Valuing New Cash Flows with Bootstrapped Factors

Valuing Cash Flows at Their Own Dates

A bootstrapped curve provides discount factors for particular future dates. To value a new liability or project, the central task is to match each cash flow with the factor for its payment date. A payment at one year should not be discounted with the two-year factor merely because both dates appear in the same cash-flow schedule.

The calculation is direct when every payment date has a bootstrapped factor. If a payment falls between dates on the curve, its factor is not automatically determined by the factors on either side. The valuation then requires a clearly stated interpolation convention, or additional market information that supplies a factor for the payment date.

Definition: Let $D(t)$ be the discount factor at valuation date 0 for a certain payment of one monetary unit at time $t$. If $C_t$ is a deterministic cash flow at time $t$, its present value is $C_tD(t)$. For cash flows at dates $t_1,\ldots,t_n$, the signed present value is $\sum_{j=1}^{n} C_{t_j}D(t_j)$, with inflows positive and outflows negative.

Here $t$ is measured in years from the valuation date, and all factors are understood to refer to that same valuation date and currency. A factor is a price for a dated unit payment, not a single interest rate that must be applied uniformly to every cash flow. The factors may have been obtained from different instruments and may reflect the market’s term structure. Use the factors as supplied; do not replace them with one rate unless the problem specifically calls for that method.

This method assumes that the cash flows are known in amount and timing, and that the bootstrapped factors are appropriate for discounting those cash flows. It does not, by itself, adjust uncertain cash flows for default, liquidity, or other differences between the cash flows being valued and the instruments used to construct the curve. Such adjustments require additional assumptions or information.

Why the Present Value Is a Sum

The rule follows from pricing each dated payment separately and using the additivity of value. A certain payment of one unit at date $t_j$ has value $D(t_j)$. A payment of $C_{t_j}$ units at that date therefore has value $C_{t_j}D(t_j)$, including a negative value when $C_{t_j}$ is an outflow in a signed project calculation. Adding the values of all dated components gives the value of the cash-flow stream.

V_0=\sum_{j=1}^{n} C_{t_j}D(t_j)

This is a finite-portfolio argument: each cash flow is a separate dated claim, and the value of the combined stream is the sum of their values. It also explains why matching dates matters. The claim paying at $t_j$ is valued with $D(t_j)$, rather than with a factor selected for a different maturity.

Worked Example: Valuing a Liability at Three Curve Dates

A hypothetical organization must pay $4,000 at year 0.5, $7,000 at year 1, and $3,000 at year 2. The valuation date is today. The bootstrapped discount factors are $D(0.5)=0.97$, $D(1)=0.93$, and $D(2)=0.86$. Assume the liability amounts and payment dates are certain, and use no additional risk adjustment. Find the present value of the liability.

Method. Treat each payment as a positive liability amount and multiply it by the factor at its own payment date. All three dates have supplied factors, so no interpolation is needed.

At year 0.5, the payment’s present value is $4{,}000(0.97)=3{,}880$.

At year 1, the payment’s present value is $7{,}000(0.93)=6{,}510$.

At year 2, the payment’s present value is $3{,}000(0.86)=2{,}580$.

V_0=3{,}880+6{,}510+2{,}580=12{,}970

The present value of the liability is $12,970.00. As a check, each factor is applied to exactly one payment date and the three discounted payments sum to the stated result. The common mistake avoided here is using $D(2)$ for all three payments just because year 2 is the final maturity.

Signed Values for Projects

A project may have both receipts and payments. For a signed present value, record receipts as positive and payments as negative before summing. A positive result means the discounted inflows exceed the discounted outflows under the stated curve and assumptions; a negative result means they fall short. This is a valuation calculation, not a claim that the project’s uncertain future outcome is guaranteed.

Worked Example: A Project with Cash Flows Between Curve Dates

A hypothetical project receives $2,500 at year 0.5 and $8,000 at year 1.5, and pays $9,000 at year 2.5. Its valuation date is today. The available factors are $D(0.5)=0.985$, $D(1)=0.96$, $D(2)=0.90$, and $D(3)=0.84$. Assume deterministic cash flows. For this calculation, use linear interpolation of discount factors between adjacent quoted dates. Find the signed present value.

Method. Use the supplied factor directly at year 0.5, then interpolate a factor at each unmatched date. Linear interpolation means the factor changes by a constant amount per year over each interval.

The year 1.5 payment is halfway between years 1 and 2. Its factor is $0.96+0.5(0.90-0.96)=0.96-0.03=0.93$.

The year 2.5 payment is halfway between years 2 and 3. Its factor is $0.90+0.5(0.84-0.90)=0.90-0.03=0.87$.

Now apply the factors to the signed cash flows. The receipt at year 0.5 contributes $2{,}500(0.985)=2{,}462.50$. The receipt at year 1.5 contributes $8{,}000(0.93)=7{,}440.00$. The payment at year 2.5 contributes $-9{,}000(0.87)=-7{,}830.00$.

V_0=2{,}462.50+7{,}440.00-7{,}830.00=2{,}072.50

The signed present value is $2,072.50. The arithmetic checks because the two discounted receipts total $9,902.50, and subtracting the discounted payment of $7,830.00 leaves $2,072.50. This result depends on the stated linear interpolation convention; the given curve factors alone do not require that particular convention.

