A Student s Primer on the Fixed-Rate Mortgage Loan: More Than Meets the Eye

Transcription

1 A Student s Primer on the Fixed-Rate Mortgage Loan: More Than Meets the Eye Joseph W. Trefzger Illinois State University Roger E. Cannaday University of Illinois Working Paper December 20, 2017 Abstract. Students often have difficulty understanding financial values generated with calculator functions and computer software they rely on. Even the fixed payment, fixed-interest rate mortgage (FRM) loan turns out to be an instrument more complex than it initially may seem. This discussion gives a back-to-basics coverage of the mechanics applied when electronic tools compute FRM values such as monthly payment, remaining principal, borrower s NPV of refinancing, and lender s rate of return. Instructors can use it as a focused numerical guide to accompany the broader loan coverage textbooks present, or to assure a common foundation in a course whose enrollment brings varied computational backgrounds. Key Lessons in Fixed Rate Mortgage Loan Payment Mechanics, a shorter work that incorporates a number of the ideas presented in this working paper, has been accepted for publication in the Journal of Economics and Finance Education (forthcoming).

2 A Student s Primer on the Fixed-Rate Mortgage Loan: More Than Meets the Eye I. Introduction The standard fixed-payment, fixed-interest rate mortgage (FRM) loan can seem quite vanilla on its surface, but a deeper examination reveals this important financial instrument s surprising complexities. Mastering the FRM s payment mechanics is essential for understanding other products offered in the real estate financing environment, and provides insights on the structures of more complicated fixed-income contracts encountered in the investments and corporate finance settings. This primer is offered not as a critique of textbook coverage, but rather as a unified start-to-finish explanation of the numbers that can supplement text authors need to intersperse analysis of various loans numerical features with related descriptive passages over multiple chapters. It also is a tool that can help instructors assure common foundations for students in real estate principles, investment, or finance courses, whose backgrounds with financial computations might vary considerably. Finally, it is a tutorial whose diligent user should genuinely grasp the origins and meanings of FRM values generated by otherwise-opaque calculator function key sequences or programmed computer software. Borrower and lender viewpoints are alternated to some extent; the two are related but are not always mirror images, in that the lender s rate of return generally differs from the borrower s percentage cost. The footnotes contain expanded explanations that the student can ignore without loss of continuity. II. How the Simplest Loans Are Structured Any loan requires the eventual repayment of all principal borrowed, plus the eventual payment of interest on principal that has remained owed. On a one-year loan, with a single year-end payment to be made, the interest component of that single payment is simply the annual interest rate multiplied by the principal borrowed, and it is tendered along with repayment of the full principal at the end of the year. For example, if $200,000 is borrowed for one year at a 6% annual interest rate, then the lone year-end payment to the lender is $200,000 plus 6% of $200,000, for $200,000 + $12,000 = $212,000 total. The lender earns $12,000 in interest, for a 6% annual rate of return on the $200,000 in principal that was lent for the year. It would be unusual, of course, for an ordinary household that borrows a large sum to have the ability to repay everything after just one year. By repaying over a greater number of periods a borrower keeps some or all payments smaller than they would be if the loan term were shorter. But more in total interest is paid when principal remains owed over a longer time span, 1 and the structuring 1 Although more in interest is paid over a longer term, it does not follow that borrowers always should negotiate shorter loan maturities and/or repay principal more quickly than their contracts specify. In return for the extra interest paid the borrower gains the continued use of money (the remaining principal owed) that otherwise might have to come from some other source, so opportunity cost is the issue. Income tax advantages available to U.S. home mortgage borrowers, coupled with the low interest rates that usually accompany loans safely collateralized 1

3 of the interest charges and total payments within such a multi-period loan can become far more complicated than the one-payment example suggests. A. Straight-Term (Interest-Only) Loan Begin with the simplest multi-period lending arrangement, the straight-term or interest-only loan, with a payment structure similar to that for the type of bond that has coupon interest payments, as discussed in introductory business finance courses. In this straightforward plan each payment prior to the maturity date consists solely of interest. An individual borrowing $200,000 for four years under an interest-only arrangement with a 6% annual interest rate pays $12,000 in interest at the end of each of years 1 through 4 on the $200,000, tendering the full $200,000 in principal along with the last interest outlay at the end of year 4. A distinguishing feature of the interestonly loan is zero amortization of principal; 2 each pre-maturity payment is sufficient to cover just interest owed for the period, and all principal repayment occurs through a single large balloon payment (a payment of principal much larger than the principal portions of other payments in a series) 3 that is combined with the final payment of interest at the maturity date: (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $212, $200, $0.00 $12,000 in interest, which is 6% of the $200,000 outstanding principal balance, is collected every year of the loan s term. The lender thereby earns a 6% rate of return each year, although the compounded average annual rate of return for the loan s four-year term is 6% only if each year 1 through 3 $12,000 receipt can be reinvested until the maturity date to earn a 6% annual return. 4 by real estate, tend to make the mortgage loan on a residence a comparatively cost-effective way for a household to obtain money. It would make little sense for a family to run up balances on a credit card with an 18% annual interest rate, for example, to free up funds toward speedier repayment of a mortgage loan with a much lower 4% annual interest rate that also potentially could reduce the family s U.S. federal income tax bill. 2 Dictionaries include extinguish and reduce to the point of death among definitions of amortize, a word derived from the Latin mors and later French mort, meaning death; the debt obligation on an amortizing loan essentially dies off over the specified term. A different real estate-related application of the word involves the expectation that an owner will steadily disinvest in a property whose nature conflicts with impending zoning changes; see The Wall Street Journal (2016). Mortgage reflects the same root in a different context; under the medieval live pledge a lender confiscated portable items serving as collateral, whereas the dead pledge (mort gage) facilitated securing a loan with real estate, which the lender could not physically possess and which, in turn, the borrower could continue to use. See Clauretie and Sirmans (2014), 49. While people casually use the word mortgage to mean a loan, technically the borrowing of money is evidenced by a promissory note; the accompanying mortgage is the document that pledges real estate or other property as security to better assure the lender that the note will be repaid. 3 Bullet payment is a phrase sometimes used for a balloon payment that covers repayment of all the borrowed principal at maturity; see Green and Wachter (2005), This issue relates to the internal (IRR) and modified internal (MIRR) rate of return measures presented in basic finance or investment courses (bond analysts call IRR the yield to maturity and MIRR the realized compound yield; 2

