Bond Cash-Flow Ledgers With Unequal Dates

Why Unequal Dates Need a Ledger

A bond transaction may involve an initial purchase, one or more coupons, a sale between coupon dates, and possibly redemption at maturity. These amounts do not necessarily occur at equally spaced dates. Adding the dollar amounts directly would ignore both their different timing and the investor’s perspective. A dated cash-flow ledger keeps each amount attached to its date, so that every amount can be moved to a common valuation date before comparison.

The ledger is a method of organizing and valuing cash flows, not a new bond-pricing convention. State whose perspective is being used, record every amount with its sign, and state the interest model or discount factors. Then discount or accumulate each entry consistently. The same approach applies whether the final cash flow is a coupon plus redemption, a coupon plus sale proceeds, or a payment made by the investor.

Definition: A dated cash-flow ledger is a list of signed cash flows $C_k$, each occurring at time $t_k$, measured from a stated time origin. From the bondholder’s perspective, cash paid to acquire or hold the bond is negative; coupons, redemption proceeds, and sale proceeds received are positive. A cash flow is recorded once at the date it occurs.

The date convention matters. If a bond is purchased at $t=0.25$, the purchase price belongs at $t=0.25$, not at $t=0$. A coupon received at $t=0.75$ belongs at $t=0.75$. If the investor sells the bond at $t=1.5$, the sale proceeds belong at $t=1.5$; they are not redemption proceeds unless the bond is actually redeemed then. If a coupon is also paid on the sale date, record both amounts, or combine them into one total for that date.

Valuing the Ledger at a Common Date

Let $D(t)$ denote the discount factor from the origin $t=0$ to time $t$, with $D(0)=1$. When the stated model is a constant effective annual rate $i$, with annual compounding extended to fractional years, $D(t)=(1+i)^{-t}$. This fractional-year convention must be stated; it is not implied by every possible bond-market quotation. Alternatively, a problem may supply discount factors directly. Use those factors as given rather than substituting a constant-rate model.

For cash flows $C_k$ at dates $t_k$, the value at the origin is the sum of their discounted values. If the requested valuation date $T$ is another date and the same consistent discount-factor curve applies, the value at $T$ is obtained by expressing all cash flows relative to $D(T)$. This comparison assumes the stated yield or discount factors are applicable to all the ledger entries and that the cash flows are valued on the same basis. If the question specifies a different yield for a cash flow, or a different transaction convention, follow that specification rather than applying one factor indiscriminately.

V_0=\sum_k C_kD(t_k) \qquad\text{and}\qquad V_T=\frac{V_0}{D(T)} =\sum_k C_k\frac{D(t_k)}{D(T)}

The value $V_0$ is the net present value from the stated perspective. It is not necessarily the bond’s quoted price: it may include the purchase, subsequent coupons, and proceeds from a later sale. A positive result means the discounted receipts exceed the discounted payments under the model; a negative result means the reverse. The sign is meaningful only after the perspective and signs are made explicit.

Fix the perspective and origin.
For a bondholder ledger, treat amounts paid as negative and amounts received as positive. Measure every date from one stated origin.

List the actual events.
Record the purchase price, each coupon, and any redemption or sale proceeds on their actual dates. Keep distinct events distinct even when they share a date.

Choose the valuation basis.
Use the specified effective rate and time unit, or use the discount factors supplied. Do not mix yield conventions and factor sets.

Move every entry to one date.
Discount all entries to the origin, or use relative factors to value them at another date. Add only after all entries are on the same basis.

Check the ledger.
Confirm that each event appears once, that the signs match the perspective, and that the valuation-date result agrees when translated from the origin.

Why the Relative-Factor Formula Holds

Under a consistent discount-factor model, a dollar due at time $t$ has value $D(t)$ at the origin. Its value at $T$ must be the amount that, when discounted from $T$ back to the origin, gives $D(t)$. If that value is $W$, then $WD(T)=D(t)$, so $W=D(t)/D(T)$. Multiplying by the signed amount $C_k$ gives its value at $T$; summing over entries gives the ledger value.

