Refinancing a Loan With Bond Sale Proceeds

Why the Transaction Date Matters

A bond sale can provide cash to pay off a loan, but the comparison is meaningful only when both amounts are measured at the same date. The loan payoff is not necessarily its original principal, and the bond’s sale value is not necessarily its face amount or accounting book value. Each must be calculated from the cash flows and terms that apply at the transaction date.

The basic decision is direct: subtract the loan payoff amount, including any payoff fee, from the net bond sale proceeds. A positive difference is a surplus; a negative difference is a shortfall. If the sale and payoff occur on different dates, first move one amount to the other date using the stated accumulation assumption.

Definition: At a specified transaction date, the net bond sale proceeds are the bond’s market sale price less sale costs. The loan payoff amount is the amount required to discharge the loan on that date, including any stated payoff fee. The refinancing surplus or shortfall is net bond sale proceeds minus the loan payoff amount.

The model in the examples assumes that the stated loan payments are made as scheduled up to the payoff date, that the stated interest rates apply to the relevant periods, and that fees are paid on the transaction date. Unless a problem says otherwise, there are no taxes, other transaction costs, or additional cash flows. If the bond is sold between coupon dates, use the transaction-date price convention given in the problem; accrued interest may affect the cash paid by the buyer. Do not compare a clean price with a payoff amount that includes a different treatment of accrued interest.

Valuing Both Sides on One Date

For a loan with $n$ level payments of $R$ remaining, payable at the end of each period, and effective interest rate $i$ per payment period, the outstanding balance immediately after a payment date is the present value of the remaining payments. If a payoff fee $F_L$ is due at that date, the cash needed to discharge the loan is the balance plus $F_L$.

\text{Loan payoff}=R a_{\overline{n}|}+F_L, \qquad a_{\overline{n}|}=\frac{1-v^n}{i}, \qquad v=\frac{1}{1+i}.

The timing convention matters: this formula values payments that occur after the balance date. A payment already made is not included. If the date is between scheduled payment dates, the balance must be brought to that date under the loan’s interest convention.

For a bond sold on a coupon date with face amount $C$, level coupon $K$, $n$ remaining coupon periods, and market yield $j$ per coupon period, the market price is the present value of its remaining coupons and redemption amount. If the sale cost is $F_B$, net proceeds are the market price less that cost.

\text{Net bond proceeds} = K a_{\overline{n}|}^{(j)}+C(1+j)^{-n}-F_B, \qquad a_{\overline{n}|}^{(j)} = \frac{1-(1+j)^{-n}}{j}.

Here $j$ is the bond’s market yield per coupon period, not necessarily its coupon rate. The coupon rate determines the coupon amount; the market yield discounts the remaining cash flows. This distinction is central to a sale-based refinancing calculation.

Why These Present-Value Formulas Work

The formulas follow from the same value principle: cash flows due in the future are discounted to the date at which the transaction is being measured. For a loan, each remaining payment $R$ due in periods $1,\ldots,n$ has present value $Rv^t$. Therefore the balance is the sum of those values:

R\sum_{t=1}^{n}v^t = R\frac{v(1-v^n)}{1-v} = R\frac{1-v^n}{i} = R a_{\overline{n}|},

because $1-v=i/(1+i)=iv$. For a bond, each coupon is discounted in the same way at the bond’s market yield, and the redemption amount is discounted from its maturity date. Thus the bond’s price is the sum of the present values of its promised remaining cash flows. Fees are then added to the cash requirement or deducted from sale proceeds according to who pays them.

Worked Example: A Sale and Loan Payoff on the Same Semiannual Date

A loan has five semiannual payments of $1,800 remaining. Its nominal annual interest rate is 8% convertible semiannually, and the payoff fee is $50. On the same date, a bond is sold immediately after a coupon payment. It has face amount $9,000, semiannual coupons of $270, six coupon payments remaining, and market yield 5% nominal convertible semiannually. The sale fee is $35. Find the refinancing surplus or shortfall.

Method. Value each set of cash flows at the transaction date using its own periodic rate, then include the stated fee on the correct side.

1. Calculate the loan’s periodic rate and balance. The nominal rate is convertible twice per year, so the semiannual rate is $0.08/2=0.04$. The remaining payment annuity has value

a_{\overline{5}|} = \frac{1-(1.04)^{-5}}{0.04} = 4.45182233.