4 An extreme interest-only/zero amortization variation would be a perpetuity, in which the lender would receive only interest every period and the full principal would remain outstanding forever: (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $12, $0.00 $200, $200, $12, $212, $12, $0.00 $200, B. Pure Discount Loan A bit more complicated is a loan that matures after multiple periods have passed but that has no intermediate payments; repayment of principal, and providing for interest on principal that has remained owed, is handled entirely through one payment that occurs at the maturity date. This pure discount loan, structured like the zero-coupon bonds presented in basic finance courses, is characterized by negative amortization of principal. A negative amortization loan has at least one period when there is not a payment sufficient to meet the periodic interest obligation, such that the shortfall is added to principal owed, interest subsequently is charged on the larger resulting balance, and then the initial principal plus added principal ultimately must be repaid. With a pure discount loan the single large outlay at maturity achieves that repayment (it is highly unlikely that this type of arrangement ever would be used to finance a home purchase, although it is used for some reverse mortgage loans through which older borrowers tap their residences equity). If someone gets a $200,000 pure discount loan with a four-year maturity and a 6% annual interest rate, the debt is settled with a $252, balloon payment at the end of year 4: 5 see Homer et al. [2013], ). MIRR is the geometric average periodic rate that blends an investment project s IRR with an explicitly specified periodic rate at which pre-maturity cash flows are reinvested until the maturity date, whereas IRR is computed based solely on a project s internal cash flows and thus is not affected by any assumption regarding reinvestment; see Cannaday et al. (1986), 18. When a textbook or other source states that the IRR computation incorporates an implicit assumption that pre-maturity cash flows are reinvested at the IRR, the interpretation is that for a project s MIRR to equal its IRR the IRR must be the specified reinvestment rate. 5 Because balloon usually means a large one-time repayment of principal we might debate whether the entire $252, payment is a balloon or if it consists of a $200,000 principal balloon and $52, in accumulated interest. In one sense the entire total is a principal balance that has grown from the original $200,000 with the accumulation of unpaid interest. But U.S. federal income tax law generally treats any scheduled increase in the value of a pure discount instrument in a particular year as received interest on which the lender must pay income tax, even though no cash has been collected. In tax parlance the $52, is an original issue discount. Attributed interest computed according to the constant yield method should be the figures running from $12,000 to $14, shown in Column B of the table. The borrower can deduct that attributed interest each year, however, only if it would qualify as a business expense and the borrower reports on an accrual basis. For a pure discount loan secured by a residence, interest typically could be deducted only when the repayment is made. See Internal Revenue Service publications 535 (2016), 5, 13; 550 (2016), 13, 32; and 936 (2016), 5. Non-deductibility of each year s interest is but one reason why a pure discount home mortgage loan, other than a reverse mortgage, would be quite unlikely. 3

5 (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $0.00 ($12,000.00) $212, $212, $12, $224, $0.00 ($12,720.00) $224, $224, $13, $238, $0.00 ($13,483.20) $238, $238, $14, $252, $252, $238, $0.00 Because $200,000 (1.06) 4 = $252,495.39, receiving that latter amount at the end of year 4 gives the lender a 6% compounded average annual rate of return for the loan s four-year term. 6 C. A Less Extreme Negative Amortization Case Now consider a $200,000 four-year loan with a 6% annual interest rate, but carrying year 1 through 3 payments of $9,000 each. Once again there is negative amortization (though to a lesser degree than with no pre-maturity periodic payments), because $9,000 is less than the (.06) ($200,000) = $12,000 in interest owed for year 1, and shortfalls continue in the two following years. But because the negative amortization is not as severe as in the previous example, the large payment at the end of year 4 is only $222, rather than the higher $252,495.39: 7 (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $9, ($3,000.00) $203, $203, $12, $215, $9, ($3,180.00) $206, $206, $12, $218, $9, ($3,370.80) $209, $209, $12, $222, $222, $209, $0.00 This series of receipts also provides the lender with a 6% compounded average annual rate of return, but again only if each intermediate $9,000 payment can be reinvested until the maturity date to earn its own 6% compounded average annual rate of return. D. Constant Partial or Full Amortization of Principal In a loan with multiple repeated periodic payments and positive amortization, on the other hand, at least one payment made before maturity exceeds the interest owed for the period, such that some principal is being repaid prior to the maturity date. Positive amortization over the loan s 6 Because a pure discount instrument has no cash flows that can be reinvested between the original investment and the maturity date, its geometric average compounded periodic rate of return is determined solely by the initial investment and the single payment to be received at maturity, and therefore its MIRR equals its IRR. 7 Again the negative amortization could open a debate on whether principal being repaid at maturity is $222, or $209, or merely $200,000, with the remainder constituting accumulated interest. As in the prior example we would expect attributed interest each year, including from an income perspective for the lender s U.S. federal income tax purposes, to be as listed in Column B of the table. 4

