Real Estate Finance: Chapter 15 Mathematics of Real Estate Finance
Interest Factor
(IF)
-Add-on Interest
Although the simple interest method is used for most real estate loans, some lenders occasionally employ and add-on rate. This technique involves the computation of interest on the total amount of the loan for the entire time period. This amount of interest is then added to the principal owed for repayment over the term of the loan before the monthly payments are calculated. This form of interest computation is used for some home improvement loans and many junior liens created by private mortgage companies. Add-on interest has the effect of almost DOUBLING the simple interest rate. Example: Using the same figures as those in the previous example, it the add-on method of computation were employed, first the total interest on the $1,200 for the one-year period would be figured ($1,200 x 0.08 = $96). Next, this amount would be added to the total principal owed ($96 + $1,200 = $1,296). Finally, this sum would be divided by the number of required payments to derive the monthly amount due of $108 ($1,296 ÷ 12 = $108). The borrower then pays a level $108 per month for 12-months until the loan is satisfied. A moment's reflection at this point will reveal that, although the first month's payment is a true reflection of the 8% per annum simple interest charge, the second and subsequent months' payments are not. The interest charged in these months did not stop on that portion of the principal which had been repaid. The accuracy of this observation is illustrated by examining the 6th payment of $108. The $100 principal portion of this payment repays half the total loan ($100 x 6 = $600), but the $8 interest portion still reflects the interest charge on the entire $1,200. Thus, the 8% add-on interest rate is, in reality, about 15%. The formula for computing the ADD-ON INTEREST RATE is: AIR = 2IC ÷ P(n + 1) Where: AIR = Add-on Interest Rate I = Number of installment payments per year C = Total loan charge P = Principal n = Number of installments in the contract NOTE: Wherever parentheses ( ) appears, it means multiply [e.g., 2 (50) = 100]. Substituting the figures from the previous example in this formula, the add-on interest rate would be computed as follows: AIR = 2IC ÷ P(n + 1) AIR = 2(12)($96) ÷ $1,200 (12 + 1) AIR = $2,304 ÷ $15,600 AIR = 0.14769 or 14,77% rounded The interest rate for a $3,000 home improvement loan to be repaid in monthly installments for 3-years at 10% add-on interest per year would be calculated as follows: AIR = 2IC ÷ P(n + 1) AIR = 2(12)($900) ÷ $3,000(36 + 1) AIR = $21,800 ÷ $111,000 AIR = 0.19459 or 19.46% rounded, almost double the 10% rate listed on the loan agreement.
-Compound Worth of an Annuity
An investor is also concerned with the compounding of a series of deposits or earnings. Any series of regular payments or receipts is termed an annuity. The compound interest formula for calculating the future worth of a series of regular deposits, or an annuity, each made at the BEGINNING of a period, is: CS = RD[(1 + i)ˆn-1 + (1 + i)ˆn-2 + (1 + i)ˆn-n or 1] Where: CS = Compound sum RD = Regular deposit i = Interest rate per period n = Number of periods Example: A regular deposit of $1 made at the beginning of each year for 3-years at 6% compound interest will be worth $3.18 at the beginning of the 3rd year, as shown below: CS = $1[(1 + 0.06)ˆ3-1 + (1 + 0.06)ˆ3-2 + 1] CS = $1[(1.o6)ˆ2 + (1.06)ˆ1 + 1] CS = $1[(1.1.236) + (1.06) + 1] CS = $1[3.1836] CS = $3.1836 or $3.18 rounded Using the same formula, the future worth of $1,000 deposited at the beginning of each year for 10-years at 6% interest compounded annually is: ...CS $1,000[(1.06)ˆ9 = 1.68947 ..................+ (1.06)ˆ8 = 1.59384 ..................+ (1.06)ˆ7 = 1.50363 ..................+ (1.06)ˆ6 = 1.41851 ..................+ (1.06)ˆ5 = 1.33822 ..................+ (1.06)ˆ4 = 1.26247 ..................+ (1.06)ˆ3 = 1.19101 ..................+ (1.06)ˆ2 = 1.12360 ..................+ (1.06)ˆ1 = 1.06000 .................+ (1.06)ˆ0 = 1.00000] ..........................Total = 13.18079 CS $1,000(13.18079) = $13,180.79 These mathematical calculations of the compound future worth of a single sum of money or of an annuity involve the use of an interest factor (IF). In the previous example of the compound sum of $1,000 at 6% for 10-years, the beginning deposit was multiplied by an IF of 1.79084. Similarly, in the example of an $1,000 annuity for 10-years at 6%, the regular deposit was multiplied by an IF of 13.18079. Thus, the (1 + i)ˆn portion of these equations represent the interest factor (IF). These interest factors are a function of interest rates and time and can be derived for any combination of these inputs. A tabular example of the future worth of $1 and the future worth of an annuity of $1, both compounded at the rate of 6% per year, is shown in Figure 15.2, Page 433. Note the 10-year IF of 1.79084 for the future worth of $1 and the IF of 13.18079 for the future worth of an annuity of $1, both of which were developed in the previous examples. Table such as those illustrated on the following pages are available for any interest rate and time combinations.
