Why Integrated Loan-and-Bond Problems Need a Consistent Ledger
A loan-and-bond transaction can involve several amounts that are all described as a “balance” or “proceeds,” even though they are not interchangeable. A loan balance is the amount required to settle or continue the debt at a specified date. A bond’s book value is determined by the yield used to carry the bond. Its market sale price is determined by the yield available to buyers at the sale date. Cash received from a coupon is a separate transaction from cash received by selling the bond.
The central discipline is to put every amount on the correct focal date and use the rate that belongs to that calculation. Loan payments and extra principal reductions change the loan balance. A refinancing payment is calculated using the new loan rate and the new balance. A bond’s book value follows its original carrying yield, while a sale price reflects the market yield at sale. This integrated approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
A Reliable Diagnostic Sequence
Before combining amounts, write down the date, the rate, and the meaning of each amount. For a loan, identify whether an extra payment occurs before or after a scheduled payment. For a bond, identify whether the transaction occurs just before or just after a coupon date. Then list receipts and payments from one consistent perspective, such as the borrower’s or the investor’s.
State exactly when the loan payment, extra principal payment, coupon, sale, or payoff occurs. Do not compare amounts from different dates without accumulating or discounting them.
Accumulate the opening balance at the loan rate, subtract payments made by the date, and subtract any extra principal reduction made by that date.
Use the carrying yield for book value and the market yield for sale price. At a coupon date, identify separately any coupon already received and the remaining bond cash flows.
Record sale proceeds and coupons as receipts, and loan payments or payoff amounts as outflows. Check that the loan balance reaches zero when it is fully settled.
Loan Payments, Extra Principal, and Refinancing
Under an annual effective loan rate \(i\), an opening balance \(B\) accumulates to \(B(1+i)\) over one year if no payment is made. A payment at year-end reduces that amount. Thus, if a scheduled payment \(R\) and an additional principal payment \(E\) are both made at the end of the year, the new balance is \(B(1+i)-R-E\). The additional payment reduces principal; it is not a payment to be added to the balance.
If the resulting balance \(L\) is refinanced into \(n\) equal annual payments at a new effective annual rate \(j\), the payment \(Q\) satisfies an equation of value at the refinancing date:
This calculation assumes payments are made at the end of each year and the new rate applies to each interval between payments. It does not apply unchanged if payments are due at the beginning of each year, if payment intervals differ, or if fees are added to the refinanced principal.
Worked Example: Refinance After an Extra Principal Payment
A borrower takes a $20,000 loan at an effective annual rate of 5%. The borrower pays $6,000 at the end of year 1 and $6,000 at the end of year 2. Immediately after the year-2 payment, the borrower pays an additional $2,000 toward principal and refinances the remaining balance for two annual end-of-year payments at 4%. Find the refinanced payments.
Method. First carry the original loan balance to the date after the second scheduled payment. Then subtract the extra principal payment and value the new two-payment loan using the new rate.
After two years, before subtracting payments, the original balance would have accumulated to \(20{,}000(1.05)^2=22{,}050\). The year-1 payment accumulates for one more year, so its value at the end of year 2 is \(6{,}000(1.05)=6{,}300\). The year-2 payment is already at that date. Therefore the balance after the scheduled year-2 payment is
The additional $2,000 principal payment reduces the balance to \(9{,}750-2{,}000=7{,}750\). At the refinancing date, the value of the two new payments is
Thus \(Q=7{,}750/1.8860947\ldots=4{,}109.0196\ldots\), or $4,109.02 per year to the nearest cent. As a check, after the first new payment the balance is \(7{,}750(1.04)-4{,}109.0196\ldots=3{,}950.9804\ldots\). Accumulating that amount one year at 4% gives $4,109.0196…, exactly the second payment required to clear the loan.
Common mistake avoided: Using 5% in the refinancing annuity equation would carry the old loan rate into the new contract. The refinanced balance is $7,750 at the refinancing date, and its two new payments are based on 4%, not 5%.
Bond Book Value and Market Sale Price
For a bond with annual coupons, the book value at a coupon date can be calculated prospectively as the present value of the remaining coupons and redemption value at the carrying yield. The market price at that same date uses the market yield instead. If a sale takes place just after a coupon has been paid, that coupon is not part of the sale price: only future cash flows are included.
The book-value recurrence also explains how the carrying yield works. If \(B_k\) is the book value just after a coupon date, \(C\) is the next coupon, and \(i\) is the carrying yield per coupon period, then the next book value just after the following coupon is
To justify the recurrence, \(B_k\) equals the present value at the carrying yield of all remaining payments. One period later, the same future payments have been discounted for one fewer period, so their value before the next coupon is \(B_k(1+i)\). The coupon \(C\) is then paid and is no longer part of the bond’s remaining value. Subtracting it gives \(B_{k+1}\). This relation describes book value under a fixed carrying yield; it does not say that the market price must follow the same path if market yields change.
Worked Example: Selling a Bond to Pay Down a Loan
An investor owns a $10,000 bond with annual coupons at 8% of face value. The bond was purchased at par when its yield was 8%. Immediately after a coupon date, two years remain. The investor sells the bond when the market yield is 10% per year and uses the proceeds to settle a separate loan balance of $9,000 on that same date. Find the bond’s book value, its sale price, and the cash remaining after the loan is settled. Ignore transaction costs and taxes.