6 term could be either partial or full. One or more of the outlays on a partially amortizing loan with regular periodic payments exceeds interest owed for the respective period, but not by enough to fully repay principal over the term, so a balloon payment of less than the full initial principal must be part of the amount paid at maturity. Periodic payments before the maturity date all could be equal but would not have to be. We might illustrate with a partially amortizing version of a plan called the constant amortization mortgage (CAM) loan, with payments that change from period to period while each pre-maturity payment includes the same amount of principal. Under this unconventional approach to paying off a 6%, $200,000 loan over four years each year 1 through 3 payment could include $35,000 in principal, along with interest equal to 6% of the principal still owed at the year s start: (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $47, $35, $165, $165, $9, $174, $44, $35, $130, $130, $7, $137, $42, $35, $95, $95, $5, $100, $100, $95, $0.00 Since there is only partial amortization, a considerable amount of principal still is owed that must be retired as part of the final payment; here $100,700 (which includes a $95,000 principal balloon, much more than the earlier $35,000 principal pieces) must be paid at maturity. But the lender still is earning 6% yearly on any principal that remains owed, although as in earlier cases it earns a 6% compounded average annual return only if each payment received before maturity is reinvested until the maturity date to earn its own 6% compounded average annual return. Now consider a loan with full amortization, for which each periodic payment contains an interest portion and a principal portion, with sufficient principal retired in each pre-maturity payment that no large balloon payment is owed at maturity. 8 In a fully amortizing version of the CAM loan the payments still change from period to period but include the same amount of principal every time: (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $62, $50, $150, $150, $9, $159, $59, $50, $100, $100, $6, $106, $56, $50, $50, $50, $3, $53, $53, $50, $ It would be possible for a loan to have one or more periods of negative amortization yet still to partially amortize, or even to fully amortize with no ending balloon payment needed, through sufficiently high payments in earlier or subsequent periods. This feature could characterize repayments on a line of credit, or on the flexible-payment option adjustable rate home mortgage loan that became problematic during the 2000s mortgage lending crisis. 5

7 Each successive payment consists of $50,000 in principal repaid and 6% interest on the year s initial principal owed, with interest thus declining by (.06) ($50,000) = $3,000 per year as each added $50,000 block of principal has been repaid. The lender once again earns 6% each year, and a 6% compounded average annual rate of return if each amount received before maturity can be reinvested to earn a 6% compounded average annual rate of return until the maturity date. E. Fully Amortizing Loan with Equal Payments: the FRM Finally, consider the case of full amortization and periodic payments that all remain equal throughout the loan s term, with no balloon payment occurring at maturity. These features characterize the fixed-payment, fixed-interest rate mortgage (FRM) loan, which is the type of home mortgage loan most commonly extended 9 and the basis for most of the remainder of this discussion. The more complicated yearly breakdown for a fully amortizing $200,000 FRM loan with a 6% annual interest rate and payments that are to be exactly equal at the end of each year for four years appears as follows: (A) (B) (C) (D) (E) (F) Principal 6.0% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Year Year of Payment of Year Year Year Principal (A + B) Payment (D B) (C D = A E) 1 $200, $12, $212, $57, $45, $154, $154, $9, $163, $57, $48, $105, $105, $6, $112, $57, $51, $54, $54, $3, $57, $57, $54, $0.00 It is easy to see how each year 1 to 3 payment on the interest-only loan is $12,000 and how each payment on the fully amortizing CAM loan is $50,000 plus the year s interest obligation. But the process for computing the unchanging $57, FRM loan payment surely is less evident. III. Computing an FRM Loan s Unchanging Periodic Payment The key ideas in understanding how the equal periodic outlays on an FRM loan serve to fully repay principal and applicable interest over the multiple included time periods are as follows: As with every loan, interest is charged each period only on the amount of principal that still was owed at the period s beginning 9 Before the mortgage meltdown occurred a decade ago some borrowers buying primary residences obtained loans with negative/zero/partial amortization and thus with planned eventual increases in their payment schedules. While loans with provisions like those, such as the option ARM mentioned in footnote 8 above, still could, in theory, finance families home purchases, it is unlikely that an American bank or other federally regulated lender would offer them now. The U.S. Consumer Financial Protection Bureau s qualified mortgage rule, the first version of which debuted early in 2014, exposes lenders to potential legal penalties and financial losses unless home mortgage loans they make are fully amortizing (so balloon payments definitely are problematic) and with unchanging periodic payments, features CFPB deems safest for borrowers. See U.S. Consumer Financial Protection Bureau (a) (2016). 6

