Zero-Coupon Debt And Discount-Note Costs

Why Zero-Coupon Debt Still Has a Cost

A zero-coupon instrument makes no periodic interest payments, but the borrower still receives less at issuance than it must repay at maturity. That difference is the financing cost. For cost-of-debt analysis, the key is to relate the maturity payment to the cash proceeds actually received and to annualize the difference using a stated time and compounding convention.

A short-term discount note may also be described using a bank discount rate. That rate is a quoting convention, not generally the borrower’s rate of return on the funds received. Keeping the two measures separate avoids a common error when comparing borrowing alternatives. The calculations below use hypothetical debt issues and assume no fees, taxes, or interim cash flows.

Definition: A zero-coupon debt issue has one cash inflow to the borrower at issuance, $P_0$, and one contractual repayment, $F$, at maturity, with no coupon payments between them. Here $P_0$ is the issue proceeds received, $F$ is the redemption amount paid, and $t$ is the term in years. With no fees or other cash flows, $P_0$ is the issue price relevant to the borrowing-cost calculation.

Effective Annual Cost for a Zero-Coupon Issue

Suppose the debt’s annual cost is expressed as an effective annual rate, $k_d$. This model assumes that the cost compounds annually, the promised redemption amount is paid in full, the term is measured in years, and the only cash flows are $P_0$ at time zero and $F$ at time $t$. The annual rate is found by equating the initial proceeds accumulated at that rate to the maturity payment.

$$ P_0(1+k_d)^t=F $$

k_d=\left(\frac{F}{P_0}\right)^{1/t}-1

This result applies to a single-payment, no-coupon issue under the stated annual effective convention. If the debt has coupons, fees, or other cash flows, the cost must instead reflect all relevant cash flows. A different compounding or day-count convention also requires a consistent calculation; the effective annual rate should not be confused with a simple annualized quote.

Why the Formula Works

The formula follows directly from the cash-flow equivalence at the maturity date. Investing the proceeds $P_0$ at the annual effective rate $k_d$ for $t$ years gives $P_0(1+k_d)^t$. For that accumulation to equal the contractual payment $F$, the equation must hold. Dividing by $P_0$, taking the $t$th root, and subtracting one isolates the annual rate:

P_0(1+k_d)^t=F \quad\Longrightarrow\quad (1+k_d)^t=\frac{F}{P_0} \quad\Longrightarrow\quad 1+k_d=\left(\frac{F}{P_0}\right)^{1/t} \quad\Longrightarrow\quad k_d=\left(\frac{F}{P_0}\right)^{1/t}-1

The exponent $1/t$ matters. A discount over two years is not itself an annual rate: the formula finds the constant annual rate that compounds to the observed ratio over the full term. If $F>P_0$, then $k_d>0$, as expected for a positive borrowing cost under this model.

Worked Example: A Two-Year Zero-Coupon Issue

A hypothetical company receives $8,100 today from a zero-coupon issue and must repay $10,000 exactly two years later. There are no other cash flows. Calculate the effective annual borrowing cost and check the result.

Method. Use the effective annual zero-coupon formula. The term is $t=2$ years, and the relevant issue proceeds are $P_0=\$8{,}100$, not the maturity amount.

k_d=\left(\frac{10{,}000}{8{,}100}\right)^{1/2}-1 =\sqrt{1.2345679}-1 =1.1111111-1 =0.1111111

The effective annual borrowing cost is 11.11%, rounded to two decimal places. To check, accumulating the proceeds for two years at the unrounded rate gives $8{,}100(1.1111111)^2$, which is approximately $10,000. The check reproduces the required repayment. A common mistake is to report the total two-year increase, $(10{,}000-8{,}100)/8{,}100$, as an annual rate; that increase is about 23.46% over the full two years, not per year.

Discount-Note Quotes and Borrowing Cost

For a short-term discount note, a bank discount rate is often quoted using the face or redemption amount as its denominator. Under a 360-day convention, let $D$ be the number of days to maturity. The bank discount rate $b$ is the discount from face value, annualized on that convention:

b=\frac{F-P_0}{F}\frac{360}{D}

Because the denominator is $F$, this quote is not the annualized return on the cash the borrower receives. A simple annualized borrowing cost based on proceeds uses $P_0$ instead:

c_{\text{simple}}=\frac{F-P_0}{P_0}\frac{360}{D}

These simple annualized measures scale the term-period cost by $360/D$; they do not compound that cost. If the desired measure is an effective annual rate and the term is $D/360$ years, use the compound formula instead:

k_d=\left(\frac{F}{P_0}\right)^{360/D}-1

The day-count basis must be stated and used consistently. For example, a 365-day basis changes the annualization factor to $365/D$. The bank discount quote and the proceeds-based simple rate answer different questions, while the effective rate additionally accounts for compounding. When comparing costs, first identify which convention is being requested.

Worked Example: Separating a Bank Discount Quote From the Cost of Funds

A hypothetical 180-day note has a face value of $100,000 and is issued for $97,000. Use a 360-day convention. Calculate (1) the bank discount rate, (2) the simple annualized borrowing cost based on proceeds, and (3) the effective annual cost. State which figure measures the actual annualized cost on funds received under each convention.

Method. First find the discount: $F-P_0=\$100{,}000-\$97{,}000=\$3{,}000$. The annualization factor is $360/180=2$.