-Nominal & Effective Rates of Interest
At the simple rate of 8% interest, the nominal rate, which is the actual rate paid by the borrower. The add-on rate of 8% interest has a nominal, or contracted, rate of 8% but results in an actual, of effective, rate of approximately 15%. Even in simple interest rate loan agreements, the nominal rate may not always be the effective rate. When the executing a new mortgage loan for $30,000, at a 3-point discount fee plus other costs totaling $1,000, a borrower will only receive $29,000. The borrower will have to pay interest on the full $30,000 contracted for, effectively raising the interest impact. The entire secondary mortgage market, as well as the bond and government securities markets, is based upon the concept of effective interest rates, or YIELDS, as they are generally termed. When the nominal interest rate on the security instrument is BELOW market interest rates, the security is usually sold at a discount, less than its face value, to develop the required yield. Conversely, when the nominal rate for the security is above the market rate, the security is sold for a premium.
-Compound Interest
Compound interest is generally described as interest paid on interest earned. This is demonstrated in a savings account, in which $1 deposited at the beginning of the year at 6% interest will have a $1.06 balance at the end of that year ($1.00 x 0.06 = $0.06 +$1.00 = $1.06). If this new $1.06 balance is allowed to remain on deposit and the interest rate continues at 6%, the balance in the account at the end of the second year will be $1.1236 ($1.06 x 0.06 = $0.0636 + $1.06 = #1.1236) and so on. Compound interest may be computated by the following formula: CS = BD(1 + i)ˆn Where: CS = Compound sum BD = Beginning deposit i = Interest rate per period n = Number of periods Example: Using this formula, the compound sum of $1,000 left on deposit for 10-years at 6% interest compounded annually would be computed as follows: CS = BD(1 + i)ˆn CS = $1,000(1 + 0.06)ˆ10 CS = $1,000(2.06)ˆ10 CS = $1,000(1.79084) CS = $1,790.84 If interest is compounded more frequently, for instance, monthly or even daily as offered by many banks, the the EFFECTIVE amount of interest earnings would be slightly higher than the nominal rate. Thus, advertisements soliciting for insured deposit at a 6% annual rate indicate 6.06% effective rate because of more frequent compounding periods. The concepts of compounding forms the basis for most investment decisions. Investors reason that a savings account at the current interest rate is a constant investment alternative for their money, and this interest rate constitutes a safe return. In other words, the compound interest earned on a savings account is a safe and viable investment yield against which potential earnings from other investment opportunities can be measured. Within this framework, an investor will measure the risk of alternative investments, assigning a required return to each alternative as a function of its specific risk. Thus, an investment in an apartment project might require at least a 10% return, or yield, as a function of a 6% SAFE RATE and a 4% RISK RATE. This analysis is premised upon the investor's ability to reinvest profits repeatedly at the required 10% yield rate.