8 As with every fully amortizing loan that has no negative amortization periods the remaining principal balance declines as time passes, so the amount of interest in each payment gets smaller over time as repayment progresses And because each FRM loan payment is the same while the interest component of the unchanging payment gets smaller with each successive period, the amount of principal repaid within each successive equal payment gets larger A. The Standard Case: End-of-Period Payments Start by imagining a borrower who asks a mortgage loan officer, If I can budget to pay $57, at the end of each year for four years and you quote a 6% annual interest rate, how much principal can I afford to borrow? From a time value of money standpoint, the amount that can be borrowed or lent is the sum of the present values (PVs) of the payments that will be made or received. So from a perspective of knowing the payments and solving for initial principal: Amount borrowed today that would correspond to a payment of $57, in 1 year is PV (1.06) 1 = $57, $57, (1.06) 1 or $57, ( )1 = $54, PV Amount borrowed today that would correspond to a payment of $57, in 2 years is PV (1.06) 2 = $57, $57, (1.06) 2 or $57, ( )2 = $51, PV Amount borrowed today that would correspond to a payment of $57, in 3 years is PV (1.06) 3 = $57, $57, (1.06) 3 or $57, ( )3 = $48, PV Amount borrowed today that would correspond to a payment of $57, in 4 years is PV (1.06) 4 = $57, $57, (1.06) 4 or $57, ( )4 = $45, PV When someone applies for a loan, the lender can justify lending the total of the PVs of all the payments it expects to receive from the borrower. The fact that the amount of principal owed, on a given date, is the sum of the PVs of the remaining payments discounted at a rate corresponding to the interest rate specified in the borrower/lender contract is perhaps the most important thing to understand about loans. In this example the PVs of the expected equal payments add to $54, $51, $48, $45, = $200,000. So $200,000 is the principal that the borrower can afford to borrow, and in turn that the lender is willing to extend, if the interest rate is 6% per year and the loan is to be serviced with four year-end payments of $57, each. The steps shown above can be summarized as $57, ( )1 + $57, ( )2 + $57, ( )3 + $57, ( )4 and, through the distributive property, 10 presented even more succinctly as 10 The distributive property holds that y (a) + y (b) + y (c) = y (a + b + c); 10 (3) + 10 (4) + 10 (6) = = 130, which equals the product 10 ( ) or 10 (13). The distributive property is a mathematical foundation for the present value of annuity and future value of annuity factors used extensively in financial computations. 7

9 $57, [( )1 + ( )2 + ( )3 + ( )4 ] = $57, ( ) = $57, ( ) = $200,000 Now reverse the order; an institution that lends $200,000 at a 6% annual interest rate, with repayment to be achieved with four equal year-end payments, can divide $200,000 by to compute the $57, annual payment. This is the present value of a level ordinary annuity factor for a 6% periodic interest rate and four time periods. The PV of annuity situation equates a series of equal or related payments to a large dollar sum that exists intact in the present but will decline over time; repaying a loan is the classic example. The factor for the PV of a level ordinary annuity, with ordinary indicating end-of-period payments and level identifying all payments as equal, is simply the sum of the present values of the single dollar amount factors for the same periodic discount rate and same number of time periods: [( )1 + ( )2 + ( )3 + ( )4 ] = ( 1 ( )4.06 ( ) = so FRM Payment = $200,000 ( 1 ( )4.06 ) = $200, = $57, (The present value of an annuity factor s magnitude is smaller than the number of payments. A borrower could not submit a mere $200,000 4 = $50,000 at the end of each year to repay this $200,000 loan by the end of year 4; four payments of $200, = $57, each, which together account for interest along with principal repayment, would be needed.) What more typically is actually used in computing the steady periodic payment on an FRM loan, however, is the PV of a level ordinary annuity factor s reciprocal (here = ); we multiply borrowed principal by this loan payment factor, also called the mortgage loan constant, to compute the periodic payment: FRM Payment = $200,000 (.06 1 ( )4 ) = $200,000 ( ) = $57, Repaying an FRM loan, with its equal payments made at the end of each period, clearly is a PV of a level ordinary annuity application. A large dollar amount that exists intact in the present (the principal borrowed) is financially equivalent to a series of equal end-of-period cash flows that relate to the number n of those payments and the interest rate r charged each period: FRM Payment ( 1 ( 1 1+r )n r ) = Principal Owed or ) Principal Owed ( 1 ( 1 1+r )n r ) = Principal Owed ( 8 r 1 ( 1 1+r )n ) = FRM Payment

