Fixed-Rate Mortgages

Transcription

1 Fixed-Rate Motgages 1 A Motivating Example We begin with a motivating example. Example 1. Smith takes out a 5-y fixed ate motgage fo 10,000 at a loan inteest ate of 10% (compounded annually). Smith makes equal annual payments to the bank ove the life of the loan. We adopt the following notation: N = the numbe of peiods of the loan. i = the pe-peiod loan inteest ate. LB(n) = the outstanding loan balance at time n, n = 0, 1,..., N. IP (n) = the inteest payment made at time n fo peiod n. P P (n) = the pincipal payment made at time n. T P (n) = the total payment made at time n. Table 1 shows the espective cash flow steams fo this example, calculated as follows: Table 1: Cash flows fo Example 1. Loan Balance (LB) Total Payment (TP) Inteest Payment (IP) Pincipal Payment (PP) Total Payment. The bank detemines the total payment (TP) so that the value of the total payment cash flow steam (T P (1), T P (2),..., T P (N)) = (T P, T P,..., T P ) (1)

2 at the ageed-upon loan inteest ate equals the initial loan balance. Thus, [ ] 1 (1 + i) N LB(0) = T P, (2) i T P = Inteest Payments. IP (n) = ilb(n 1). ilb(0) 0.10(10000) = = (3) 1 (1 + i) N 1 (1.1) 5 Pincipal Payments. Fo this type of loan P P (n) = T P IP (n). Obsevations: 1. The loan balance deceases by the pincipal payment amount. Bank eans inteest on the outstanding loan balance ove the peiod. The new loan balance equals the old loan balance plus inteest eaned minus the cash the bank eceives (the total payment made by Smith). That is, 2. Pincipal payments gow exponentially. LB(n) = LB(n 1) + IP (n) T P (n) (4) = LB(n 1) + IP (n) [IP (n) + P P (n)] (5) = LB(n 1) P P (n). (6) P P (n) = T P IP (n) (7) = T P ilb(n 1) (8) = T P i[lb(n 2) P P (n 1)] using (6) (9) = [T P ilb(n 2)] + ip P (n 1) (10) = [T P IP (n 1)] + ip P (n 1) using (7) (11) = P P (n 1) + ip P (n 1) (12) = (1 + i)p P (n 1). (13) 3. The outstanding loan balance at time n always equals the pesent value of the futue total payment cash flow steam. Hee, total payment cash flow steam is the one associated when the loan was oiginated. Fo example, the loan balance at time 3 equals = At time n thee ae N n payments emaining. Thus, [ ] 1 (1 + i) (N n) LB(n) = T P i (14)

3 [ ] ilb(0) [ 1 (1 + i) (N n) ] = 1 (1 + i) N (15) i [ ] 1 (1 + i) (N n) = LB(0) 1 (1 + i) N. (16) Thus, the atio of the outstanding loan balance at time n to the initial loan balance is given by: LB(n) LB(0) 1 (1 + i) (N n) =. (17) 1 (1 + i) N To calculate the outstanding loan balance fo this type of loan, you can eithe (i) fist calculate the total payment (TP) fo the oiginal loan in (3) and then substitute it into (14) o (ii) you can diectly use (16). Once you know LB(n) you can calculate the upcoming peiod s inteest and pincipal payments. 4. The loan balance at the end of the loan always equals 0. Since the loan balance deceases each peiod by the pincipal payment amount, the sum of the pincipal payments always equals the initial loan balance. (Can you show this?) 2 Conventional Fixed-Rate Motgages Typically, a home motgage has monthly payments with tems of eithe 15, 20 o 30 yeas. Inteest is compounded monthly. The equations above still apply but the pe-peiod inteest ate i equals the loan inteest ate divided by 12. Do not foget this! Example 2. Conside a 30-yea, fixed-ate motgage fo 125,000 at 6.00%. Hee i = 0.06/12 = What is the monthly total payment? T P = (0.005)(125, 000) 1 (1.005) 360 = (18) What is the loan balance afte 10 yeas and 8 months? Hee, n = 128 peiods and thee ae = 232 peiods emaining. Keep in mind you have to convet to monthly peiods! Using (14), [ ] 1 (1.005) 232 LB(128) = = 102, (19) Altenatively, using (17), LB(128) LB(0) which gives the same loan balance. = 1 (1.005) 232 = , 1 (1.005) 360

