Present Values of Future Liabilities

Why Future Liabilities Need Maturity-Specific Discounting

A liability schedule can contain payments of different sizes due at different times. Its present value is not found by adding the future amounts or by automatically discounting every payment at one rate. Each payment must be valued using the discount factor that corresponds to its own maturity. This gives a consistent value today for a stream of promised payments, provided the payment amounts and dates are known.

The setting here is a valuation at time 0 of certain, fixed payments. Assume the relevant spot discount factors are available for every payment date and that the payments can be valued on a consistent market basis. Taxes, transaction costs, and differences in credit or liquidity risk are ignored. If a payment is uncertain, its value is not generally obtained by simply substituting its expected amount into the deterministic formula; the uncertainty and appropriate pricing assumptions must also be addressed.

Definition: The discount factor $D_t$ for maturity $t$ is the time-0 value of one unit of currency payable with certainty at time $t$. If a certain liability payment of $C_t$ is due at time $t$, its present value is $C_tD_t$. For a schedule of payments, the present value is the sum of the present values of the individual payments.

The maturity $t$ is measured in years from the valuation date unless another time unit is explicitly specified. A factor for a payment due in 18 months must therefore be the factor for $t=1.5$ years, not for $t=18$ years or $t=1$ year. The factor may be supplied directly or calculated from an annual-effective spot rate. Write $i_t$ for the annual-effective spot rate applying to a single payment at maturity $t$. Under that quotation convention,

D_t=\frac{1}{(1+i_t)^t}=(1+i_t)^{-t}.

The exponent is the number of years to maturity. In particular, $i_t$ is the spot rate for maturity $t$, not necessarily the same rate as the spot rate for another maturity. If discount factors are supplied directly, use those factors rather than converting them through a rate unnecessarily.

Why the Present Values Add

The sum rule follows from valuing the cash-flow components separately. A payment of $C_t$ at time $t$ is $C_t$ units of the claim that pays one unit at that date. Since one such unit has value $D_t$, the claim paying $C_t$ has value $C_tD_t$. A schedule of certain payments is the combination of these individual claims. By additivity of value, the total value is the sum of their values.

\text{PV}=\sum_{k=1}^{n} C_{t_k}D_{t_k}.

Here $t_k$ is the date of payment $k$, and $C_{t_k}$ is the amount due on that date. The dates need not be equally spaced, and the amounts need not be level. The formula assumes that each amount and date is known and that the corresponding discount factor is available. It is not an annuity formula: no regular-payment pattern is required.

List the liabilities.
Record each payment amount and its exact time from the valuation date.

Match each date to its factor.
Use the factor for that maturity, or calculate it from the spot rate quoted for that maturity.

Discount and add.
Multiply each amount by its own factor, then sum the resulting present values.

Worked Example: Mixed Payments at Irregular Dates

A hypothetical organization must pay $18,000 in 6 months, $6,000 in 15 months, $14,000 in 2 years, and $32,000 in 3 years. At the valuation date, the applicable spot discount factors are $D_{0.5}=0.985$, $D_{1.25}=0.955$, $D_2=0.920$, and $D_3=0.875$. Find the present value. Report the final amount to the nearest cent.

Method. Convert the payment dates to years and multiply each liability by the discount factor for exactly that maturity. The payment dates are $0.5$, $1.25$, $2$, and $3$ years.

At 6 months:

$$ 18{,}000(0.985)=17{,}730. $$

At 15 months:

$$ 6{,}000(0.955)=5{,}730. $$

At 2 years:

$$ 14{,}000(0.920)=12{,}880. $$

At 3 years:

$$ 32{,}000(0.875)=28{,}000. $$

Adding the four present values gives $17{,}730+5{,}730+12{,}880+28{,}000=64{,}340$. The present value is $64,340.00. The common mistake avoided here is using an integer-year factor for the 6-month or 15-month payment; their stated fractional maturities determine which factors apply.

Calculating Factors from Spot Rates

When a problem gives annual-effective spot rates instead of discount factors, calculate one factor for each maturity before valuing the payments. A spot rate $i_t$ is the annual-effective rate for a single payment at time $t$; it is not an instruction to use the same rate for all the other dates. The factor $D_t=(1+i_t)^{-t}$ converts a certain amount due at $t$ into its time-0 value.

Worked Example: A Liability Schedule with a Flat Spot Curve

A hypothetical fund owes $10,000 in 1 year, $25,000 in 2 years, and $40,000 in 3 years. The annual-effective spot rate is 4% at each of the three maturities. Find the present value, reporting the final amount to the nearest cent.

Method. Apply the annual-effective spot-rate formula separately at each maturity. Here the curve is flat, so the same 4% rate applies at all three dates; that is a feature of these data, not a general rule.

D_1=\frac{1}{1.04},\qquad D_2=\frac{1}{(1.04)^2},\qquad D_3=\frac{1}{(1.04)^3}.