10 It is not unusual for an analyst to compute an FRM loan s regular payment based on given principal, r, and n values (dividing principal by the PV of a level annuity factor or multiplying principal by the reciprocal payment factor); or to compute instead the principal that could be lent based on specified r, n, and periodic payment values (multiplying the payment by the PV of a level annuity factor). 11 The essentially universal practice is for loans to have end-of-period payments, as shown above, allowing the borrower s use of the full initial principal for a period before the first payment is required. There is no free lunch, of course; interest is added to the beginning principal, leaving the borrower owing more at the end of the first period than was initially borrowed ($212,000 is owed by the end of year 1 for the annual-payment FRM loan, indeed for all of the $200,000 loans with annual payment plans, as shown in the repayment tables above). But the borrower s productive efforts during that first period normally would be expected to create income, which would enable paying the period s interest obligation, plus something toward principal if the loan has FRM-type full amortization features. B. The Unlikely Case of Beginning-of-Period Payments A borrower required to make beginning-of-period payments would receive the loan proceeds and then make an immediate payment back, thus lacking use of the full borrowed principal from the outset. If the plan had been to buy a $250,000 house with $50,000 from savings and $200,000 borrowed, that purchaser would be unable to meet the transaction price after instantaneously repaying part of the $200,000. (A larger grossed up loan could be structured to leave a desired remainder after an up-front payment, but this arrangement s features hardly seem less awkward than simply prescribing end-of-period payments.) If equal payments on a $200,000 loan with a 6% annual interest rate are to be made at the starts of four consecutive years, with the first payment made immediately and thus discounted for zero periods, the factor for the PV of a level annuity due, with due indicating beginning-of-period payments, is computed as [( )0 + ( )1 + ( )2 + ( )3 ] = ( 1 ( )4.06 ) (1.06) or ( 1 ( )4 (1.06).06 ( ) = (1.06) = ) Its reciprocal, the accompanying loan payment factor, is = , and each level beginning-of-year payment is a smaller $200,000 [( 1 ( )4 (1.06).06 )] = $200,000 [(.06 1 ( )4 (1.06) )] = $200,000 ( ) = $54,451.22: 11 A later section of this discussion covers computing n, the number of periods it would take to repay a loan based on a given regular payment, principal borrowed, and r; and the more common but also more complex computation of r, the periodic percentage rate of return to the lender based on a given payment, principal borrowed, and n. 9

11 (A) (B) (C) (D) (E) Principal Remaining 6.0% Principal Owed at Beginning- Principal Interest Owed at End Start of of-year Owed on That of Year Year Year Payment (A B) Principal (C + D) 1 $200, $54, $145, $8, $154, $154, $54, $99, $5, $105, $105, $54, $51, $3, $54, $54, $54, $0.00 $0.00 $0.00 It should make sense that if even a small amount of principal is repaid right away then less remains to be charged interest in the first period, and systematically throughout the loan s contract term, than would be seen in an otherwise similar loan with end-of-period payments, so the borrower s obligations can be fulfilled with smaller equal periodic outlays. But as noted earlier, a loan with beginning-of-period payments would be almost unheard of, so the remainder of this discussion will deal solely with the end-of-period payments case. 12 C. Makeup of the FRM Loan Payment Factor (Mortgage Loan Constant) Because an FRM s loan payment factor, or mortgage loan constant, is merely the reciprocal of the attendant PV of a level ordinary annuity factor, computing the FRM s unchanging periodic payment as the borrowed principal divided by the annuity factor produces the same result as multiplying principal by the payment factor. However, using the payment factor can provide an intuitive advantage. It turns out that the annual loan payment factor or mortgage loan constant computed for the year-end payments case as described above is the sum of two significant components: the 6% annual interest rate, and the annual sinking fund factor for a 6% annual discount rate and four yearly periods. Just as the PV of a level ordinary annuity factor is the sum of the PVs of the single dollar amount factors for the same periodic discount rate and number of time periods, the future value (FV) of a level ordinary annuity factor, with ordinary again denoting end-of-period payments, is the sum of the FVs of the single dollar amount factors for the same periodic rate of return and number of periods. 13 For the case of a 6% periodic rate and four time periods: [(1.06) 3 + (1.06) 2 + (1.06) 1 + (1.06) 0 ] = ( (1.06)4 1 ).06 ( ) = Grossed-up principal for four beginning-of-year payments is the payment based on three year-end installments multiplied by (1 + the 3-year PV of a level ordinary annuity factor): ($74,821.96) ( ) = $274, After immediately paying $74, the borrower has the remaining $200,000, and she amortizes it by paying the same $74, at the beginning of each of years 2 through 4 (which is the end of each of years 1 through 3). 13 The simplest way to demonstrate that the PV of annuity factor is the sum of the corresponding PV of single dollar amount factors, and that the FV of annuity factor is the sum of the corresponding FV of single dollar amount factors, is by taking the difference of two related equations involving the single dollar amount time value factors; see, e.g., Rao (1995), 80, 85. A presentation of applicable steps is provided in Appendix 5 at the end of this working paper. 10

