Financial Math FM/Loans

add more examples and the missing stuffs Learning objectives The Candidate will understand key concepts concerning loans and how to perform related calculations. Learning outcomes The Candidate will be able to: Define and recognize the definitions of the following terms: principal, interest, term of loan, outstanding balance, final payment (drop payment, balloon payment), amortization. Calculate: The missing item, given any four of: term of loan, interest rate, payment amount, payment period, principal. The outstanding balance at any point in time. The amount of interest and principal repayment in a given payment. Similar calculations to the above when refinancing is involved. Introduction In this chapter, two methods of repaying a loan will be discussed, namely amortization method and sinking fund method. In particular, for each of these two methods, we will discuss how to determine he outstanding loan balance at any point in time, and the amount of interest and principal repayment in each payment made by borrower. Amortization method (Amortization method) For amortization method, the borrower repays the lender by a series of payments at regular intervals. Each payment is applied first to interest due on the outstanding balance at the time just before the payment is made to pay the interest, and after deducting the amount of interest from each payment, the amount left in each payment is going as the principal repayment to reduce the loan balance (i.e. how much the borrower owes). Payments are made to reduce the loan balance to exactly zero. Amortization of level payment The series of payments made by borrower is level in this subsection, and payments form annuity-immediate in our discussion For annuity-due, a payment is made immediately after receiving the loan, which is unusual. Even if this is the case, the situation is the same as that for annuity-immediate, except that the amount of loan is and payments last for periods (see the following for explanation of notations). . To illustrate this, consider the following diagrams. Borrower's perspective: L R R ... R ... R ↑ ↓ ↓ ↓ ↓ ---|-----|-----|-------|----------|--- 0 1 2 ... k ... n Lender's perspective: L R R ... R ... R ↓ ↑ ↑ ↑ ↑ ---|-----|-----|-------|----------|--- 0 1 2 ... k ... n in which ↑ means the amount is received, ↓ means the amount is paid; is the amount borrowed (i.e. the amount of loan); is the number of payments; is the level payment made by the borrower (this is return from the lender's perspective). Let be the outstanding balance at time , just after the th payment (, which is the initial balance). Let be the effective interest rate during each interval for payments. (Recursive method to determine outstanding balance (level payment)) . First, will accumulate to from time to . The interest due on is . So, the reduction of outstanding balance from the payment of at is . It follows that the outstanding balance at time is . (Fundamental relationship between amount of loan and payments) . Using the above recursive method, ; ; ; ... . Since for amortization method (the loan balance is reduceed to zero at the end by definition), we have Because of this relationship, we can calculate by pressing PMT N I/Y CPT PV to get . In particular, is inputted since payments are cash outflow (we use borrower's perspective here). (Prospective method to determine outstanding balance (level payment)) . From the proof of fundamental relationship between and , we have . (Retrospective method to determine outstanding balance (level payment)) From the proof of fundamental relationship between and , we have Another method to determine outstanding balance (and also principal and interest paid in different payments) is using BA II Plus. Procedure: Input into PMT (if is unknown, it should be determined first). Input into PV We can also compute PMT or PV given sufficient information. Press 2ND PV Press the starting payment number (for th payment, press ) and press ENTER ↓. Press the ending payment number (for th payment, press ) and press ENTER ↓ (press the same number as the starting payment number for selecting exactly one payment you may press other numbers for selecting multiple payments.). Then, outstanding balance just after the selected payment(s) is displayed (BAL=... is displayed). Press ↓ and loan (or "principal") paid in the selected payment(s) is displayed (PRN=... is displayed). Press ↓ and interest paid in the selected payment(s) is displayed (INT=... is displayed). Suppose a loan of 2000 is repaid by 15 annual payments of in