*Interest
From the borrower's point of view, interest can be described somewhat simply as RENT PAID for the use of money. A lender views interest as money received or EARNED from a loan investment. Money is borrowed (leased) at a certain interest rate (rent) for a specified time period, during which the amount borrowed is repaid.
-Present Worth of an Annuity
Just as the present worth of $1 is the reciprocal of its compound rate, so also is the present worth of an annuity reciprocally related to its compound formula. The present worth of an annuity is expressed as follows: PWA = RA [(1 ÷ (1 + i)ˆn) + (1 ÷ (1 + i)ˆn-1) ... +(1 ÷ (1 + i)ˆn-2)] Where: PWA = Present Worth of Annuity RA = Regular Amount i = Interest rate per period n = Number of periods Example: The present worth of $1 to be received at the end of each year for 3-years at 6% interest would be $2.67, as shown below: PWA = $1[(1 ÷ (1 + 0.06)ˆ3) + (1 ÷ (1 + 0.06)ˆ2) + (1 ÷ (1 + 0.06)ˆ1)] PWA = $1[(1 ÷ 1.1910) + (1 ÷ 1.1236) + (1 ÷ 0.9433)] PWA = $1(0.8396 + 0.8899 + 0.9433) PWA = $1(2.6729) PWA = $2.6729 or $2.67 rounded Similarly, the present worth of an annuity of $1,000 to be received at the END of each year for 10-years at 6% interest would be $7,360. PWA $1,000[(1.06)ˆ10 = 1/1.790 = 0.558395 ...................+ (1.06)ˆ9 = 1/1.689 = 0.591898 ...................+ (1.06)ˆ8= 1/1.593 = 0.627412 ...................+ (1.06)ˆ7 = 1/1.503 = 0.665057 ...................+ (1.06)ˆ6 = 1/1.418 = 0.704961 ...................+ (1.06)ˆ5 = 1/1.338 = 0.747258 ...................+ (1.06)ˆ4 = 1/1.262 = 0.792094 ...................+ (1.06)ˆ3 = 1/1.191 = 0.839619 ...................+ (1.06)ˆ2 = 1/1.123 = 0.889996 ...................+ (1.06)ˆ1 = 1/1.060 = 0.943396 ............................................Total = 7.360087 PWA = $1,000(7.360087) = $7,360.08 or $7,360 rounded Thus, a contract such as a lease, or any annuity designed to develop a $1,000 net annual cash flow at a 6% interest over the next 10-years, would be worth $7,360 today. In order to better understand the present worth of a long-term lease agreement, it must be recognized that rent is usually paid at the BEGINNING of each rental period. A 10-year lease contract established at a rate of $1,000 per year would most likely require the tenant to make a $1,000 rental payment to the landlord on the first day of the lease period. This payment would cover the rent for the first full year and would be immediately available to the landlord for reinvestment during that year. Because the other nine rental payments would not be received until stipulated future dates, the remaining $9,000 rent contracted for under the lease would not be worth as much today as it would be if the tenant paid the entire 10-years' rent in advance. The landlord cannot invest this money until it is received in yearly installments from the tenant. Thus, the present worth of a lease for 10-years where the landlord would receive $1,000 at the beginning of each rental period, which would be invested at 6%, would be worth the full present amount of the $1,000 for the first year's payment, plus the present worth of an annuity of $1,000 at 6% for nine years or a total of $7,801.70, rounded [$1,000 + ($1,000 x 6.801692) = $7,801.70]. The IF of 6.801692 is derived by subtracting the last year's IF of 0.558395, found at the top of the previous example, from the total IF of 7.360087. This reduces the analysis to $1,000 cash in advance plus a 9-year, $1,000 annual annuity at 6%. Paralleling the tables of interest factors for future worth, as illustrated in Figure 15.1 Page 423, table have also been developed to derive the present worth of $1 or the present worth of an annuity of $1, as shown in Figure 15.3 Page 435. Note that in Figure 15.3, the IF for the present worth of $1 to be received 10-years in the future is 0.558395, the same factor that was derived mathematically in the previous example. Also note that the IF of 7.360087 that appears is Figure 15.3 as the present worth of $1 to be received as an annuity for 10-years in the same as the IF calculated in the preceding example, and the IF of 6.801692 is the amount derived for the 10-year lease where the rent is to be received at the beginning of each period.