12 The FV of annuity situation equates a series of equal or related payments to a large dollar amount that grows over time and will not be fully intact until some future date; saving up for retirement is perhaps the best example. If someone wants to build progressively to a $200,000 future total with equal deposits at the end of each year for four years (such that the final deposit would earn no interest, as shown by the 0 exponent on the [1.06] 0 term above), while earning 6% per year on the account s growing balance, 14 the unchanging deposit to be made at the end of each year is computed as Deposit ( (1.06)4 1 ) = Deposit ( ) = $200, so Deposit = $200, = $45, (The future value of an annuity factor s magnitude is larger than the number of payments. It would not be necessary to deposit $200,000 4 = $50,000 at the end of each year to accumulate $200,000 by the end of year 4; four deposits of $200, = $45, each, plus the accompanying interest buildup, would bring the desired result.) A step equivalent to dividing $200,000 by the FV of a level ordinary annuity factor is multiplying it by that annuity factor s reciprocal, which is called the sinking fund factor. A sinking fund is the series of deposits that will grow, along with financial returns earned, to reach a targeted future total by the end of a specified number of time periods. The annuity factor s reciprocal here is the sinking fund factor of = So the required annual deposit can be found instead as $200,000 (.06 (1.06) 4 1 ) = $200,000 ( ) = $45, Recall that the loan payment factor can be computed as the periodic interest rate plus the periodic sinking fund factor; for the 6% annual interest rate, four-year FRM loan that sum would be = (computed earlier as the PV of a level ordinary annuity factor s reciprocal). D. How FRM Loan Equates to Interest-Only Loan Plus Savings Plan Observing that the loan payment factor is the sum of the interest rate and the sinking fund factor opens the door to some deeper insights on loan payment mechanics. Consider two borrowers. Individual A obtains a $200,000 FRM loan that has a 6% annual interest rate and calls for equal payments at the end of each year for four years. The loan is fully amortizing: every equal payment contains the period s interest and some principal, and all principal will be repaid within the four regular payments. As seen earlier, the annual payment on this loan is Total payment made directly to lender: $200,000 ( ) = $57, Just as the PV of a level annuity due factor is the PV of a level ordinary annuity factor multiplied by (1 + r), the FV of a level annuity due factor is the FV of a level ordinary annuity factor times (1 + r). Equal payments made at the start of each period cause interest to be applied to remaining principal one differential number of times over the term relative to the outcome for end-of-period payments (one less time on the declining principal in the PV of annuity case, one more time on the growing principal in the FV of annuity case). 11

13 Borrower B gets a four-year interest-only loan at a 6% annual interest rate. Nothing beyond interest is to be paid at the end of each of years 1 through 3, such that the entire principal borrowed remains owed until the loan s maturity date, and then a balloon payment of the full $200,000 must be made along with the final interest payment at the end of year 4. B will accumulate the $200,000 needed to make full repayment at the end of year 4 by putting equal deposits into a savings plan (a sinking fund) at the end of each year for the four years. If the savings plan administrator pays a 6% average annual rate of return on the account s growing balance, then individual B s yearly outlays will consist of the following: Interest paid to the lender: $200,000 ( ) = $12, Savings plan sinking fund: $200,000 ( ) = $45, Year s total outlay: $200,000 ( ) = $57, An FRM loan therefore is mathematically equivalent to an interest-only loan accompanied by a balloon payment for the full principal to be made at the end of the loan s term, if that balloon amount can be amassed through a savings plan that earns an average periodic rate of return equal to the periodic interest rate charged on the interest-only loan (or that would be charged on the corresponding FRM). A borrower would be indifferent between paying interest amounts that start at $12,000 and then decline each subsequent year, or paying $12,000 in yearly interest and at the end of year 4 paying $200,000 that had been accumulated through a sinking fund earning a 6% average compounded annual return. 15 A lender would be indifferent between receiving $12,000 in yearly interest and $200,000 at the end of year 4, or receiving interest amounts that start at $12,000 and then decline each subsequent year as long as the growing stream of accompanying principal payments received could be reinvested for a compounded annual return averaging 6%, such that they would grow together to total $200,000 by year 4 s end. 16 IV. The More Realistic Monthly Payment Case The examples above reflect time periods that are full years. One reason yearly time periods work well for an introductory loan discussion is that details of all payments for an entire multi-year 15 This analysis ignores income tax issues. A borrower who itemizes deductions can claim interest paid on a home mortgage loan, while one with a sinking fund to amass a principal balloon would be taxed on interest earned on that fund s growing balance. So a particular individual might benefit more over the years, or at least in specific years, by keeping the loan s deductible interest payments at the higher $1,000 monthly level while being taxed on the savings plan s interest receipts, especially since the loan s interest rate would likely exceed that on the savings plan. But the balance on which sinking fund interest is earned would increase with each passing year. So interest rates, potential early repayment of the loan, and other aspects of the borrower s overall finances and income tax picture all could affect the optimal choice between the FRM option and the interest-only loan paired with a sinking fund. 16 U.S. federal income tax issues also might cause the lender to place different values on the interest-only loan s unchanging $1,000 monthly interest receipts vs. the declining interest earnings on an amortizing loan. In addition, a lender expecting interest rates to rise might favor the FRM, with payments containing principal portions that could be re-lent at higher interest rates without increased risk to generate a compounded average annual return on principal greater than 6%. A lender anticipating a decline in interest rates might prefer to make the interest-only loan, with its assured 6% annual return received from the borrower s pocket on all principal while it remained outstanding (although refinancing also would be more likely), rather than the fully amortizing FRM whose principal payment components would have to be reinvested, through loans to others at lower interest rates if the risk level remained the same, for a compounded return on principal averaging less than 6% per year. 12