W D(T)=D(t) \quad\Longrightarrow\quad W=\frac{D(t)}{D(T)} \quad\Longrightarrow\quad V_T=\sum_k C_k\frac{D(t_k)}{D(T)}

This also explains an important check: if the origin value is $V_0$, then $V_T=V_0/D(T)$. A direct calculation at $T$ should agree, apart from rounding. For dates earlier than the origin or for a valuation basis that changes over time, the problem must provide enough information to value those past cash flows; the simple future-cash-flow interpretation should not be extended without checking the specified model.

Worked Example: Purchase, Coupons, and a Later Sale

A hypothetical investor buys a bond for $\$970$ at $t=0.25$, receives a $\$35$ coupon at $t=0.75$, then receives another $\$35$ coupon and sells the bond for $\$980$ at $t=1.5$. Discount factors from $t=0$ are $D(0.25)=0.98$, $D(0.75)=0.94$, and $D(1.5)=0.88$. Find the net present value at the origin and the equivalent value at the purchase date.

Method. Use the bondholder’s perspective and first value every ledger entry at the common origin. The purchase is negative. At $t=1.5$, the coupon and sale proceeds are both receipts, so their combined amount is $\$35+\$980=\$1{,}015$.

V_0=(-970)(0.98)+(35)(0.94)+(1{,}015)(0.88)

The three contributions are $-\$950.60$, $\$32.90$, and $\$893.20$. Therefore $V_0=-950.60+32.90+893.20=-\$24.50$. The net present value is negative $\$24.50$.

To express the same value at the purchase date $T=0.25$, divide by $D(0.25)=0.98$:

V_{0.25}=\frac{-24.50}{0.98}=-\$25.00

Check by valuing each entry at $t=0.25$. The purchase remains $-\$970$. The coupon at $t=0.75$ contributes $35(0.94/0.98)=\$33.571428\ldots$. The final combined receipt contributes $1{,}015(0.88/0.98)=\$911.428571\ldots$. Thus $-970+33.571428\ldots+911.428571\ldots=-\$25.00$, agreeing with the origin calculation.

Common mistake avoided: Dividing by the purchase-date factor changes the valuation date; it does not mean that every receipt is separately rounded before the ledger is added. Keep adequate precision until the final amount.

Worked Example: Unequal Dates Under a Stated Effective Yield

A hypothetical bondholder pays $\$925$ at $t=0$ for a bond that pays a $\$100$ coupon at $t=0.5$, followed by a $\$100$ coupon and $\$1{,}000$ redemption at $t=1.5$. Value the ledger at $t=0$ using an effective annual rate $i=21\%$, with annual compounding extended to fractional years. Round the final amount to the nearest cent.

Method. Place the purchase at the origin and combine the coupon and redemption that occur together at $t=1.5$. Discount each receipt using the actual elapsed time. The total at $t=1.5$ is $\$1{,}100$.

V_0=-925+\frac{100}{(1.21)^{0.5}}+ \frac{1{,}100}{(1.21)^{1.5}}

Since $(1.21)^{0.5}=1.1$, the first coupon has present value $100/1.1=\$90.90909\ldots$. Also, $(1.21)^{1.5}=1.21(1.1)=1.331$, so the final combined receipt has present value $1{,}100/1.331=\$826.44628\ldots$. Therefore,

V_0=-925+90.90909\ldots+826.44628\ldots =-7.64462\ldots

The ledger’s net present value is $-\$7.64$, rounded to the nearest cent. The discounted receipts total $\$917.36$, which is less than the $\$925$ purchase payment. As a check, $1{,}331(826.44628\ldots)=1{,}100$, and $1.1(90.90909\ldots)=100$.

Common mistake avoided: Using one period for both receipts would ignore the half-year and one-and-a-half-year dates. The stated fractional-year convention requires exponents $0.5$ and $1.5$, respectively.

Worked Example: Revaluing an Irregular Ledger at an Intermediate Date

A hypothetical investor buys a bond for $\$1{,}020$ at $t=0$, receives a $\$40$ coupon at $t=0.4$, and sells the bond for $\$1{,}005$ at $t=0.9$. A consistent discount-factor curve gives $D(0.4)=0.97$ and $D(0.9)=0.93$. Find the net present value at the origin and the equivalent value at $t=0.4$.

