Bootstrapping the First Discount Factor

Why the First Discount Factor Matters

Bootstrapping a term structure starts with the earliest future payment date for which a market price is available. The first discount factor is especially direct to recover when the shortest-maturity instrument has just one future payment date: the bond’s coupon and redemption are then valued using the same factor. Once recovered, that factor can be checked by repricing the bond.

This method uses the bond’s actual contractual cash flow and its market price at the valuation date. It does not treat the coupon rate as a market interest rate, and it does not require a constant rate across all maturities. The procedure is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $D(t)$ be the discount factor at the valuation date for a payment due at future time $t$. It is the current market value of one unit of currency payable with certainty at $t$. Under the usual no-arbitrage pricing model for known bond cash flows, a cash flow of amount $C$ due at $t$ has present value $C D(t)$.

The model assumes that the bond’s future payments are specified, the quoted price is the full market price at the valuation date, and the same discount factor applies to any claim with the same payment date and relevant payment characteristics. Taxes, transaction costs, default risk, and embedded options are excluded. If the cash flows are uncertain, or the quoted price needs an adjustment before it represents the value of the stated cash flows, this simple calculation does not by itself recover the appropriate discount factor.

The One-Date Coupon-Bond Equation

Suppose a bond has face amount $F$, annual coupon rate $c$, and one coupon payment at its maturity date $t_1$. The coupon at that date is $cF$, and the redemption payment is $F$. Thus the entire future cash flow is $(1+c)F$. If its market price is $P$, then

P = \bigl(cF+F\bigr)D(t_1) = (1+c)F D(t_1).

Solving for the first discount factor gives

D(t_1)=\frac{P}{(1+c)F}.

This equation applies only when all cash flows in the bond occur at $t_1$. If a coupon is paid earlier, its value uses the discount factor for that earlier date. Unless that earlier factor is already known, a single price equation contains more than one unknown factor and cannot isolate $D(t_1)$.

Why the Formula Holds

The bond’s terminal payment can be viewed as a claim to $cF+F$ dollars payable at $t_1$. By the definition of $D(t_1)$, one dollar due at that date is worth $D(t_1)$ today. Multiplying by the number of dollars due gives the value of the whole claim:

\underbrace{(cF+F)}_{\text{dollars due at }t_1} \underbrace{D(t_1)}_{\text{value today per dollar due at }t_1} = \underbrace{P}_{\text{market price today}}.

If the market price differed from this value while the cash flows and assumptions were unchanged, an investor could compare the quoted bond with a claim paying the same amount on the same date. In the idealized no-arbitrage model, they must have the same value. Rearranging this equality yields the bootstrap formula above.

Identify the first payment date.
List every promised payment and its date. Do not assume that maturity is the only payment date.

Combine payments on that date.
For a one-date coupon bond, add the coupon and redemption to find the total maturity cash flow.

Divide price by that cash flow.
The quotient is the discount factor for the payment date, provided no other future payment date is involved.

Reprice the bond.
Multiply the recovered factor by the full cash flow. The result must equal the given market price.

Worked Applications

Worked Example: A One-Year Coupon Bond

A hypothetical bond has face amount $1,000, an annual coupon rate of 4%, and a single payment at the end of one year. Its full market price today is $1,000. Find the one-year discount factor and check it by repricing. Assume the coupon and redemption are paid together at maturity and that the bond has no default risk or embedded options.

Method: combine the cash flows at the common date. The annual coupon is

$$ 0.04(1000)=40. $$

The bond therefore pays $40+1000=1040$ dollars at $t_1=1$. Since there is only this one future payment date, the price equation is

$$ 1000=1040D(1). $$

Solving,

D(1)=\frac{1000}{1040}=\frac{25}{26} =0.961538\ldots.

To check, reprice the full cash flow using the unrounded factor:

1040\left(\frac{25}{26}\right)=40(25)=1000.

The repriced amount is exactly the observed price. Rounded only at the final step, the discount factor is 0.961538 to six decimal places.

Common mistake avoided: the coupon rate of 4% is not itself the discount factor. The factor comes from dividing the observed price by the total maturity payment, including both coupon and redemption.

Worked Example: A Zero-Coupon Instrument

A hypothetical zero-coupon instrument promises $2,500 in one year and pays no coupons. Its market price today is $2,375. Recover the one-year discount factor and verify the price. Assume the promised payment is certain and there are no other payments.

Method: use the sole promised payment. The only future cash flow is $2,500 at $t_1=1$, so

$$ 2375=2500D(1). $$

Therefore,

D(1)=\frac{2375}{2500}=0.95.