The outstanding loan balance is $1{,}800(4.45182233)=\$8{,}013.2802$. Adding the $50 payoff fee gives a total payoff requirement of $\$8{,}013.2802+\$50=\$8{,}063.2802$.

2. Calculate the bond’s market price and net proceeds. The semiannual market yield is $0.05/2=0.025$. The present-value annuity factor for the six coupons is

a_{\overline{6}|}^{(0.025)} = \frac{1-(1.025)^{-6}}{0.025} \approx 5.508124.

The coupon present value is $270(5.508124)\approx \$1{,}487.19$. The redemption present value is $9{,}000(1.025)^{-6}\approx \$7{,}760.67$. Thus the gross market price is approximately $\$1{,}487.19+\$7{,}760.67=\$9{,}247.87$, and net sale proceeds after the $35 fee are $\$9{,}247.8656-\$35=\$9{,}212.8656$.

3. Compare proceeds with the payoff.

\$9{,}212.8656-\$8{,}063.2802 = \$1{,}149.5854.

Rounded to the nearest cent, the transaction produces a $1,149.59 surplus. As a check, net proceeds equal the payoff plus the surplus: $\$8{,}063.2802+\$1{,}149.5854=\$9{,}212.8656$. This method avoids the mistake of comparing only the bond’s face amount with the loan balance.

Fees, Market Price, and the Direction of the Difference

The subtraction should be written in a fixed order so that the sign has a consistent meaning:

\text{surplus or shortfall} = \text{net bond sale proceeds} - \text{loan payoff amount}.

If the result is positive, the bond sale leaves cash after the loan is paid. If it is negative, the absolute value is the additional cash required. A payoff fee increases the amount needed, while a bond sale fee reduces the amount available. Keep each fee with the side of the transaction on which it is paid.

The bond’s book value can differ from its market sale price. Book value reflects a yield and accounting method used to track the bond; the sale price is based on the market yield at the sale date and the remaining cash flows. For a refinancing transaction, use the actual sale price or the market yield specified in the problem, not a historical book value, unless the problem explicitly asks for book value.

Worked Example: A Premium Bond Sale Covers a Loan Payoff

A loan has five annual payments of $2,200 remaining. The effective annual loan rate is 6%, and a payoff fee of $80 is due. On the same date, an annual-coupon bond with face amount $10,000 has three coupons of $600 remaining and is sold at a market yield of 5% effective annually. The sale fee is $25. Find the surplus or shortfall.

Method. Calculate the loan’s date-specific balance prospectively, price the bond at its market yield, and apply each fee once.

1. Loan payoff. The payment rate is 6% per year. The annuity factor is

a_{\overline{5}|} = \frac{1-(1.06)^{-5}}{0.06} \approx 4.2123638.

The balance is $2{,}200(4.2123638)=\$9{,}267.20$. Including the payoff fee, the loan payoff amount is $\$9{,}267.20+\$80=\$9{,}347.20$.

2. Bond sale proceeds. The bond’s yield is 5% per year, so

a_{\overline{3}|}^{(0.05)} = \frac{1-(1.05)^{-3}}{0.05} \approx 2.7232480, \qquad (1.05)^{-3}\approx 0.8638376.

The coupons have present value $600(2.7232480)=\$1{,}633.95$. The redemption amount has present value $10{,}000(0.8638376)=\$8{,}638.38$. Gross market price is approximately $\$10{,}272.32$; after the $25 sale fee, net proceeds are $\$10{,}247.32$.

3. Find and check the result. Net proceeds exceed the payoff by $\$10{,}247.32-\$9{,}347.20=\$900.12$, a surplus. The check is $\$9{,}347.20+\$900.12=\$10{,}247.32$. The bond sells above face value here because its coupon rate is 6% of face while its market yield is 5%; the price used is still the discounted value of its remaining cash flows.

When the Sale and Payoff Dates Differ

A sale amount and a payoff amount from different dates cannot be subtracted directly. Move one amount to the other date under the accumulation or discount convention specified in the problem. For example, if net proceeds are received at time 0 and retained until time 1 at an effective annual rate $r$, their value at time 1 is the original proceeds multiplied by $1+r$. If the cash is used for some other purpose in the meantime, that cash flow must also be included when the problem supplies it.