1. Bank discount rate. Use face value in the denominator:

b=\frac{3{,}000}{100{,}000}\times 2 =0.03\times 2 =0.0600 =6.00\%

2. Simple annualized borrowing cost. Use the $97,000 actually received:

c_{\text{simple}}=\frac{3{,}000}{97{,}000}\times 2 =0.03092784\times 2 =0.06185567 =6.19\%

3. Effective annual cost. The note runs for $180/360=0.5$ years:

k_d=\left(\frac{100{,}000}{97{,}000}\right)^{1/0.5}-1 =\left(\frac{100}{97}\right)^2-1 =\frac{10{,}000}{9{,}409}-1 =0.062812

The effective annual cost is 6.28%, rounded to two decimal places. The three figures differ because the bank quote uses face value, the simple borrowing rate uses proceeds without compounding, and the effective rate uses proceeds with annual compounding. The calculation also checks directionally: since $P_0<F$, each measure is positive, and the proceeds-based rates exceed the bank discount quote.

Common mistake avoided. Calling $3{,}000/100{,}000\times2=6.00\%$ the actual borrowing cost treats the bank discount quote as though it were calculated on proceeds. The correction is to divide by $97,000 for the simple annualized cost, or to apply the effective-rate formula when an effective annual cost is required.

Annualizing a Non-Whole-Year Term

For a term that is not a whole number of years, the same effective annual formula applies with the actual term expressed in years. For example, nine months is $t=9/12=0.75$ years under a year-based convention. Do not use the total term discount as if it occurred over one year, and do not substitute a simple annualization when the requested measure is effective annual.

Worked Example: A Nine-Month Discount Issue

A hypothetical borrower receives $9,450 from a zero-coupon issue and pays $10,000 nine months later. Assume no fees or interim cash flows and report the effective annual borrowing cost to two decimal places.

Method. Convert the term to years: $t=9/12=0.75$. Then use the effective annual formula, with proceeds as the denominator inside the ratio.

\frac{F}{P_0}=\frac{10{,}000}{9{,}450}=1.05820106

k_d=(1.05820106)^{1/0.75}-1 =(1.05820106)^{4/3}-1 \approx 0.0784

The effective annual borrowing cost is 7.84%, rounded to two decimal places. As a check, compounding the proceeds at the unrounded annual rate for 0.75 years gives $9{,}450(1+k_d)^{0.75}$, approximately $10,000. A simple annualization would instead multiply the nine-month percentage increase by $12/9$; that is a different convention and does not produce the effective annual rate requested here.

Interpreting the Result and Avoiding Traps

A zero coupon describes the timing of the contractual payments; it does not imply that the borrower receives financing for free. When the issue price is below the redemption amount, the difference is economically equivalent to financing compensation embedded in the maturity payment. For cost-of-debt analysis, a rate based on proceeds and the time pattern of payments is more informative than the dollar discount alone.

The formula is a promised borrowing cost under the stated cash-flow assumptions. It does not incorporate default risk, taxes, issue fees, or changes in market value after issuance. It also should not be treated as an observed realized return if the borrower fails to pay as promised. If the financing includes additional cash flows, the single-payment equation no longer describes the full cost.

Check the denominator. Use $P_0$ for a return or cost measured on funds received. A bank discount quote conventionally uses $F$.
Check the term unit. Convert months or days to years consistently before applying an effective annual formula.
Check the annualization convention. Simple annualization, an effective annual rate, and a bank discount rate are not interchangeable.
Check the cash flows. If coupons or other payments exist, a zero-coupon calculation is not sufficient.

Key takeaway: For a no-coupon issue with one payment at maturity, the effective annual borrowing cost is $\left(F/P_0\right)^{1/t}-1$. A discount-note quote may use a different denominator and annualization convention, so label the measure before comparing rates.

Check Your Understanding

For each problem, use the convention stated and distinguish the requested rate from any other annualized measure.

A hypothetical one-year zero-coupon issue provides $48,500 today and requires $50,000 in one year. Find its effective annual borrowing cost, to two decimal places. (Answer: $(50{,}000/48{,}500)-1=3.09\%$.)
A hypothetical three-year zero-coupon obligation provides $7,290 and requires $9,000 at maturity, with no interim cash flows. Find the effective annual cost, to two decimal places. (Answer: $(9{,}000/7{,}290)^{1/3}-1=7.28\%$.)
A 120-day hypothetical discount note has face value $250,000 and issue proceeds of $246,000. Using a 360-day basis, find both its bank discount rate and its simple annualized borrowing cost on proceeds, to two decimal places. (Answer: Bank discount rate $=(4{,}000/250{,}000)(360/120)=4.80\%$; simple borrowing cost $=(4{,}000/246{,}000)(360/120)=4.88\%$.)
A hypothetical zero-coupon issue provides $4,900 and repays $5,000 after 240 days. Use a 360-day year and find the effective annual cost, to two decimal places. (Answer: $(5{,}000/4{,}900)^{360/240}-1=3.08\%$.)
Compare two hypothetical no-fee zero-coupon issues. Issue A provides $98,200 and repays $100,000 after one year; Issue B provides $96,000 and repays $100,000 after two years. Calculate each effective annual cost to two decimal places and identify the lower-cost alternative. (Answer: A: $100{,}000/98{,}200-1=1.83\%$; B: $(100{,}000/96{,}000)^{1/2}-1=2.06\%$; Issue A has the lower effective annual cost.)

What You'll Learn