Repricing gives

$$ 2500(0.95)=2375. $$

The calculated price matches the quoted price exactly, so the recovered one-year discount factor is 0.95. A zero-coupon instrument reaches the same present-value equation as a one-date coupon bond; its sole cash flow is simply its redemption amount.

Common mistake avoided: do not add a coupon that the instrument does not pay. The numerator is the full market price, and the denominator is the actual promised payment at the date being discounted.

Worked Example: Separating Coupon and Redemption

A hypothetical shortest-maturity bond has face amount $5,000, an annual coupon rate of 5.5%, and a single payment at its maturity date $t_1$. Its full market price is $5,125. Find $D(t_1)$ and confirm the result by repricing. Assume the coupon and principal are both paid at $t_1$, with no earlier coupon payments.

Method: calculate the actual terminal cash flow before discounting. The coupon is

$$ 0.055(5000)=275. $$

Adding the $5,000 redemption gives a total maturity payment of

$$ 275+5000=5275. $$

The price equation is $5125=5275D(t_1)$. Hence

D(t_1)=\frac{5125}{5275}=\frac{205}{211} =0.971563981\ldots.

Reprice using the exact fraction:

5275\left(\frac{205}{211}\right) =25(205)=5125.

The check reproduces the observed price exactly. Rounded only at the final step, $D(t_1)=\mathbf{0.971564}$ to six decimal places.

Common mistake avoided: dividing the price by face amount alone would give $5125/5000=1.025$, which does not account for the $275 coupon. The denominator must be the complete cash flow due on the date.

When the First Factor Is Not Isolated

The phrase “shortest maturity” does not guarantee that a bond has only one payment date. For example, a bond maturing at 18 months might pay coupons at 6 months and 12 months as well as a final coupon and redemption at 18 months. Its price would involve $D(0.5)$, $D(1)$, and $D(1.5)$. If none of those factors is known, one price equation cannot uniquely determine all three.

The first factor is directly recoverable from a coupon bond when the coupon and redemption are paid together at the earliest date being bootstrapped, or when every earlier-date factor in the price equation is already known. This is the practical reason to inspect payment dates before writing a formula. A bond’s label, coupon rate, and maturity alone do not establish that its price has only one unknown discount factor.

Common Exam Traps

Using the coupon rate as a market rate. The coupon rate determines the coupon amount, not the discount factor. Obtain the factor from the market price and all cash flows.
Omitting the coupon or the redemption. For a one-date coupon bond, the terminal amount is coupon plus principal.
Discounting cash flows at the wrong date. A coupon paid before maturity cannot be combined with redemption as though both were paid at maturity.
Mixing price and cash-flow units. If the price is stated per $100 of face amount, either calculate the payment per $100 or convert both quantities to matching dollar amounts.
Stopping before the repricing check. Substituting the factor back into the complete cash-flow equation catches arithmetic and input errors.

Key takeaway: When the shortest-maturity instrument has one future payment date, divide its full market price by its complete cash flow on that date to recover the first discount factor. Then multiply the cash flow by the factor and confirm that the price is reproduced.

Check Your Understanding

For each problem, assume the stated payments are certain, the quoted price is the full price today, and there are no taxes, transaction costs, or embedded options. Where appropriate, calculate the discount factor from the complete cash flow and check that the instrument has only one future payment date.

A hypothetical bond has face amount $1,000, a 4% annual coupon, and a single payment at maturity. Its market price is $975. Find the discount factor for maturity. (Answer: $D(t_1)=0.9375$.)
A zero-coupon instrument promises $3,200 in one year and costs $3,008 today. Find its one-year discount factor and state the repriced value. (Answer: $D(1)=0.94$; repriced value is $3,008.)
A one-date bond has face amount $2,500, a 6% annual coupon, and a market price of $2,525. Find the factor for its payment date and verify the price. (Answer: $D(t_1)=101/106\approx0.952830$; the repriced value is $2,525.)
A hypothetical bond has face amount $5,000, a 5% annual coupon, and one payment at maturity. The full market price is $4,925. Calculate the terminal cash flow and the discount factor. (Answer: terminal cash flow is $5,250; $D(t_1)=197/210\approx0.938095$.)
A bond is the shortest-maturity instrument available and matures at 18 months. It pays coupons at 6 months and 12 months, then pays its final coupon and redemption at 18 months. No discount factors are known. Can its single market price alone determine the 18-month discount factor? Explain. (Answer: No; its price equation involves three unknown factors, $D(0.5)$, $D(1)$, and $D(1.5)$.)

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