arrears. With annual interest rate of 10%, . {Calculate with annual interest rate of 10% payable semiannually. |type="()"} - 192.68 || What is the annual effective interest rate? - 262.95 || What is the annual effective interest rate? + 266.71 || The annual effective interest rate is . || Then, - 804.23 || 10% is not semiannual effective interest rate. - 964.35 Suppose a loan of is repaid by 20 monthly payments of in arrears. Calculate the annual effective interest rate . Solution: The monthly effective interest rate is calculated by (using BA II Plus). So, the annual effective interest rate . {Calculate if there are 30 such monthly payments. |type="()"} + 190.93% || The monthly effective interest rate is calculated by . || So, . - 217.56% - 274.23% - 290.93% - 317.56% A loan of 1000 is repaid by 12 annual level payments in arrears. The annual interest rate is 8%. Then, the outstanding balance at the end of 5th year is 690.86 by pressing 1000 PV 12 N 8 I/Y CPT PMT 2ND PV 5 ENTER ↓ 5 ENTER ↓, and outstanding balance is displayed. {Calculate the outstanding balance at the end of 8th year. |type="()"} - 42.38 || This is the interest paid in the 8th payment. - 90.31 || This is the principal repaid in the 8th payment. - 341.97 + 439.50 || Without clearing memory, press 2ND PV 8 ENTER ↓ 8 ENTER ↓ and the outstanding balance is displayed. - 529.81 {A loan is repaid by 10 annual payments at the end of each year. In the first 3 years, payments of 1000 are made at the end of each year. In the remaining 7 years, payments of 2000 are made at the end of each year.} {Choose the correct expression for the amount of loan. |type="()"} + || We can split the payments of 2000 to two payments of 1000. Then, a series of payments of 1000 are made at the end of each year throughout the 10 years (), || and another series of payments of 1000 are made at the end of each year from year 3 to year 10 (). - || The payments in the remaining 7 years are 2000 each. - || Payments of 1000 are made in the first 3 years. - || Different payments are made at different time, and you cannot just add them up. - || Payments of 2000 at the end of each year only in the remaining 7 years. {Choose the correct expression for the amount of outstanding balance at year 3. |type="()"} - || What is the amount of loan? - || What is the amount of loan? + || Get this by prospective method. - || We are calculating the outstanding balance at year 3, not amount of loan. - None of the above || Consider prospective and retrospective methods. {Choose the correct expression for the amount of outstanding balance at year 7. |type="()"} + || Use retrospective method. Amount of loan is . || Accumulated value of past payment paid is by splitting each of the payments of 2000 at the end of each of year 4,5,6,7 to two payments of 1000, || then a series of payments of 1000 is paid at the end of each of year 1,2,3,4,5,6,7, (the corresponding accumulated value is ), and || another series of payment is paid at the end of each of year 4,5,6,7 (the corresponding accumulated value is ). - || What is accumulated value of past payments paid? - || What is accumulated value of past payments paid? - || The three future payments are 2000 each. - None of the above || Consider prospective and retrospective methods. Now, we consider the amount of interest and principal repayment in each payment made by borrower. (Splitting an installment into principal and interest repayments (level payment)) Let be the principal repaid in the th installment (i.e. th payment made by borrower), and be the amount of interest paid in the th installment. (This notation has different meaning in the context of Chapter 1.) Suppose the installments made by borrower is level, and each of them equals . Then, First, by definitions, because the installment is first deducted by the interest due (), and the remaining amount () is used to repay principal. Therefore, . It remains to prove the formula for . By definition, because the interest is due on the outstanding balance (before the th installment). We can use BA II Plus to determine and , which is discussed before. After splitting each installment, we can make an amortization schedule which illustrates the splitting of each repayment in a tabular form. An example of amortization schedule is as follows: + Amortization schedule for a loan of repaid over periods at rate Period Payment Interest paid Principal repaid Outstanding loan balance 0 0 0 0 (prospective) 1 1 2 1 ... ... ... ... ... 1 ... ... ... ... ... 