4 We now addess two pactical questions often asked by a motgagee. (1) If I want to pay off my loan in M months, then how much moe do I have to pay pe month? The pesent value of the new total payment cash flow steam (each total payment now equals T P + A) ove the emaining M months must equal the cuent outstanding loan balance. That is, fom which one can solve fo A. LB(0) = (T P + A)(1 (1 + i) M ), (20) i (2) If I pay an additional A dollas pe month, then how many months M will it take to pay off my loan? To answe this question, we can use (20), except hee the value of A is known and we seek the value of M. Example 3. Conside once again the loan of Example 2, and suppose the emaining duation of the loan is 19 yeas and 4 months. Recall that T P = Since the emaining duation equals 232 months we also know that the outstanding loan balance equals 102, If 100 is added to the monthly payment, then we have 102, = [1 (1.005) M ], (21) which implies that M = months (o yeas). How much must be added to the monthly payment to pay off the loan in 10 yeas (o 120 payments peiods)? Let A denote the additional amount. We have that (T P + A) = 102, (0.005) 1 (1.005) 120 = , (22) which implies that A = Example 4. Conside a 15-yea conventional fixed-ate motgage fo 200,000 at 6.25%. (The solution is in backets.) What is the monthly payment? [T P = ] What is the loan balance afte 4 yeas, 3 months? [LB(51) = 160, ] Suppose the emaining duation of the loan is 10 yeas and 9 months. If we pay 2000 each month how quickly will the loan be paid off? [ months o 8.70 yeas.] Suppose the emaining duation of the loan is 10 yeas, 9 months. How much do we have to add to ou monthly payment to pay off the loan in 5 yeas? [ ]

5 3 Refinancing The US govenment allows one to deduct inteest payments fom thei tax obligations. Consequently, the afte-tax cash flow steam is diffeent than the constant total payment cash flow steam. This diffeence should be consideed when deciding on whethe to efinance. Fist, we look at the motivating example s afte-tax cash flow steam, which is summaized in Table 2. The tax ate τ = Table 2: Afte-tax cash flows fo Example 1 when i = 10%. Loan Balance (LB) Total Payment (TP) Inteest Payment (IP) Pincipal Payment (PP) Tax Shield (TS) Afte-tax cash flow Remak 1. The effective afte-tax loan inteest ate equals (1 τ)i = 0.60(0.10) = 0.06 o 6%. What is the pesent value of the afte-tax cash flow steam using the afte-tax loan inteest ate? Is this always tue? Can you pove it? [Hint: Apply the poject balance concept and the fact that the sum of the pincipal payments equals the initial loan balance (Obsevation 4).] The loan of Example 1 could have been oiginated many yeas ago; howeve, thee ae only five yeas left now. At the time of oigination the pevailing maket ate of inteest (fo loans of this type of isk) was 10%. Let us suppose that inteest ates have come down and the pevailing maket ate of inteest is now 8%. Should Smith efinance? To answe this question, we fist compute the afte-tax cash flow steam assuming Smith did efinance, as shown in Table 3. If Smith wee to efinance then the incemental cash flow steam is given in Table 4. What is the value of this cash flow steam? It depends on the choice of the discount ate. An appopiate discount ate is the afte-tax loan inteest ate equal to (1-0.40)(0.08) = o 4.8%, which epesents the cost of funds to Smith. In this case, the value of efinancing equals = (23) If Smith took out a hypothetical loan at 8% in the amount of , then the epayment schedule associated with the incemental afte-tax cash flows would be as shown in Table 5. By constuction, the value of the afte-tax cash flows associated with the hypothetical loan is zeo.

