Financing a Bond Purchase With a Loan

Why the Combined Cash-Flow Ledger Matters

Borrowing to purchase a bond creates two sets of cash flows: receipts from the bond and payments on the loan. A coupon is not automatically a gain of that amount, because loan interest may be due on the same date. Likewise, the bond’s redemption proceeds must be compared with the loan balance due, not treated as a free terminal receipt. The correct comparison depends on when each cash flow occurs.

A dated ledger makes those timings explicit. It begins with the loan advance and bond purchase cost, records every later receipt and payment on its due date, and then—if a terminal surplus or shortfall is required—moves the net flows to one stated focal date at one stated accumulation rate. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A financed bond-purchase ledger records, at each relevant date, the bond purchase cost and loan advance, followed by bond coupons and redemption proceeds and loan interest and principal repayments. The net cash flow at a date is the sum of all receipts minus all payments on that date. A terminal surplus or shortfall is the accumulated value at a chosen terminal date of the ledger’s net cash flows. A positive value is a surplus; a negative value is a shortfall.

Set Up the Dates Before Doing the Arithmetic

Use a consistent sign convention: receipts are positive and payments are negative. At the purchase date, include both the loan advance and the full bond acquisition cost. If the advance is less than the cost, the difference is an additional cash outlay from the purchaser. At later dates, enter each bond receipt and each loan payment once.

A loan payment may contain both interest and principal. For a loan with an effective rate $j$ per payment period, the interest due for a period is $j$ times the outstanding balance at the start of that period. The remainder of a payment that reduces the balance is principal. For an interest-only loan, the periodic payment before maturity is just the interest; principal is repaid at maturity.

To find a terminal value, choose an accumulation rate and apply it consistently from each cash-flow date to the focal date. The chosen rate is a valuation assumption for comparing dated amounts; it need not be the bond’s coupon rate. If a problem specifies the loan rate as the accumulation rate, use that rate. If it does not specify a rate or another comparison rule, state the assumption you adopt.

Record the purchase-date flows.
Enter the loan advance as a receipt and the bond’s full purchase cost as a payment. The resulting net flow shows any cash contribution needed at the outset.

Schedule each later receipt and payment.
List bond coupons and redemption proceeds separately from loan payments. Use the actual due dates, including any dates on which only one side has a cash flow.

Net each date, then accumulate.
Combine flows only when they occur on the same date. Accumulate each resulting net amount to the chosen focal date, or accumulate bond receipts and loan payments separately as a check.

Why Accumulating Net Flows Works

Suppose the net cash flow at date $t_k$ is $C_k$, and the terminal date is $T$. If the accumulation factor from $t_k$ to $T$ is $A(t_k,T)$, the terminal value is the sum of each dated flow multiplied by its own factor:

V_T=\sum_k C_k A(t_k,T).

This follows directly from the ledger. At each date, receipts and payments are added with opposite signs. Accumulating the combined amount is equivalent to accumulating the receipts and payments separately, because multiplication by the same accumulation factor distributes over addition:

(C_k)A(t_k,T) = \bigl(\text{receipts at }t_k-\text{payments at }t_k\bigr)A(t_k,T).

Summing over dates therefore gives the accumulated bond receipts less the accumulated loan payments, with the initial net purchase-date flow included. This equivalence also provides a useful arithmetic check. It requires that the same dates, signs, and accumulation assumptions be used in both calculations.

Worked Example: Annual Principal Repayments and a Maturing Bond

A purchaser borrows $900 at an effective annual loan rate of 6% and uses the entire advance to buy a bond for $900 at time 0. The bond pays a $60 coupon at the end of each of years 1 and 2, then pays a $60 coupon and $1,000 redemption amount at the end of year 3. The loan repays $300 of principal at the end of each year, with interest charged at 6% on the balance at the start of that year. Accumulate the dated net flows to year 3 at 6%. Find the terminal surplus or shortfall.

Method. First calculate the loan payment on each date; then net the loan payment against the bond receipt on that date. The purchase-date advance and cost cancel, so there is no initial net cash flow.

At the start of year 1 the balance is $900. Interest is $0.06(900)=\$54$, so the total payment is $300+54=\$354$, leaving $600. At the start of year 2 the balance is $600. Interest is $0.06(600)=\$36$, so the payment is $300+36=\$336$, leaving $300. At the start of year 3 the balance is $300. Interest is $0.06(300)=\$18$, so the payment is $300+18=\$318$, leaving zero.