1 1 Total not important (You may verify the recursive method to determine outstanding balance using this table, e.g. ) It can be seen that total payment () equals total interest paid () plus total principal repaid (), and each payment equals the interest paid plus principal repaid in the corresponding period (read horizontally), as expected, because the payment is either used for paying interest, or used for repaying principal. It can also be seen that the total principal repaid equals the amount of loan (i.e. outstanding loan balance in period 0) (), as expected, because the whole loan is repaid by the payments in periods. A loan of 1000 is repaid by seven annual payments of in arrears. The annual interest rate is 5%. The principal repaid in 3rd payment is approximately 135.41 (press 1000 PV 7 N 5 I/Y CPT PMT 2ND PV 3 ENTER ↓ 3 ENTER ↓ ↓); the interest paid in 3rd to 6th payment is approximately 107.65 (press ↑ ↑ 6 ENTER ↓ ↓ ↓, continuing from the above pressing sequence). {A loan is being repaid with a series of payments at the end of each month for 5 years, and amount of each payment is 100. The annual interest rate is 10%.} {Choose the correct expression for the amount of principal repaid in the 2nd payment. |type="()"} - || Payment is made at the end of each month. - || Payment is made at the end of each month. Also, consider the above amortization schedule. + || The equivalent monthly interest rate is . So, the amount is || - || Consider the above amortization schedule. - None of the above {Calculate the sum of amount of principal repaid in 3rd payment and payments after this payment. |type="()"} - 63.09 || This is the principal repaid in 3rd payment. - 124.68 || This is the sum of principal repaid in 1st and 2nd payment. - 126.68 || This is the sum of principal repaid in 3rd and 4th payment. + 4629.17 || The amount of loan is which equals the total amount of principal repaid. || Therefore, the sum is . || Alternatively, using BA II Plus, press 100 PMT 60 N I/Y CPT PV 2ND PV 3 ENTER ↓ 60 ENTER ↓ ↓, and the answer is displayed. - 4691.76 || This is the sum of principal repaid in 2nd payment, 3rd payment, etc. {Choose the correct expression for the outstanding balance at the end of year 4. |type="()"} - || - || - || + || || by retrospective method - || Amortization of non-level payment In this subsection, we will consider amortization of non-level payment. The ideas and concepts involved are quite similar to the amortization of non-level payment. Borrower's perspective: L R_1 R_2 ... R_k ... R_n ↑ ↓ ↓ ↓ ↓ ---|-----|-----|-------|----------|--- 0 1 2 ... k ... n Lender's perspective: L R_1 R_2 ... R_k ... R_n ↓ ↑ ↑ ↑ ↑ ---|-----|-----|-------|----------|--- 0 1 2 ... k ... n in which are non-level payments, and the other relevant notations used in amortization of level payment have the same meaning. Because the payments are now non-level, we need formulas different from that for the amortization of level payment to determine amount of loan and outstanding balance at different time, and to split the payment into interest payment and principal repayment. They are listed in the following. (Relationship between amount of loan and payments (non-level payment)) Omitted since the main idea is identical to the proof for the level payment version. (Prospective method to determine outstanding balance (non-level payment)) Omitted since the main idea is identical to the proof for the level payment version. (Retrospective method to determine outstanding balance (non-level payment)) Omitted since the main idea is identical to the proof for the level payment version. (Recursive method to determine outstanding balance (non-level payment)) Omitted since the main idea is identical to the proof for the level payment version. Recursive method is more useful for non-level payment than level payment. (Splitting an installment into principal and interest repayments (non-level payment)) : It follows from definition of . : It follows from which is true by definition. {A loan is repaid with repayments that start at 100 at the 1st year, and increase by 100 per year until a payment of 1000 is made, and thereafter no payments will be made. The annual effective interest rate is 5%.