*Real Estate Math Formulas
Real estate professionals have a number of formulas and rules to use to determine interest rates, selling prices, and costs. Following are the 3-way formula concepts used in the field. I. 3-Way Formulas: When using this 3-way formula system in solving a variety of math problems, one time is equal to the other two. Place one item in the top half of the circle, according to the formula. Place each of the other two items in adjacent quarters of the bottom half of the circle. Divide the upper half by one of the lower quarters and you will arrive at the solution for the adjacent quarter and vice versa. When multiplying a lower quarter times the adjacent quarter, you will arrive at the solution in the upper half of the circle. A. Interest Formula: Interest = Rate x Principal .....Interest..... Rate | Principal B. Income Formula: Income = Rate x Value .....Income..... Rate | Value C. Area Formula: Area = Width x Length ........Area........ Width | Length D. Commission Formula: Commission = Rate x Sales Price .....Commission..... Rate | Sales Price E. Appraisal Formula: Net Operating Income = Capitalization Rate x Value .....Net Operating Income..... Capitalization | Value F. Made-Percent-Paid Formula: Made = Percent x Paid .......Made....... Percent | Paid II. Selling Price Rule: To find the selling price of an item, subtract the percentage of profit desired from 100% and then divide the cost by the remainder. Remember that the letter s is in both SELLING and SUBTRACT. This will help you recall to subtract from 100%, then divide, when it is the selling price you are seeking. A. Selling price rule applied: Joe wishes to earn 10% profit over the sales price on a lot he bought for $60,000. Joe will have to pay closing costs of $460. What should he sell the lot for? 100% - 10% = 90%. Add the closing costs to the cost of $60,000 to equal $60,460. Divide $60,460 by 90% to arrive at the answer of $67,177.78. B. Selling price rule applied to discounted note problems: Norman bought a note for 4% discount and paid $16,800. What is the face amount of the note? 100% - 4% = 96%. Divide the cost of the note of $16,800 by 96% and the answer is $17,500. III. Cost Rule: To find the cost or net of an item, add the percentage of profit desired to 100% and then divide the selling price by the total percentage. Remember there are are 3-letters in the word NET, which is for this discussion the same as cost, and 3-letters in the word ADD. This will help you to recall to add to 100% and then divide when it is the cost you are seeking: A. Cost rule applied: Mayme sold her property at $90,000 and made a 15% profit. What did the property cost? 100% + 15% = 115%. Divide $90,000 by 115%, and the answer is $78,260.87. B. Cost rule applied when closing costs are included: Mike sells his vacant lot for $45,000. He makes an 18% profit after paying closing costs of $350. What did the lot cost? 100% + 18% = 118%. Deduct closing costs of $350 from the selling price of $45,000 to equal $44,650. Divide $44,650 by 118%, and the answer is $37,838.98. IV. Principal-Interest-Rate-Time-Rule: PRINCIPAL is the loan amount; INTEREST represents the cost of borrowing money; RATE is the cost of borrowing expressed as a percentage of the loan amount paid in interest for one year; and TIME refers to the length of the loan, usually expressed in years. A. Principal unknown: Principal = Interest ÷ (Rate x Time). What is the loan amount necessary to receive $1,200 interest at 12% if the money is loaned for 3-years? Solution: P = $1,200 ÷ 0.12 x 3 P = $1,200 ÷ 0.36 P = $3,333,33 B. Interest unknown: Interest = Principal x Rate x Time. Find the interest on $3,333.33 for 3-years at 12%. I = $3,333.33 x 3 x 0.12 I = $1,200 C. Rate unknown: Rate = Interest ÷ (Principal x Time). Find the rate on $3,333.33 that earns $1,200 interest for 3-years. Rate = $1,200 ÷ $3,333.33 Rate = 0.12 or 12% D. Time unknown: Time = Interest ÷ (Rate x Principal). Find the time necessary to return $1,200 on a principal amount of $3,333.33 at an annual rate of 12%. Time = $1,200 ÷ (0.12 x $3,333.33) Time = $1,200 ÷ 39.99996 Time = 3 years