14 term can be illustrated easily in a table, as shown earlier for the various four-year loans. Another reason to begin with annual outlays is that payments made yearly permit a simpler presentation of interest rates. Financial transactions can play out over periods ranging from milliseconds to decades, so having a common time frame of reference facilitates making at least big-picture comparisons of percentage returns and costs across a range of lending or other investing situations. If lender C made a loan on January 1 and then on March 31 got back 2.36% more than the original principal, while lender D lent on January 1 and then on June 30 got back 4.75% on top of the original principal, we would ask both to restate their returns, for discussion purposes, based on the same time period. Because so many important life events occur yearly, it is not surprising that the financial world s convention is to use years as the common time reference for discussing percentage rates of return or cost. Even if parties C and D both insist that they would not have been able to continue their investments for an entire year, we still ask them to convert their results to annual figures toward allowing a general comparison of risk-based returns. So we talk about interest rates, and other percentage rates of return or cost, on an annual basis. But in financial computations we must work with a rate that corresponds to the timing of the payments and compounding. If payments and compounding occur annually, then the annual interest rate talked about is also the rate used in the computing activity: 6% in the earlier illustrations. Consider also that the amortization period for an FRM home loan almost always is longer than a few years; 15 and 30-year maturities have been common in the past. If the loan introduced above called for year-end payments to be made for 30 years rather than four, each would be Payment = $200,000 ( 1 ( )30.06 ) = $200,000 (.06 1 ( )30 ) = $200,000 ( ) = $14,529.78, much less than the $57, paid each time if the loan had to be completely retired with just four equal payments. [But far more in total interest is paid over 30 years than over four years: ($14,529.78) (30) = $435, exceeds the $200,000 principal by $235, in total interest, 17 while ($57,718.30) (4) = $230, exceeds that principal by only $30, in total interest.] Here the sinking fund factor is the reciprocal of the FV of a level ordinary annuity factor for 6% per period and 30 periods: 1 ( (1.06) ) = ( ) = ,.06 (1.06) 30 1 such that the loan payment factor can be decomposed as = A. Computing Equal Monthly Payments But along with long contractual repayment periods it also almost always is the case that home mortgage loan borrowers, like those with automobile and student loans, make their payments 17 Money values shown throughout this discussion are rounded to whole cents, but the supporting work is more precise. So while (30) ($14,529.78) = $435,893.40, computations in this example actually involve the payment that would more exactly amortize $200,000 over 30 years: (30) ($14, ) = $435,

15 monthly. If $200,000 in principal is borrowed and the interest rate the lender and borrower talk about is a 6% stated annual rate, but payments are to be made at the end of each month for 30 years, the monthly periodic rate to work with in the time value of money computations is =, or.5% per period. With equal payments to occur at the end of each month, (30) (12) = 360 in total, the unchanging monthly payment is computed with the PV of a level ordinary annuity factor, or its reciprocal, for 360 months and a.5% monthly periodic interest rate as $200,000 ( 1 ( 1 1 )360 ) = $200,000 ( 1 ( 1 1 )360 ) = $200,000 (996) = $1, The sinking fund factor for a.5% periodic discount rate and 360 periods (the reciprocal of the related FV of a level ordinary annuity factor) is 1 ( (1)360 1 ) = ( ) = (1) For this loan the monthly loan payment factor can be broken down into the monthly interest rate and the monthly sinking fund factor: = 996. This total indicates that the first month s payment, which is equal to all the other payments, includes.5% interest on principal originally borrowed, plus sinking fund dollar amount ($200,000) ( ) = $ as the first month s small proportion toward paying back principal. (A naïve expectation might be that a repayment of 1/360 =.2778% of the principal should be provided for in each monthly payment; however, because the buildup in a savings plan would be supplemented by interest, only a much smaller.0996% of principal need be accounted for in the first monthly payment.) [Here we could equate 1) an FRM loan with a total monthly payment made directly to the lender of $200,000 (996) = $1, with 2) an interest-only loan combined with a sinking fund savings plan that earns a.5% average monthly rate of return: Interest paid to the lender $200,000 (000) = $1, Sinking fund deposit $200,000 ( ) = $ Total monthly outlay of $200,000 (996) = $1, ] A quick estimate of an FRM loan s monthly payment therefore should be the monthly interest rate multiplied by principal borrowed, plus a little extra. Someone expecting the monthly payment on a $200,000 FRM loan with a.5% monthly interest rate and any maturity to be less than () ($200,000) = $1,000 would be missing the obvious. The extra amount added to the in the first payment grows larger for shorter maturities (the sinking fund factor is for a loan repaid over 480 months, for 300 months, for 240 months, for sixty months, for thirty-six months, and for twelve months). The payment schedule for the first twelve months and last twelve months of this monthlypayment loan proceeds as: 14

