Comparing Refinancing Choices by Present Value

Why Compare Loan Choices by Present Value?

A refinancing offer can lower the payment but extend the repayment period, or reduce the total number of payments while requiring a larger installment. Fees and payment timing add further complications. Comparing only the payment amounts does not put these choices on a common basis.

The present-value method converts each choice’s borrower cash flows to a value at the same date, using one stated comparison rate. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus. Its conclusion is conditional on that rate and on the cash flows included: a lower present value means a lower value of the specified payments, not necessarily a better choice under every personal or contractual consideration.

Definition: The present value of a loan choice at a selected valuation date is the sum of the present values, at a stated comparison rate, of the borrower’s net cash outflows associated with that choice. A borrower cash outflow is counted positively; cash received by the borrower is counted negatively. Compare alternatives using the same valuation date, rate, and scope of cash flows.

Set Up a Fair Comparison

First state the valuation date. In the examples below it is time 0, immediately before the next scheduled payment. Then state the comparison rate and how it applies to the cash-flow intervals. This rate is the common basis for comparing timing; it need not equal either loan’s contractual rate.

For an effective rate $i$ per payment period, a payment $C_k$ due $k$ periods after the valuation date has present value $C_kv^k$, where $v=1/(1+i)$. For a finite sequence of payments and fees, add the discounted amounts using their actual dates.

\operatorname{PV}_0=\sum_{k=0}^{n} C_kv^k,\qquad v=\frac{1}{1+i}

Here $C_k$ is a net borrower outflow at time $k$; a receipt is represented by a negative $C_k$. The index $k=0$ represents a cash flow at the valuation date, which is not discounted. If payments are level and occur at the ends of periods, their present value can also be found using $C a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$. That annuity factor applies only when there are $n$ equal payments at the ends of consecutive periods, with the same effective rate per period.

Fix the comparison basis.
Choose one valuation date and state the comparison rate and period. Convert a quoted rate to the matching period rate before discounting.

List net borrower cash flows.
Record each payment and fee at its actual date. If replacement-loan proceeds fund the old-loan payoff, include both the proceeds and payoff if using a full transaction cash-flow list; they offset when equal.

Discount and add.
Bring each net outflow to the common date and sum. Do this separately for every alternative.

Compare the totals.
The lower present value is lower on the stated comparison basis. Report the difference and explain what the comparison includes.

Why the Present-Value Formula Works

The formula follows from the definition of an effective interest rate. One unit invested at time 0 accumulates to $1+i$ after one period, so an amount due at time 1 has time-0 value $1/(1+i)=v$ per unit. Repeating this argument, one unit due at time $k$ has time-0 value $v^k$. Therefore an amount $C_k$ due then has present value $C_kv^k$. Present values are additive: the value of a collection of cash flows is the sum of the values of its individual flows. This gives the stated sum.

This reasoning assumes the comparison rate is applied consistently over the stated periods. It does not make the comparison rate a prediction of future investment returns, nor does it determine whether a borrower qualifies for an offer. It is a valuation convention for comparing the listed cash flows.

Worked Comparisons

Worked Example: Compare Keeping a Loan With a Three-Payment Refinance

A borrower has three annual payments of $4,000 remaining on an existing loan. A replacement loan would provide exactly the amount needed to settle the old loan at time 0. The new loan requires three annual payments of $3,700 and an additional $300 fee paid at time 0. Compare keeping the loan with refinancing at an effective annual comparison rate of 5%. Assume all payments occur at year-end and there are no other costs or cash flows.

Method. Value the scheduled payments and fee at time 0. The old-loan payoff is funded by equal replacement-loan proceeds, so these two time-0 transaction flows cancel. Do not add the payoff to the replacement payments without also accounting for the proceeds.