The dated net flows are:

\begin{aligned} C_0&=900-900=0,\\ C_1&=60-354=-294,\\ C_2&=60-336=-276,\\ C_3&=(60+1{,}000)-318=742. \end{aligned}

Accumulate each flow to year 3. The year 1 flow accumulates for two years, the year 2 flow for one year, and the year 3 flow needs no accumulation:

\begin{aligned} V_3 &=(-294)(1.06)^2+(-276)(1.06)+742\\ &=(-294)(1.1236)-292.56+742\\ &=-330.3384-292.56+742\\ &=119.1016. \end{aligned}

Rounded to the nearest cent, the result is a $119.10 terminal surplus. The purchase-date flows cancel because the loan exactly funds the purchase. The calculation uses the year 3 coupon as well as the redemption amount.

Check. Accumulated bond receipts are $60(1.1236)+60(1.06)+1{,}060=67.416+63.60+1{,}060=\$1{,}191.016$. Accumulated loan payments are $354(1.1236)+336(1.06)+318=397.7544+356.16+318=\$1{,}071.9144$. Their difference is $1{,}191.016-1{,}071.9144=\$119.1016$, agreeing with the ledger.

Common mistake avoided: treating the third-year bond receipt as only the $1,000 redemption omits the $60 coupon due on that same date. The year 3 receipt is $1,060.

Unequal Dates and the Double-Counting Trap

A bond and its financing need not have cash flows on every common date. The ledger should still show the loan payment on each due date, even when the bond pays no coupon then. One safe procedure is to calculate the net flow on every date and accumulate those net flows directly. A second, equivalent procedure is to accumulate all bond receipts and all loan payments separately and subtract the totals. Do not combine the procedures by subtracting loan payments again after they are already included in the net flows.

Worked Example: Quarterly Loan Payments and Semiannual Bond Coupons

A $600 interest-only loan charges an effective rate of 2% per quarter. Its interest is paid at each quarter-end, and its $600 principal is repaid at the end of quarter 8. The advance funds a $600 bond purchase. The bond pays $50 at the end of quarters 4 and 8, and repays $600 principal at the end of quarter 8. Find the terminal value at quarter 8, accumulating at 2% per quarter.

The quarterly loan interest is $0.02(600)=\$12$. At quarters 1, 2, 3, 5, 6, and 7, the bond pays nothing, so the net flow is $-\$12$. At quarter 4, the bond pays $50 and the loan requires $12 interest, giving a net flow of $50-12=\$38$. At quarter 8, the bond pays $50+600=\$650$, while the loan requires $12+600=\$612$, giving another net flow of $650-612=\$38$. The time-0 advance and purchase cost cancel.

Bring each flow to quarter 8. The quarter 1, 2, and 3 flows accumulate for 7, 6, and 5 quarters; the quarter 4 flow for 4 quarters; and the quarter 5, 6, and 7 flows for 3, 2, and 1 quarter:

\begin{aligned} V_8={}&-12\bigl[(1.02)^7+(1.02)^6+(1.02)^5 +(1.02)^3+(1.02)^2+1.02\bigr]\\ &+38(1.02)^4+38. \end{aligned}

The powers are $1.1486856676,\ 1.1261624193,\ 1.1040808032,\ 1.061208,\ 1.0404,$ and $1.02$, respectively. Their sum is $6.5005368901$. Also, $38(1.02)^4=38(1.08243216)=\$41.13242208$. Therefore,

$$ V_8=-12(6.5005368901)+41.13242208+38 =1.12597940. $$

Rounded to the nearest cent, the terminal result is a $1.13 surplus.

Check. Accumulated bond receipts are $50(1.02)^4+650=54.121608+650=\$704.121608$. Accumulated loan payments are $12[(1.02)^7+(1.02)^6+(1.02)^5+(1.02)^4+(1.02)^3+(1.02)^2+1.02]+612$. The seven powers sum to $7.5829690501$, so the loan total is $12(7.5829690501)+612=\$702.99562860$. The difference is $1.12597940, agreeing with the net-flow method.

Common mistake avoided: after using the net flows of $38 at quarters 4 and 8, do not subtract those dates’ $12 interest payments again. They are already included in $50-12$ and $650-612$. Similarly, when using the separate-total check, include each interest payment exactly once.

When the Loan Does Not Fund the Full Purchase Cost

If the loan is smaller than the bond’s purchase cost, the difference is an initial outflow of the purchaser’s own funds. That amount remains part of the terminal calculation. Omitting it would report the financing-and-bond result while ignoring the cash invested at time 0. The following example includes that initial contribution explicitly.

