950.262960. \end{aligned}

The third-year payment is $\$80+\$1{,}000=\$1{,}080$, because the final coupon and face value are both paid then. The calculated value at 10% exceeds the issue price by $\$950.262960-\$950.26=\$0.002960$, so the exact yield is slightly above 10%. The difference is less than half a cent, and the yield rounded to two decimal places is 10.00%.

Check. At the reported yield, the present value rounds to $\$950.26$, matching the proceeds to the nearest cent. The estimated before-tax cost is therefore 10.00% per year under the annual-payment and no-fee assumptions. A common mistake is to use the 8% coupon rate as the cost: because the bond is issued below its $1,000 face value, the investor’s yield also reflects the gain from paying $950.26 and receiving $1,000 at maturity.

Use the Current Rate for New Financing

The rate on existing debt describes a contractual obligation already in place. It can be useful for understanding the firm’s current interest payments, but it does not necessarily measure what the firm would pay to borrow more today. For an incremental financing decision, use the current quote for comparable proposed borrowing or the issue yield on a proposed bond, with the relevant payment and compounding conventions.

For example, a firm may have outstanding bonds with a 5% coupon and receive a current loan offer at 8%. The 5% coupon remains relevant to the promised payments on those bonds. It is not evidence that an additional loan is available at 5%. Conversely, a current quote should not be substituted for the rate actually specified in an existing debt contract when calculating that contract’s scheduled interest.

When the question asks for the cost of a particular new borrowing, the rate should match that borrowing’s terms and the amount being raised. A company-wide estimate may require more information about all sources of financing; a single existing coupon or new-loan quote does not automatically answer every broader cost-of-capital question.

Worked Example: Separating an Existing Coupon from a New Loan Rate

A firm has $5,000,000 face value of existing bonds with a 6% annual coupon. It is considering a separate $1,200,000 loan at a lender’s quoted effective annual rate of 8.25%. Assume the new loan’s balance remains $1,200,000 for the year, its interest is charged at that annual rate, and there are no fees. Estimate the rate and annual interest dollars relevant to the new borrowing.

Method. Use the lender’s current effective annual quote for the proposed loan. Do not substitute the existing bonds’ coupon, which applies to a different borrowing contract.

The before-tax marginal cost estimate for the new loan is 8.25%. Its annual interest under the stated constant-balance assumption is

\$1{,}200{,}000(0.0825)=\$99{,}000.

For comparison, the existing bonds’ annual coupon payment is $\$5{,}000{,}000(0.06)=\$300{,}000$. That $300,000 is the contractual coupon amount on the existing bonds, not the rate or dollar interest cost of the proposed loan. The new loan’s estimate is 8.25% and $99,000 of annual interest under the stated assumptions. The calculation excludes taxes, fees, and changes in the loan balance.

Common Mistakes and Practical Checks

Before reporting a borrowing-cost estimate, check that the rate refers to the borrowing being evaluated, the periods are consistent, and the cash flows used in the calculation match the stated contract. In particular:

Do not confuse coupon rate with yield. A coupon determines the bond’s promised interest payments; the issue price and all promised payments together determine the issue yield.
Do not treat a nominal quote as an effective annual rate. Convert it using the stated compounding frequency before comparing it with an effective annual quote.
Do not use an old contractual rate as the current marginal rate. Existing debt terms govern that debt; a current proposed borrowing rate estimates the cost of additional debt.
Do not overlook payment timing. Use the actual coupon dates and a yield convention consistent with their frequency.
Do not imply that the estimate includes items excluded from the data. The calculations here are before tax and omit fees. If those amounts are supplied and relevant, they require a separate adjustment.

A useful final check is economic direction. Holding the promised payments fixed, a lower bond issue price implies a higher yield, because the buyer pays less for the same future cash flows. For lender quotes, more frequent compounding at the same positive nominal annual rate produces a higher effective annual accumulation. If a result moves in the opposite direction, recheck the conversion or cash-flow equation.

Check Your Understanding

For each problem, use the stated conventions and distinguish a current borrowing estimate from a rate applying to existing debt.

A lender quotes 9% nominal annual interest compounded monthly. Find the effective annual rate, to four decimal places. Assume the quoted monthly rate compounds for 12 months. (Answer: 9.3807%)
A firm proposes a two-year bond with face value $1,000, annual coupons of $70, and an issue price of $947.93. Estimate the annual issue yield to two decimal places, assuming no fees. (Answer: 10.00%)
A firm pays a 4.5% coupon on outstanding debt and receives a new loan offer at an effective annual rate of 7.25%. The proposed loan principal is $800,000 and remains outstanding for the full year. What rate estimates the new borrowing cost, and what is the year’s interest? (Answer: 7.25%; $58,000)
Compare a lender’s 7.2% nominal annual quote compounded quarterly with a second lender’s 7.35% effective annual quote. Convert the first quote and identify the lower effective annual rate. (Answer: The first quote is approximately 7.3967% effective annually; the second lender’s 7.35% rate is lower.)
A proposed three-year bond has face value $1,000, annual coupons of $60, and an issue price of $1,000. Find its annual yield and explain why the coupon is or is not a suitable estimate of the issue yield. Assume no fees. (Answer: 6.00%; because the bond is issued at face value, the annual coupon yield and issue yield are both 6.00%.)

What You'll Learn