For the existing loan, the discount factor is $v=1/1.05=0.952380952$. Its powers are $v^2=0.907029478$ and $v^3=0.863837598$. Thus:

\operatorname{PV}_{\text{keep}} =4{,}000(0.952380952+0.907029478+0.863837598) =4{,}000(2.723248028) =\$10{,}892.99

For refinancing, the present value of the three payments is $3{,}700(2.723248028)=\$10{,}076.02$. Add the time-0 fee:

\operatorname{PV}_{\text{refinance}} =\$300+\$10{,}076.02 =\$10{,}376.02

Refinancing has the lower present value by $\$10{,}892.99-\$10{,}376.02=\$516.97$. On the stated comparison basis, refinancing is lower by $516.97. This difference includes the $300 fee.

Check. The replacement payments are each $300 below the existing payments, giving a present-value reduction of $300(2.723248028)=\$816.97$. Subtracting the $300 fee leaves the same $516.97 reduction.

Worked Example: Rank Offers With Different Terms

A borrower can keep a loan requiring four annual payments of $2,800. Offer A requires a $250 fee at time 0 followed by three annual payments of $3,600. Offer B requires a $500 fee at time 0 followed by five annual payments of $2,300. Assume each new loan’s proceeds settle the existing balance, so the payoff and proceeds offset. Compare all three choices at time 0 using an effective annual comparison rate of 6%. Assume no other cash flows.

Method. Use the common rate for all three choices, even though their loan terms differ. At 6%, $v=1/1.06=0.943396226$, $v^2=0.889996440$, $v^3=0.839619283$, $v^4=0.792093663$, and $v^5=0.747258173$.

Keeping the loan means four payments. The relevant factor is $v+v^2+v^3+v^4=3.465105612$:

\operatorname{PV}_{\text{keep}} =2{,}800(3.465105612) =\$9{,}702.30

Offer A has three payments, with factor $v+v^2+v^3=2.673011949$. Its present value, including the time-0 fee, is:

\operatorname{PV}_{A} =\$250+3{,}600(2.673011949) =\$250+\$9{,}622.84 =\$9{,}872.84

Offer B has five payments, with factor $v+v^2+v^3+v^4+v^5=4.212363785$. Therefore:

\operatorname{PV}_{B} =\$500+2{,}300(4.212363785) =\$500+\$9{,}688.44 =\$10{,}188.44

The ranking from lowest to highest present value is keeping the loan ($9,702.30), Offer A ($9,872.84), then Offer B ($10,188.44). Offer A has the largest installment, but the shortest payment period among the two offers. Offer B has the smallest installment but the longest term and largest fee. Neither installment size nor total undiscounted payments alone is enough to rank these choices.

Check. The five-year offer’s present value exceeds the four-payment keep option by $\$10{,}188.44-\$9{,}702.30=\$486.14$, consistent with the ranking.

Worked Example: Use a Semiannual Comparison Rate and Irregular Payments

A borrower’s existing loan requires three semiannual payments of $2,500 at 0.5, 1.0, and 1.5 years, followed by a $1,000 final payment at 2.0 years. A refinance requires a $180 fee at time 0 and four semiannual payments of $1,800 at 0.5, 1.0, 1.5, and 2.0 years. Assume the replacement proceeds settle the old loan, so the time-0 payoff and proceeds offset. Compare at a nominal annual rate of 8% convertible semiannually, with no other cash flows.

Method. The stated nominal comparison rate gives an effective rate of $0.08/2=0.04$ per half-year. Use this half-year rate because every payment date is a whole number of half-years after time 0. The discount factors are $v=1/1.04=0.961538462$, $v^2=0.924556213$, $v^3=0.888996359$, and $v^4=0.854804191$.

The existing loan’s three level payments have factor $v+v^2+v^3=2.775091034$. The final $1,000 is a separate payment at the fourth half-year:

\operatorname{PV}_{\text{keep}} =2{,}500(2.775091034)+1{,}000(0.854804191) =\$6{,}937.73+\$854.80 =\$7{,}792.53

The refinance payments have factor $v+v^2+v^3+v^4=3.629895225$. Add the fee at time 0:

\operatorname{PV}_{\text{refinance}} =\$180+1{,}800(3.629895225) =\$180+\$6{,}533.81 =\$6{,}713.81

The refinance has the lower present value by $\$7{,}792.53-\$6{,}713.81=\$1{,}078.72$. The calculation accounts for the old loan’s final payment at year 2 separately; it does not treat all four periods as identical payments under the old loan.