Method. Calculate the book value at the 8% carrying yield, then independently price the remaining bond cash flows at the 10% market yield. Compare the sale proceeds with the loan balance only after placing both at the sale date.
The annual coupon is \(0.08(10{,}000)=\$800\). Since the bond was purchased at par at an 8% yield, its book value immediately after a coupon date remains $10,000: the remaining $800 and $10,800 payments have a present value of
At the sale date, the market yield is 10%. The market price is therefore
The sale produces $9,652.89 to the nearest cent. After settling the $9,000 loan balance, the investor has \(9{,}652.8926\ldots-9{,}000=652.8926\ldots\), or $652.89 remaining. The check is that the market sale price is below the $10,000 book value, as expected when the market yield has risen above the carrying yield.
Common mistake avoided: Applying the $10,000 book value as sale proceeds would overstate available cash by $347.11. Book value is not a promise that the bond can be sold for that amount.
Combining a Bond Purchase, Loan, and Sale
An integrated ledger should distinguish the initial financing from later cash flows. Borrowing to fund part of a bond purchase does not change the bond’s contractual cash flows or carrying yield. At a later sale, the coupon received on that date, the sale proceeds, and the loan payoff are separate entries. A useful check is to add receipts and subtract outflows at the transaction date, while also confirming that the bond price uses only cash flows remaining after that date.
Worked Example: Loan-Financed Bond Purchase and Early Sale
An investor buys a two-year, $10,000 bond with annual coupons of 6%, paying its par price of $10,000. The bond’s purchase yield is 6%. The investor borrows $6,000 at 5% with interest-only payments and principal due at the end of year 2; the remaining $4,000 of the purchase price comes from the investor. Immediately after the year-1 coupon is paid, the investor sells the bond at a market yield of 8% and repays the loan in full. Find the bond book value, sale proceeds, loan payoff, and net cash received at the sale date. Ignore transaction costs and taxes.
Method. Keep the bond valuation and loan settlement separate until the year-1 transaction date. Assume annual effective rates, coupon and interest payments at each year-end, and sale immediately after the year-1 coupon, so no coupon is accrued but unpaid.
The annual bond coupon is \(0.06(10{,}000)=\$600\). At the 6% carrying yield, the book value immediately after the first coupon is the present value of the remaining year-2 payment:
At the sale date, the market yield is 8%, so the sale price of the remaining $10,600 payment is
The coupon of $600 has already been received and is not included in this sale price. The loan has accumulated one year of interest of \(6{,}000(0.05)=\$300\). Since it is an interest-only loan, the $300 interest payment is due at year 1, and the $6,000 principal is also repaid when the loan is settled. The total loan outflow at the sale date is therefore
The receipts at the sale date are the coupon and the sale proceeds: \(600+9{,}814.8148\ldots=10{,}414.8148\ldots\). After paying off the loan, net cash received is
or $4,114.81 to the nearest cent. As a transaction check, the initial $10,000 purchase was funded by $6,000 of borrowing and $4,000 of investor cash. At year 1, the $600 coupon plus the $9,814.81 sale proceeds, less the $6,300 loan settlement, gives the stated $4,114.81 net cash.
Common mistake avoided: Subtracting only the $6,000 principal omits the $300 interest due at year 1. A second error would be to add the $600 coupon to the bond’s market price and then also record it as a separate receipt.
Common Diagnostic Errors
The arithmetic in an integrated problem is often less difficult than keeping the assumptions straight. Use these checks before accepting a result:
- Wrong focal date: A loan balance and bond sale price can be compared directly only when both refer to the same date. A coupon paid earlier is a separate dated receipt.
- Wrong rate: Use the loan’s contractual rate to update the loan balance, the new rate to calculate refinanced payments, the carrying yield for book value, and the market yield for sale price.
- Wrong payment order: A principal reduction made after a scheduled payment is subtracted after that payment. If it occurs earlier, the balance earns interest on a smaller principal for the remaining interval.
- Wrong sign or double count: From the investor’s perspective, coupons and sale proceeds are receipts, while loan interest and principal payoff are outflows. Do not count a coupon both as a separate receipt and as part of the price of a bond sold after that coupon date.
- Assuming book value equals sale proceeds: Book value follows the carrying yield; market price can differ when the market yield differs.
Check Your Understanding
For each problem, identify the relevant date and rate before calculating. Answers are rounded to the nearest cent where needed.
- A $15,000 loan earns 4% effective annually. At the end of year 1, the borrower pays $5,000 and immediately makes an additional $1,000 principal payment. What is the balance immediately afterward? (Answer: $9,600.00)
- A $5,000 bond pays annual coupons at 6% of face value. Two years remain, and the market yield is 8% effective annually. The sale occurs just after a coupon date. What is its market price? (Answer: $4,821.67)
- A loan balance of $9,600 is refinanced at 5% effective annually into two equal annual end-of-year payments. Find each payment. (Answer: $5,162.93)
- A $20,000 bond has one annual coupon period remaining, pays a 5% coupon, and is carried at a 5% yield. Immediately after a coupon date, the market yield is 6%, and the bond is sold to help settle a $19,000 loan balance. Find the book value, sale price, and surplus after settling the loan. (Answer: Book value $20,000.00; sale price $19,811.32; surplus $811.32)