Why a Cash-Flow Sign Must Have a Viewpoint
A loan payment is an outflow to the borrower and an inflow to the lender. A bond coupon is an outflow to the issuer and an inflow to the bondholder. The amount and date are the same; the signs differ because the perspectives differ. In a multi-step problem, changing perspective without changing every sign can produce a plausible-looking but incorrect present value.
The first step is therefore not to write a formula. It is to name the person or entity whose cash flows are being recorded. Then use one rule consistently: cash received by that entity is positive, and cash paid by that entity is negative. The rule applies to advances, repayments, purchases, coupons, redemptions, fees, and sale proceeds.
In this tutorial, a present value is the sum of the signed cash flows discounted to the chosen focal date. Unless stated otherwise, assume cash flows occur at the stated integer year-ends, the effective annual rate is constant, and there are no taxes, transaction costs, or default. These assumptions allow discounting each dated amount using \(v^t=1/(1+i)^t\). If the rate varies over time, the appropriate discount factor for each date must replace this expression; the sign rule itself does not change.
Signed Present Values and Perspective Reversal
Let \(C_t\) denote the signed cash flow at time \(t\), from one fixed perspective, and let \(i\) be the effective annual rate used for discounting. The signed present value at time 0 is
The time-0 flow is not discounted, because \(v^0=1\). A positive or negative result is the signed value from the selected perspective; it is not automatically a “gain” or “loss” without context. For example, the sign tells whether the discounted receipts exceed the discounted payments at the stated rate.
This follows directly from the definition of present value. Discount factors do not depend on the perspective, so changing the sign of each cash flow changes the sign of the sum:
This is the basic justification for the perspective-reversal rule. It also shows why the signs must be assigned before adding or discounting: discounting does not repair a ledger that mixes viewpoints.
Worked Examples
Worked Example: A Loan from the Borrower’s and Lender’s Viewpoints
A borrower receives a loan advance of $12,000 at time 0 and agrees to pay $4,000 at the end of year 1 and $9,000 at the end of year 2. Using an effective annual discount rate of 5%, calculate the signed present value from the borrower’s perspective and from the lender’s perspective. Assume the stated cash flows are the only transactions.
Method. Set up the borrower’s ledger first. The advance is received by the borrower and is positive; both repayments are paid by the borrower and are negative. Then discount each dated flow to time 0.
The borrower’s signed present value is \(+\$27.21\), rounded to the nearest cent. The lender’s cash flows are exactly opposite: \(-\$12{,}000\) at time 0, \(+\$4{,}000\) at year 1, and \(+\$9{,}000\) at year 2. Thus
or \(-\$27.21\) to the nearest cent. As a check, the two present values sum to zero before rounding. The method avoids treating the loan advance as a payment by the borrower, a frequent sign error.
Worked Example: A Bond Issued for a Rounded Price
An issuer receives $9,727.68 for a three-year bond. It pays a $400 coupon at the end of each of years 1 and 2. At the end of year 3, it pays a $400 coupon and $10,000 redemption. At an effective annual rate of 5%, find the signed present value of these issue cash flows from the issuer’s perspective and from the bondholder’s perspective.
Method. The issuer receives the issue price at time 0, so that flow is positive. It makes all three contractual payments, so those flows are negative. Discount the coupon and redemption amounts separately to show the full cash-flow structure.
The discounted value of the contractual payments is approximately \(380.9524+362.8118+8{,}983.9110=\$9{,}727.6752\). The issue price is $9,727.68, so the issuer’s signed present value is approximately \(+\$0.0048\). To the nearest cent, this is \(+\$0.00\).
Every issue cash flow has the opposite sign for the bondholder. The bondholder pays the issue price and receives the coupons and redemption, so the bondholder’s signed present value is approximately \(-\$0.0048\), which also rounds to \(-\$0.00\) to the nearest cent. The small difference from zero arises because the issue price is stated to the nearest cent, while the discounted contractual payments are approximately $9,727.6752.
Common mistake avoided. The discounted payments are about $9,727.68, not $9,726.68. Subtracting the correct amount from the issue price gives approximately \(+\$0.0048\), not a difference of about one dollar. Keep the unrounded amounts through the subtraction and round only at the end.
Worked Example: A Bondholder Sells After Receiving a Coupon
An investor buys a bond for $10,000 at time 0. The investor receives a $500 coupon at the end of year 1 and sells the bond immediately after that coupon is paid for $10,000. Using an effective annual discount rate of 5%, calculate the investor’s signed present value. Also identify whether the sale proceeds are a cash flow of the issuer.