Check. The refinance cash flows total $7,380 without discounting, while the existing-loan cash flows total $8,500. Discounting changes the difference, but the calculated present values preserve the same ordering.

Fees, Payoffs, and Common Traps

A refinance payoff is the amount required to settle the old debt; it is not automatically an additional cost on top of every replacement-loan payment. If the new loan provides exactly the payoff amount, the borrower receives that amount and immediately pays it to settle the old loan. At the common date, the receipt and payment cancel. The relevant incremental cash flows may then be the fee and replacement payments, compared with the existing payments.

If the borrower contributes some cash at closing, or if the new loan proceeds exceed or fall short of the payoff, the net amount at closing does not cancel. Include that net cash flow at its actual date. If a fee is financed into the replacement loan, do not also count it as an upfront cash fee: its effect is already reflected in the resulting loan payments or balance.

Wrong: discounting offers at different rates. This makes the resulting present values incomparable. Correction: use one stated comparison rate for every alternative.
Wrong: treating a payment at time 0 as if it were due in one period. This discounts it unnecessarily. Correction: a time-0 fee enters the present value at its full amount.
Wrong: comparing payment amounts alone. A lower installment may come with more installments or a fee. Correction: list and value the complete set of included cash flows.
Wrong: counting an old-loan payoff without the replacement proceeds that fund it. This can double-count the debt settlement. Correction: either compare net incremental cash flows after equal payoff and proceeds cancel, or include both flows explicitly.

Key takeaway: A refinancing comparison is only as meaningful as its common valuation date, comparison rate, and cash-flow scope. Discount every included payment and fee to that date, handle loan proceeds and payoffs consistently, and interpret the result as a comparison on the assumptions stated.

Check Your Understanding

For each problem, compare the choices at the stated date and rate. Assume no cash flows other than those listed, and account for any payoff and replacement proceeds as specified.

An existing loan has three annual payments of $1,500. A refinance has a $100 fee at time 0 and three annual payments of $1,400; replacement proceeds exactly settle the old loan. Compare at an effective annual rate of 5%. Which choice has the lower present value, and by how much? (Answer: Refinance; its present value is lower by $172.32.)
An existing loan requires two annual payments of $5,000. An offer has a $250 time-0 fee and two annual payments of $4,800. The replacement proceeds settle the old loan. Compare at an effective annual rate of 4%. (Answer: Refinance; present values are $9,430.47 to keep and $9,303.25 to refinance, so refinancing is lower by $127.22.)
An existing loan requires payments of $3,200 at 0.5 and 1.0 years. A refinance has a $90 time-0 fee and three payments of $2,150 at 0.5, 1.0, and 1.5 years. Replacement proceeds settle the old loan. Compare at a nominal annual rate of 6% convertible semiannually. (Answer: Keep the existing loan; its present value is $6,123.10 versus $6,171.51 for refinancing, a difference of $48.41.)
An existing loan requires two annual payments of $3,600. A refinance requires $3,300 at year 1, a $200 fee at year 1, and $3,300 at year 2. Replacement proceeds settle the old loan. Compare at an effective annual rate of 5%. (Answer: Refinance; its present value is $6,326.53 versus $6,693.88 to keep, so refinancing is lower by $367.35.)
An existing loan requires payments of $1,000 at year 1, $2,000 at year 2, and $3,000 at year 3. A refinance has a $150 time-0 fee and three annual payments of $1,750; replacement proceeds settle the old loan. Compare at an effective annual rate of 7%. (Answer: Refinance; its present value is $4,742.55 versus $5,130.35 to keep, a reduction of $387.80.)

What You'll Learn