Unmatched Dates and Interpolation

A bootstrapped curve usually gives factors at a set of instrument cash-flow dates. A new liability can have dates that do not coincide with that set. Before calculating, identify which of the following situations applies:

Exact match: use the factor supplied for the payment date.
Between quoted dates: apply the interpolation convention specified for the valuation. If none is specified, the factors alone may not determine a unique answer.
Outside the quoted range: interpolation is not available from the surrounding factors. Extrapolation requires an additional convention or market information and should not be silently assumed.

Different conventions interpolate different quantities. For example, linear interpolation of factors makes $D(t)$ linear in time over an interval. Log-linear interpolation makes $\log D(t)$ linear, which means the factor itself changes multiplicatively. Neither convention is implied simply by having two endpoint factors. State the convention and use it consistently.

Log-linear interpolation: If $t_a<t<t_b$, the endpoint factors are positive, and log-linear interpolation is specified, then $D(t)=D(t_a)^{1-w}D(t_b)^w$, where $w=(t-t_a)/(t_b-t_a)$. This is the exponential of the linearly interpolated logarithm of the endpoint factors.

To see why, write the interpolation condition as $\log D(t)=(1-w)\log D(t_a)+w\log D(t_b)$. Exponentiating both sides gives $D(t)=\exp((1-w)\log D(t_a)+w\log D(t_b))=D(t_a)^{1-w}D(t_b)^w$. For a midpoint, $w=0.5$, so the interpolated factor is the geometric mean of the endpoint factors. This formula requires positive endpoint factors because their real logarithms must exist.

Worked Example: A Liability with a Midpoint Payment

A hypothetical liability requires a certain payment of $5,000 at year 1.5 and $3,000 at year 2. The valuation date is today. The bootstrapped factors are $D(1)=0.96$ and $D(2)=0.90$. Assume deterministic payments and use log-linear interpolation between years 1 and 2. Find the present value, rounded to the nearest cent.

Method. The year 1.5 payment is a midpoint, so $w=(1.5-1)/(2-1)=0.5$. Apply log-linear interpolation to obtain its factor. The year 2 payment has an exact factor.

The interpolated factor at year 1.5 is $D(1.5)=0.96^{0.5}(0.90)^{0.5}=\sqrt{0.96(0.90)}=\sqrt{0.864}$. Numerically, $\sqrt{0.864}\approx0.929516003$.

The year 1.5 payment has present value $5{,}000(0.929516003)\approx4{,}647.580015$. The year 2 payment has present value $3{,}000(0.90)=2{,}700$.

V_0\approx4{,}647.580015+2{,}700=7{,}347.580015

Rounding only at the final step to two decimal places gives a liability value of $7,347.58. As a numerical check, the interpolated factor is between 0.90 and 0.96, as expected for this midpoint under log-linear interpolation. The common mistake avoided is treating the geometric mean as $\sqrt{0.729}$ or another product that does not equal $0.96(0.90)=0.864$; the product must be checked before taking the square root.

Common Exam Traps

The calculations are compact, but small setup errors can change the answer materially. Check the dates, signs, and interpolation assumption before multiplying.

Using a nearby maturity’s factor without justification. A factor for year 1 is not automatically the factor for year 1.5. Interpolate only under a stated convention.
Applying an endpoint factor to an unmatched payment. In the project example, using $D(1)=0.96$ for the $8,000 receipt at year 1.5 would give $8{,}000(0.96)=7{,}680$, rather than the correct linearly interpolated amount $8{,}000(0.93)=7{,}440$. That incorrect substitution would produce $2{,}462.50+7{,}680-7{,}830=2{,}312.50$, not the correct $2,072.50.
Reversing the sign of an outflow. For a signed project value, a payment is subtracted. For a liability value reported as a positive amount, sum the discounted obligations as positive amounts.
Rounding too early. Keep adequate precision in interpolated factors and intermediate products, then round the final monetary value as requested.
Assuming interpolation beyond the curve. A factor beyond the final quoted date needs an extrapolation rule or additional information; it is not determined by interpolation between known dates.

Key takeaway: Value each known cash flow with the discount factor for its own date. If the date has no supplied factor, state the interpolation assumption before calculating; if it lies outside the quoted range, do not treat extrapolation as automatic.

Check Your Understanding

For each problem, use the stated factors and interpolation convention. Treat receipts as positive and payments as negative when asked for a signed value.

A certain liability pays $1,500 at year 0.5 and $4,000 at year 1. The factors are $D(0.5)=0.98$ and $D(1)=0.95$. Find its present value. (Answer: $5,270.00)
A liability pays $6,000 at year 1 and $2,000 at year 2. The factors are $D(1)=0.94$ and $D(2)=0.87$. Find its present value. (Answer: $7,380.00)
A certain payment of $10,000 is due at year 1.25. The factors are $D(1)=0.97$ and $D(2)=0.91$. Use linear interpolation of factors. Find the present value. (Answer: $9,550.00)
A project receives $3,000 at year 0.5 and pays $7,000 at year 1.5. The factors are $D(0.5)=0.99$, $D(1)=0.96$, and $D(2)=0.88$. Use linear interpolation of factors for the year 1.5 payment. Find the signed present value. (Answer: -$3,470.00)
A certain liability pays $4,000 at year 2. The factors are $D(1)=0.81$ and $D(3)=0.49$. Use log-linear interpolation. Find the present value. (Answer: $2,520.00)

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