The three discounted payments are

\frac{10{,}000}{1.04}=9{,}615.384615\ldots,

\frac{25{,}000}{(1.04)^2} =\frac{25{,}000}{1.0816} =23{,}113.905325\ldots,

and

\frac{40{,}000}{(1.04)^3} =\frac{40{,}000}{1.124864} =35{,}559.854347\ldots.

Keep the unrounded values until the sum is formed:

9{,}615.384615\ldots+ 23{,}113.905325\ldots+ 35{,}559.854347\ldots =68{,}289.144287\ldots.

Rounded to the nearest cent, the present value is $68,289.14. The check is that the result is less than the undiscounted total of $75,000, as expected with positive spot rates and positive future payments. Rounding each intermediate present value to cents before adding can create a small discrepancy; retain precision and round the final sum.

Worked Example: Different Spot Rates at Different Maturities

A hypothetical sponsor has liabilities of $10,000 due in 1 year and $20,000 due in 2 years. The one-year annual-effective spot rate is 25%, and the two-year annual-effective spot rate is $1/9$, or approximately 11.111%. Find the present value.

Method. Calculate a separate discount factor using the rate and maturity for each payment. At 1 year,

D_1=(1.25)^{-1}=0.8, \qquad 10{,}000D_1=10{,}000(0.8)=8{,}000.

At 2 years, the annual-effective spot rate is $1/9$, so $1+i_2=10/9$. Thus

D_2=(1+1/9)^{-2} =(10/9)^{-2} =(9/10)^2 =0.81,

and the present value of the second liability is $20{,}000(0.81)=16{,}200$. Therefore,

\text{PV}=8{,}000+16{,}200=24{,}200.

The present value is $24,200.00. A tempting but incorrect calculation is to apply the one-year 25% rate to both payments, giving $10{,}000/1.25+20{,}000/(1.25)^2=8{,}000+12{,}800=20{,}800$. That calculation uses the wrong maturity-specific rate for the two-year payment. Using the stated two-year spot rate gives the correct value, $24,200.00.

Interpretation and Common Traps

A present value is a time-0 amount equivalent, under the stated discount factors, to the future liability schedule. It is not the sum of the future payments, and it does not imply that the organization will actually make a single payment today. It is a valuation of the scheduled payments on the stated basis.

A useful audit is to check the units and timing before checking arithmetic. Every term $C_tD_t$ is an amount of money today; discount factors are dimensionless. For positive payments and factors between zero and one, each discounted payment is less than its future amount, and so is the total present value. This inequality is only a check under those conditions: a zero or negative spot rate need not produce a factor below one.

Using one rate by habit. A single rate is valid for all dates only when the data or assumptions actually give the same applicable rate, or otherwise specify a common discounting basis. In general, use the rate or factor for each maturity.
Using the wrong time unit. Convert months to years when rates and factors are quoted on a year basis. For example, 15 months is $1.25$ years.
Applying the rate without its maturity exponent. Under an annual-effective spot-rate quote, a payment at time $t$ uses $(1+i_t)^{-t}$, not simply $1/(1+i_t)$ for every maturity.
Rounding too soon. Intermediate rounded amounts can change the final cents. Keep adequate precision through the sum and round only at the end when the question asks for a monetary answer.
Discounting an amount that is not a certain scheduled payment as if it were certain. The deterministic sum formula requires fixed payments. Uncertain benefits, costs, or payment dates require assumptions beyond this calculation.

Key takeaway: For each known liability, identify its payment date, use the discount factor for that exact maturity, multiply by the payment amount, and add the present values. The method works for uneven payment amounts and irregular dates; the maturity-specific match is essential.

Check Your Understanding

For each problem, use the stated valuation date and discounting convention. Report money to the nearest cent where applicable.

A certain liability of $8,000 is due in 1 year and $15,000 in 2 years. The discount factors are $D_1=0.96$ and $D_2=0.90$. Find the present value. (Answer: $21,180.00)
A project owes $5,000 in 6 months, $9,000 in 18 months, and $12,000 in 2 years. The respective discount factors are $0.98$, $0.92$, and $0.88$. Find the present value. (Answer: $23,740.00)
A certain liability is $20,000 in 1 year and $30,000 in 2 years. The annual-effective spot rates are 3% at 1 year and 4% at 2 years. Find the present value. (Answer: $47,154.16)
A schedule requires payments of $7,000 in 9 months, $11,000 in 18 months, and $20,000 in 30 months. The discount factors at those dates are $0.97$, $0.91$, and $0.83$, respectively. Find the present value. (Answer: $33,400.00)
A liability schedule requires $4,000 and $6,000 at the end of year 1, and $10,000 at the end of year 2. The applicable factors are $D_1=0.95$ and $D_2=0.89$. Find the present value of the complete schedule. (Answer: $18,400.00)

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