-Simple Interest
Simple interest is paid only for the amount of principal that is still owed. The formula for computing simple interest is: I = PRT Where: I = Interest P = Principal R = Rate T = Time Example: Using this formula, the interest on a $1,200 loan to be repaid in one year at an 8% annual rate is $96. I = PRT I = $1,200 x 0.08 x 1/12 I = $96/12 I = $8 The amount of the first month's payment in this example is $108 ($1,200 ÷ 12 = $100 +$8 interest = $108). The second month's payment is $107.34 ($1,100 x 0.08 ÷ 12 = $7.34 + $100 = $107.34). The amount of the third month's payment is $106.67 ($1,000 x 0.08 ÷ 12 = $6.67 + $100 = $106.67) and so on until the loan is repaid, when the last payment of $100.67 is made ($100 x 0.08 = $0.67 + $100 = $100.67). This illustrates the fact that simple interest is charged only upon the remaining amount of principal owed and only for the time that the amount is unpaid.
-Present Worth of a Dollar
The present worth of a dollar is, in mathematical terms, the RECIPROCAL of its COMPOUND WORTH. The formula for its derivation is then: PW = A [1 ÷ (1 + i)]ˆn Where: PW = Present worth A = Amount i = Interest rate per period n = Number of periods Example: PW = 1 ÷ (1 + i)ˆ10 PW = $1,000[1 ÷ (1 + 0.06)ˆ10] PW = $1,000 (1 ÷ 1.790848) PW = $1,000 (0.55839) PW = $558.39 or $558.40 rounded To verify that the $1,000 is multiplied by the reciprocal of the compound IF found in Figure 15.1: the worth of a $558.40 deposit at 6% interest for 10-years can be calculated. If the $558.40 were deposited today into a savings account with interest at the rate of 6% compounded annually, this money would grow to $1,000 in 10-years. ($558.40 x 1.790848 = $1,000)
*Time Value of Money
We can observe from the previous examples of compound interest that money grows in value over time. This growth is a function of the rate of interest (yield) being earned on a deposit (investment) during the time it is invested. By the same logic, money NOT received until some time in the future by be worth LESS today. How much less is also a function of yield coupled with time. Example: At a yield rate of 6%, a dollar that will not be received for one year is really worth only 94¢ today. If the dollar could be deposited today at 6% per annum, it would be worth $1.06 at the end of this year. By having to wait a year in order to receive the dollar, the opportunity to earn 6¢ is lost. Thus, the 6¢ can be interpreted as the opportunity cost or discount rate that diminishes the present worth of the subject dollar to be received at the end of the year to 94¢.
Figure 15.1 Future Worth of a Deposit of $1 Compounded at 6% Interest per Period
Yrs.....Growth of $1.....Growth of $1 per ....................................Period (Annuity) 1..........1.060000..........1.000000 2..........1.123600..........2.060000 3..........1.191016..........3.183600 4..........1.262477..........4.374616 5..........1.338226..........5.637093 6..........1.418519..........6.975319 7..........1.503630..........8.393838 8..........1.593848..........9.897468 9..........1.689479..........11.491316 10.........1.790848..........13.180795 11.........1.898299..........14.971643 12.........2.012196..........16.869941 13.........2.132928..........18.882138 14.........2.260904..........21.015066 15.........2.396558..........23.275970 16.........2.540352..........25.672528 17.........2.692773..........28.212880 18.........2.854339..........30.905653 19.........3.025600..........33.759992 20.........3.207135..........36.785591 21.........3.399564..........39.992727 22.........3.603537..........43.392290 23.........3.819750..........46.995828 24.........4.048935..........50.815577 25.........4.291871..........54.864512