16 Exhibit 1: Fully Amortizing FRM Loan With Monthly Payments, Years 1 & 30 of 30 (A) (B) (C) (D) (E) (F) Principal 0.5% Total Owed Principal Principal Owed at Interest by End End of Portion Owed at End Start of on That of Month Month of Payment of Month Month Month Principal (A + B) Payment (D B) (C D = A E) 1 $200, $1, $201, $1, $ $199, $199, $ $200, $1, $ $199, $199, $ $200, $1, $ $199, $199, $ $200, $1, $ $199, $199, $ $200, $1, $ $198, $198, $ $199, $1, $ $198, $198, $ $199, $1, $ $198, $198, $ $199, $1, $ $198, $198, $ $199, $1, $ $198, $198, $ $199, $1, $ $197, $197, $ $198, $1, $ $197, $197, $ $198, $1, $ $197, Year 1 Totals $11, $14, $2, $13, $69.66 $14, $1, $1, $12, $12, $64.01 $12, $1, $1, $11, $11, $58.34 $11, $1, $1, $10, $10, $52.63 $10, $1, $1, $9, $9, $46.90 $9, $1, $1, $8, $8, $41.14 $8, $1, $1, $7, $7, $35.35 $7, $1, $1, $5, $5, $29.53 $5, $1, $1, $4, $4, $23.69 $4, $1, $1, $3, $3, $17.81 $3, $1, $1, $2, $2, $11.90 $2, $1, $1, $1, $1, $5.97 $1, $1, $1, $0.00 Year 30 Totals $ $14, $13, Year Totals $231, $431, $200, Check your understanding by confirming that $ must be paid at the end of each month for twenty-five years to amortize a $160,000 FRM loan with a 4.44% stated annual interest rate. B. Annual Percentage Rates and Effective Annual Rates of Return or Cost As noted, the convention of the marketplace is to talk about percentage rates of return or cost on an annual basis to facilitate comparisons across a range of financial, including borrowing and lending, situations. But in time value of money computations it is essential to work with a rate r that corresponds to the timing of the payments and compounding. So typically an annual rate that is talked about must be converted to a periodic rate r to be used in various PV or FV factors. And this conversion is complicated somewhat by the fact that either of two values legitimately can be 15

17 discussed as an annual rate of return or cost. One has an adjustment for intra-year compounding that the other lacks, so it is important to specify which annual rate is being talked about. For example, if a savings plan provides 2% per quarter in interest, what does the saver earn on an annualized basis? The simple answer.02 per quarter corresponds to earning (.02) (4) =.08, or 8%, for the year. But if someone deposits $100 at the beginning of the year into an account that earns 2% quarterly, the total balance will grow to $100 (1.02) = $102 by the end of quarter 1, $102 (1.02) = $ by the end of quarter 2, $ (1.02) = $ by the end of quarter 3, and $ (1.02) = $ by the end of quarter 4, which is % more than the initial $100 deposit. So is the annual rate of return on a plan that earns 2% quarterly 8%, or %? The answer is that each is correct in its own context. The simple interest result, (.02) (4) =.08, is called the annual percentage rate, or APR. APR is a representation of a rate of return or cost on an annual basis in which we account for the periodic rate and number of periods in a year, but not for the way periodic returns or costs compound as the year progresses. APR, which is just periodic rate times number of periods in a year, therefore is a convenient, but not an accurate, way to present the lender s return or borrower s opportunity cost if the contract remains in place for an entire year, the time frame convention for discussing rates of return or cost. The compounded interest result, (1.02) 4 1 = , is called the effective annual rate, or EAR. EAR is 1 plus the periodic rate taken to the power of the number of periods in a year, with 1 then subtracted so the answer reflects just the return earned and not the entire year-end balance (the 1 represents 100% of the initial principal in a plan on which periodic compounding occurs). With EAR we account for the periodic rate and the number of periods in a year, and also for the way periodic returns or costs compound over the course of the year. EAR thus is an accurate representation of the lender s return/borrower s opportunity cost if the arrangement stays in place for an entire year, but it is not convenient to deal with since most people cannot manipulate exponents and roots in their heads. So we might say that APRs are easy to use but not accurate, while EARs are accurate but not easy to use. 18 Now viewed from the opposite direction: if a situation involves quarterly payments and compounding and the annual rate discussed is identified as an 8% APR, the quarterly r to use in time value computations is simply.08 4 =.02. If the annual rate discussed is identified as an % EAR we find the quarterly r to 4 work with by undoing the compounding that resulted in the EAR: = A bank generally lists two numbers when it posts savings account interest rates, such as 3.00%/3.05%. The first figure is the APR, and the second is labeled the annual percentage yield, or APY, which is what bankers call the EAR. Use of the word yield often is meant to convey the inclusion of compounding that will occur within the year if the arrangement is maintained for one or more entire years. If the bank compounds interest on a one-year (or multiple-year) certificate of deposit daily based on a 365-day year, then the 3% advertised APR is broken into 365 equal parts of = each, such that someone who keeps money in the account for the entire year will end up with ( ) = , or almost 3.05%, more than the initial deposit. It is interesting that under the federal Truth in Savings Act (Regulation DD) a bank must report the compounded APY as the saver s effective yearly return on a deposit account, while under the Truth in Lending Act (Regulation Z, discussed more fully below) it must report a non-compounded APR-based measure as the borrower s effective yearly cost on a loan. See U.S. Consumer Financial Protection Bureau (b) (2015), 14 and Federal Reserve Board (2013). 16