} {Choose a correct expression for outstanding balance at the end of 4th year. |type="()"} - || Try to draw a time diagram. - || Try to draw a time diagram. - || Try to draw a time diagram. + || We can split the payment of 500 into payments of 400 and 100, payment of 600 into payments of 400 and 200, etc. || By prospective method, the outstanding balance is . - || Try to draw a time diagram. {Choose a correct expression for principal repaid at the end of 7th year. |type="()"} - || Try to draw a time diagram. - || Try to draw a time diagram. + || Payment at the end of 7th year is . Outstanding balance at the end of 7th year is || - || Try to draw a time diagram. - || Try to draw a time diagram. Amortization of payments that are made at a different frequency than interest is convertible In this situation, we can obtain the amount of loan, outstanding balance, and principal repaid and interest paid in a payment, by calculating the equivalent interest rate that is convertible at the same frequency at which payments are made. Then, the previous formulas can be used directly, at this equivalent interest rate. This method is analogous to the method for calculating the annuity with payments made at a different frequency than interest is convertible. {A loan is repaid with level payments of 2000 each at the end of each quarter for 10 years. The semiannual effective interest rate is 5%. Calculate the interest paid in the 3rd payment. Correct your answer to two decimal places. |type="{}"} { 1208.53 } || There are 40 quarters in 10 years. || The equivalent effective interest rate per quarter is . || Then the interest paid in the 3rd payment is Sinking fund method After discussing amortization method, we discuss another way to repay a loan, namely sinking fund method. (Sinking fund method) For sinking fund method, all principal (i.e. amount of loan) is repaid by the borrower in a single payment at maturity. Interest due on the principal is paid at the end of each period and a deposit is made into a sinking fund at the end of each period (same amount of deposit for each of the time points), so that the accumulated value of sinking fund equals the amount of principal at maturity. Borrower's perspective: Loan repayment: L Li Li ... Li L ↑ ↓ ↓ ↓ ↓ ---|-----|-----|------------------|-----|--- 0 1 2 ... n-1 n \ / \ / \ / \ / \ / ... \ / i i i rate Sinking fund: D D ... D D L ↓ ↓ ↓ ↓ ↗ ---|-----|-----|------------------|-----|--- 0 1 2 ... n-1 n \ / \ / \ / \ / \ / ... \ / j j j rate Lender's perspective: (Lender do not know how the borrower repays the loan, so sinking fund is not shown) Loan repayment: L Li Li ... Li L ↓ ↑ ↑ ↑ ↑ ---|-----|-----|------------------|-----|--- 0 1 2 ... n-1 n \ / \ / \ / \ / \ / ... \ / i i i rate in which is the amount borrowed is the number of payment periods is the effective interest rate paid by borrower to lender is the effective interest rate earned on the sinking fund (which is usually strictly less than in practice) is the level sinking fund deposit Let is the level payment made by borrower at the end of each period, which equals interest paid to lender, i.e. . By definition of sinking fund method, because the accumulated value of sinking fund equals amount of loan at maturity. Using these two equation, we can have the following theorem. (Relationship between each payment made by borrower and amount of loan in sinking fund method) Because Recall that . We can observe that a similar expression compared with the right hand side appears in above equation (). In view of this, we define (we use '' because the right hand side involves both and .) Then, if the amount of loan is 1, then the payment made by borrower at the end of each period is . Naturally, we would like to know what equals. We can determine this as follows: (The right hand side also involve and , as expected, because the reciprocal of an expression involving and should also involve and ) In particular, if , as expected, and Therefore, each level payment made by borrower in the sinking fund method is the same as the levelpayment in the amortization method, because in amortization method of level payment. Using this notation, we can express the relationship between and as follows: {A borrower borrows a loan of 1000 for ten years, and repays it using the sinking fund method. The annual effective interest rate charged for the loan is 5%, and the annual effective interest rate earned for the sinking fund is 3%.} {Choose correct expression(s) for the amount of each payment made at the end of year. |type="[]"} - || Which percentage is ? Which percentage is ? - || The order matters in the notation. The first and second rate represent different rates. - || Wrong formula. + || . . - || It should be instead. If we assume that the balance in the sinking fund could be used to reduce the amount of loan, then net amount of loan after th payment is net amount of interest paid in the th period is and principal repaid in the th period is {TODO |type=}