Method. The origin ledger has entries $-\$1{,}020$, $+\$40$, and $+\$1{,}005$. Discounting the receipts to $t=0$ gives

V_0=-1{,}020+(40)(0.97)+(1{,}005)(0.93) =-1{,}020+38.80+934.65=-\$46.55

Now divide by $D(0.4)=0.97$ to value the whole ledger at the coupon date:

V_{0.4}=\frac{-46.55}{0.97}=-47.989690\ldots

Thus the value at $t=0.4$ is $-\$47.99$, rounded to the nearest cent. Check directly: the purchase contributes $-1{,}020/0.97=-1{,}051.546391\ldots$, the coupon contributes $\$40$, and the sale contributes $1{,}005(0.93/0.97)=\$963.556701\ldots$. Their sum is $-\$47.989690\ldots$, as required.

Common mistake avoided: The sale price is a receipt at $t=0.9$, not at the origin or at the coupon date. Its relative factor for valuation at $t=0.4$ is $D(0.9)/D(0.4)$.

Common Ledger Errors

A ledger is most useful when it makes the assumptions and timing visible. The following checks address errors that can otherwise produce a plausible-looking but incorrect total.

Adding amounts from different dates. Cash flows at unequal dates cannot be summed as though they were simultaneous. Move each entry to a common date first.
Misplacing the purchase price. Record it on the stated purchase date. Do not silently treat a later purchase as an origin-time payment.
Confusing sale and redemption. Sale proceeds are received when the investor sells; redemption is paid by the issuer at the specified redemption date. They are different events unless the scenario says they coincide.
Using the wrong sign. From the bondholder’s perspective, a purchase is negative and receipts are positive. Reversing perspective reverses every sign, not just the purchase.
Using the wrong time unit. Under annual compounding extended to fractional years, a cash flow at $t=0.5$ uses a half-year exponent. A nominal rate convertible $m$-thly requires its own stated conversion rule instead.
Rounding too early. Keep full precision in discount factors and intermediate values, then round the requested final amount.

Key takeaway: A reliable unequal-date ledger has one perspective, one entry for every cash-flow event, actual dates, and one consistent valuation basis. The common-date total and its equivalent value at another date must agree under the stated discount-factor model.

Check Your Understanding

For each problem, write a signed ledger before valuing it. Round dollar answers to the nearest cent when needed.

A hypothetical investor pays $\$1{,}000$ at $t=0$, receives $\$300$ at $t=0.5$, and receives $\$800$ at $t=1.25$. Given $D(0.5)=0.96$ and $D(1.25)=0.90$, find the net present value at the origin. (Answer: $\$8.00$)
An investor pays $\$960$ at $t=0.2$, receives a $\$25$ coupon at $t=0.7$, and sells the bond for $\$980$ at $t=1.2$. Given $D(0.2)=0.99$, $D(0.7)=0.95$, and $D(1.2)=0.90$, find the origin value and the equivalent value at the purchase date. (Answer: origin value $-\$44.65$; purchase-date value $-\$45.10$)
A bond costs $\$1{,}080$ at $t=0$, pays a $\$110$ coupon at $t=0.5$, and pays a $\$131$ coupon plus $\$1{,}200$ redemption at $t=1.5$. Using effective annual $i=21\%$ with annual compounding extended to fractional years, find the net present value. (Answer: $\$20.00$)
An investor pays $\$900$ at $t=0.25$, receives $\$30$ at $t=0.75$, and receives sale proceeds of $\$950$ at $t=1.25$. Given $D(0.25)=0.98$, $D(0.75)=0.94$, and $D(1.25)=0.90$, find the value at the origin and the equivalent value at the purchase date. (Answer: origin value $\$1.20$; purchase-date value $\$1.22$)
An investor pays $\$1{,}000$ at $t=0$, receives a $\$40$ coupon at $t=0.5$, and will receive sale proceeds $X$ at $t=1.5$. Given $D(0.5)=0.96$ and $D(1.5)=0.84$, determine $X$ so that the ledger has zero net present value. (Answer: $X=\$1{,}144.76$)

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