Worked Example: A Cash Contribution at Purchase

A purchaser pays $1,020 for a bond using a $1,000 loan and $20 of personal cash at time 0. The bond pays an $80 coupon at the end of each of years 1 and 2 and repays $1,000 principal with the year 2 coupon. The loan is interest-only at an effective annual rate of 4%, with interest paid at each year-end and the $1,000 principal repaid at year 2. Accumulate the full dated cash flows to year 2 at 4%. Find the terminal result.

At time 0, the loan advance is $1,000 and the purchase cost is $1,020, so the net flow is $1{,}000-1{,}020=-\$20$. The annual loan interest is $0.04(1{,}000)=\$40$. In year 1, the bond coupon is $80 and the loan payment is $40, giving a net flow of $40. In year 2, the bond pays $80+1{,}000=\$1{,}080$, and the loan payment is $40+1{,}000=\$1{,}040$, again giving a net flow of $40.

Accumulate the time-0 contribution for two years and the year 1 flow for one year:

\begin{aligned} V_2 &=(-20)(1.04)^2+40(1.04)+40\\ &=-20(1.0816)+41.60+40\\ &=-21.632+41.60+40\\ &=59.968. \end{aligned}

Rounded to the nearest cent, the terminal result is a $59.97 surplus. As a check, bond receipts accumulated to year 2 total $80(1.04)+1{,}080=\$1{,}163.20$, while loan payments accumulated to year 2 total $40(1.04)+1{,}040=\$1{,}081.60$. Their difference is $81.60. Subtract the initial $20 contribution accumulated for two years, $20(1.0816)=\$21.632$, to obtain $81.60-21.632=\$59.968$, as before.

Common mistake avoided: counting the loan advance as if it covered the entire purchase would remove the $20 initial outflow. The advance is a receipt, but the bond costs $1,020; the uncovered amount must appear in the ledger.

Practical Checks Before Reporting the Result

The sign and size of a terminal value are meaningful only if the dates and assumptions are consistent. Before rounding, check the following:

Purchase date: the net flow equals the loan advance minus the full acquisition cost. Include any unfunded amount.
Loan balance: interest is calculated on the balance at the start of the period, and principal repayments reduce that balance. At maturity, include any remaining principal due.
Bond maturity date: include both the final coupon, if payable, and redemption proceeds. Do not replace their sum with redemption alone.
Payment dates: accumulate each cash flow for the number of periods from its own date to the focal date. A quarter-end flow is not accumulated as though it occurred at the start of that quarter.
Independent check: compare accumulated net flows with accumulated bond receipts less accumulated loan payments, including any initial net purchase-date flow.

These checks are especially useful when the coupon and loan-payment frequencies differ or when the loan has a balloon balance. A zero or positive result is not a guarantee of an actual realized gain: the calculation depends on the stated payment schedule and accumulation assumption. If a payment is missed, a rate changes, or the bond is sold early, the ledger must be revised to reflect those changed cash flows and dates.

Key takeaway: Put the loan advance, purchase cost, bond receipts, and loan payments on their actual dates. Net flows only on matching dates, then accumulate once to the chosen terminal date. This keeps principal, interest, coupons, redemption, and any initial cash contribution visible and prevents double-counting.

Check Your Understanding

For each problem, use the stated accumulation rate and report the terminal result to the nearest cent.

A $500 interest-only loan at 4% effective annually funds a $500 bond purchase. The bond pays $30 at the end of each of years 1 and 2 and repays $500 principal with the year 2 coupon. Loan interest is paid annually and principal is repaid at year 2. Find the terminal value at year 2, accumulating at 4% annually. (Answer: $20.40 surplus)
A purchaser pays $950 for a bond using a $900 interest-only loan and $50 of personal cash. The bond pays $60 at the end of each of years 1 and 2 and repays $1,000 principal at year 2. The loan rate is 5% effective annually, with annual interest and principal due at year 2. Find the terminal value at year 2, accumulating at 5%. (Answer: $75.63 surplus)
A $600 interest-only loan charges 2% per quarter, with quarterly interest and principal due at quarter 8. It funds a $600 bond that pays $50 at quarter 4, then $50 plus $600 redemption at quarter 8. Find the terminal value at quarter 8, accumulating at 2% per quarter. (Answer: $1.13 surplus)
A $1,000 loan at 6% effective annually funds a $1,000 bond purchase. The loan repays $400 principal at year 1 and the remaining $600 at year 2; interest is charged on each year’s opening balance. The bond pays $70 at each year-end and repays $1,000 principal at year 2. Find the terminal value at year 2, accumulating at 6%. (Answer: $20.60 surplus)

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