Method. From the investor’s perspective, the purchase price is negative. At the end of year 1, the coupon and sale proceeds are both received, so both are positive. Since they occur on the same date, they can be added before discounting.
The investor’s signed present value is $0 at the stated rate. As a check, the year-1 receipts accumulate to \(10{,}500\), which equals \(10{,}000(1.05)\), the time-0 purchase price accumulated for one year at 5%.
The $10,000 sale proceeds are paid by the subsequent buyer to the investor; they are not a payment by the issuer. The issuer’s contractual cash flow at that date is the $500 coupon. A bond can change hands between investors without changing the issuer’s obligation to pay the coupon or redemption. The common mistake is to place the secondary-market sale price in the issuer’s ledger.
Applying the Sign Rule to Common Transactions
The following table summarizes the usual signs. “Usual” is important: the sign is determined by the entity named in the problem, not by the label attached to the transaction. For example, a “repayment” is negative for the borrower but positive for the lender.
| Transaction | Borrower or issuer | Lender or bondholder |
|---|---|---|
| Loan advance or bond issue price | Positive when received | Negative when paid |
| Loan repayment or bond coupon | Negative when paid | Positive when received |
| Bond redemption paid at maturity | Negative for issuer | Positive for bondholder |
| Bond purchase in a secondary-market sale | Not an issuer cash flow | Negative for buyer; positive for seller |
| Bond sale proceeds | Not an issuer cash flow | Positive for seller; negative for buyer |
A ledger is easiest to audit when its perspective is stated explicitly in its heading, such as “borrower” or “bondholder.” For each entry, ask two questions: who pays or receives the money, and on what date? Only after answering both should the amount be assigned a sign and discounted.
Identify whose signed present value is requested. Do not alternate between the parties while building one ledger.
Record receipts as positive and payments as negative for that entity. In particular, check the initial advance or purchase price.
Use the appropriate discount factor for each cash-flow date and the stated rate assumptions.
Reverse every sign for the other party only if the transactions are between those two parties. Do not assign an investor-to-investor sale to the issuer.
Common Sign Errors and Their Corrections
A common trap is to calculate a borrower’s value using the lender’s signs, then describe the result as the borrower’s. For example, with an advance of $8,000 and a $3,000 repayment one year later at 4%, the borrower’s signed present value is \(8{,}000-3{,}000/1.04\), not \(-8{,}000+3{,}000/1.04\). The latter is the lender’s signed present value.
Another trap is to treat every amount associated with a bond as an issuer cash flow. The issuer receives the original issue price and pays contractual coupons and redemption. A later buyer pays the seller in a secondary-market transaction; this sale price changes the buyer’s and seller’s ledgers, not the issuer’s contractual ledger.
Finally, do not infer that a positive present value is an accounting profit or that a negative value is automatically a loss. It is a signed value relative to the specified perspective, interest rate, cash-flow dates, and focal date. A different rate can change the numerical present value, although it does not change who pays or receives each amount.
Check Your Understanding
For each problem, state the requested perspective before calculating. Use the stated effective annual rate and round money to the nearest cent.
- A borrower receives $8,000 at time 0 and pays $3,000 at the end of year 1 and $5,600 at the end of year 2. At 4%, find the borrower’s and lender’s signed present values. (Answer: borrower \(-\$62.13\); lender \(+\$62.13\))
- A bond issuer receives $19,000 at time 0, pays a $900 coupon at the end of year 1, and pays a $900 coupon plus $20,000 redemption at the end of year 2. At 5%, find the issuer’s and bondholder’s signed present values. (Answer: issuer \(-\$814.06\); bondholder \(+\$814.06\))
- An investor pays $9,600 for a bond at time 0, receives a $400 coupon at the end of year 1, and immediately after the coupon sells the bond for $9,500. At 4%, find the investor’s signed present value. (Answer: \(-\$80.77\))
- A borrower receives $15,000 at time 0 and pays $5,000 at the end of year 1 and $11,000 at the end of year 2. At 5%, find the borrower’s signed present value and the lender’s signed present value. (Answer: borrower \(+\$260.77\); lender \(-\$260.77\))
- A bondholder buys a bond for $10,200 at time 0, receives a $300 coupon at the end of year 1, and sells the bond immediately after that coupon for $10,100. At 5%, find the buyer’s signed present value. Is the $10,100 sale price a cash flow of the issuer? (Answer: buyer \(-\$295.24\); no, it is a payment between investors)