Similarly, the loan balance must reflect payments and interest up to the intended payoff date. A balance immediately after a scheduled payment excludes that payment and includes only subsequent payments. State clearly whether the payoff occurs before or after a scheduled payment; shifting that timing by one payment can change the result materially.

Worked Example: Accumulating Sale Proceeds to the Loan Payoff Date

A borrower sells a bond immediately after a coupon date and receives net proceeds of $9,700.70 at time 0. The proceeds are held in an account earning an effective annual rate of 4%. The loan has five annual payments of $3,000 remaining at time 0, and its effective annual rate is 10%. The borrower makes the first scheduled payment at time 1 and then pays off the loan immediately after that payment. The payoff fee at time 1 is $100. Find the surplus or shortfall at time 1.

Method. Accumulate the sale proceeds to the payoff date, and value only the four loan payments remaining immediately after the time-1 payment.

1. Proceeds at time 1. The $9,700.70 proceeds earn 4% for one year:

\$9{,}700.70(1.04)=\$10{,}088.728 \approx \$10{,}088.73.

2. Loan balance immediately after the first payment. Four payments of $3,000 remain. At the 10% effective annual loan rate,

a_{\overline{4}|} = \frac{1-(1.10)^{-4}}{0.10} \approx 3.16986545.

The balance is $3{,}000(3.16986545)=\$9{,}509.59635$. Adding the $100 fee gives a payoff requirement of $\$9{,}609.59635$, or $9,609.60 to the nearest cent.

3. Compare at time 1. The surplus is $\$10{,}088.728-\$9{,}609.59635=\$479.13165$, which rounds to $479.13. The check is that proceeds at time 1 equal the payoff plus the surplus: $\$9{,}609.59635+\$479.13165=\$10{,}088.728$. Treating the time-0 proceeds as though they were already available at time 1 would miss the one-year accumulation.

Common Traps in a Refinancing Calculation

Using face amount instead of market price. Face amount is the redemption cash flow at maturity, not necessarily the price a buyer pays today. Discount the remaining coupons and redemption at the market yield.
Using book value as sale proceeds. Book value need not equal the market price on the transaction date. Use the sale price or the yield specified for market valuation.
Using the wrong periodic rate. Convert nominal rates to the rate per payment period before valuing payments. For example, 8% nominal convertible semiannually gives a rate of 4% per half-year, not 8% per half-year.
Putting fees on the wrong side. Subtract a sale fee from bond proceeds; add a payoff fee to the loan amount required.
Mixing transaction dates. Accumulate or discount proceeds and payoff amounts to a common date before subtracting.
Including a payment that has already been made. At a balance immediately after a scheduled payment, value only payments still due after that date.

A useful final check is an equation in which all quantities refer to the same date. For a surplus $S$, verify that proceeds equal payoff plus $S$. For a shortfall $H$, verify that proceeds plus $H$ equal payoff. This check also helps catch a reversed subtraction sign.

Key takeaway: A refinancing decision compares the bond’s net market sale proceeds with the loan’s date-specific payoff amount. Calculate both from the applicable cash flows, rates, fees, and timing assumptions before deciding whether the transaction leaves a surplus or requires additional cash. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the common transaction date, apply fees to the correct side, and report a surplus or shortfall.

A loan has three annual payments of $2,000 remaining at an effective annual rate of 8%, with a $40 payoff fee. Same-date bond sale proceeds after sale costs are $6,300. Find the surplus or shortfall. (Answer: $1,105.81 surplus)
A bond with face amount $5,000 pays annual coupons of $300, has two years remaining, and is sold at a 4% effective annual market yield. The sale fee is $10. The same-date loan payoff, including its fee, is $5,500. Find the surplus or shortfall. (Answer: $321.39 shortfall)
Net bond sale proceeds of $4,000 are received at time 0 and held for two years at an effective annual rate of 3%. The loan payoff amount at time 2, including fees, is $4,325. Find the time-2 surplus or shortfall. (Answer: $81.40 shortfall)
A bond sale produces gross proceeds of $5,500 on the loan payoff date. The loan requires $5,400 to discharge, including its payoff fee. What is the largest bond sale fee that can be paid without requiring additional cash? (Answer: $100)

What You'll Learn