6 Table 3: Afte-tax cash flows fo Example 1 when i = 8%. Loan Balance (LB) Total Payment (TP) Inteest Payment (IP) Pincipal Payment (PP) Tax Shield (TS) Afte-tax cash flow Table 4: Incemental afte-tax cash flow steam. Incemental Afte-Tax Cash Flow Remak 2. Given a cash flow steam, discount it at the afte-tax loan inteest ate. Take out a loan in this amount at the pe-tax inteest ate. Assume the pincipal payments ae such that the afte-tax cash flow steam so geneated pecisely matches the given cash flow steam. This equiement detemines the inteest and total payment cash flow steams, as shown, fo example, in Table 5. By constuction, the pesent value of the given cash flow steam at the afte-tax loan inteest ate is zeo. The loan value at the end of the loan will always be zeo, and thus the implied epayment plan is valid. Can you pove this? (Hint: Apply the poject balance concept.) Refinancing entails costs (e.g. oigination fees, appaisal, Smith s time). If the cost to Smith exceeds the , then he should not finance. What if the cost is less than this value? Actually, it is possible that the best decision is fo Smith to wait fo example, suppose Smith knew inteest ates wee about to fall shotly? This kind of analysis equies a stochastic model of inteest ates and a game plan fo how to make such decisions that involve isk ove time, a subject to be discussed late. The value of efinancing citically depends on how long the loan will last. Fo example, if Smith wee to sell the asset and pay off his loan soone than 5 yeas, then the value of efinancing will be less. Suppose, fo example, that Smith intends to pay off the loan afte 3 yeas. Then the espective afte-tax cash flow steams associated with the oiginal and new loans would be as shown in Table 6. At 4.8% the pesent value of the evised incemental afte-tax cash flow steam is

7 Table 5: Afte-tax cash flows fo Hypothetical Loan. Loan Balance (LB) Total Payment (TP) Inteest Payment (IP) Pincipal Payment (PP) Tax Shield (TS) Afte-tax cash flow Table 6: Afte-tax cash flows when hoizon equals thee yeas Afte-tax cash 10% = Afte-tax cash 8% = Incemental afte-tax cash flow Loan Valuation In this section we deive a geneal fomula fo the Loan Value, namely, the value of the afte-tax cash flows associated with a loan. We assume that the loan is valid, namely, the final loan balance is zeo. We equie one additional assumption: inteest is always paid on the cuent outstanding loan balance, i.e., IP (n) = ilb(n 1). (This cetainly holds tue fo the fixed-ate motgage type loans examined above.) The implication of this assumption is that at any time the outstanding loan balance equals the sum of the futue pincipal payments, i.e., Notation and Definitions. LB(n) = k=n+1 P P (k). (24) τ denotes the appopiate tax ate. i a = (1 τ)i denotes the afte-tax loan inteest ate. denotes the appopiate pe-peiod discount ate. LoanP incipalv alue := LB(0) N P P (n) n=1 (1+) n pincipal cash flows. equals the pesent value of the loan

8 AfteT axinteestv alue = (1 τ) N IP (n) n=1 (1+) n inteest cash flows. equals the pesent value of the afte-tax Geneal Fomula. LoanV alue = (1 i a ) LoanP incipalv alue. (25) Deivation. Since the inteest payments ae deductible LoanV alue = LB(0) In light of (27) it emains to show that n=1 P P (n) + (1 τ)ip (n) (1 + ) n (26) = LoanP incipalv alue Af tet axinteestv alue. (27) AfteT axinteestv alue = i a which we deive in the following sequence of identities: AfteT axinteestv alue = (1 τ) = (1 τ) = (1 τ) k=1 LoanP incipalv alue, (28) n=1 n=1 n=1 IP (n) (1 + ) n (29) ilb(n 1) (1 + ) n (30) i N k=n P P (k) (1 + ) n using (24) (31) k P P (k) = (1 τ)i (1 + ) k=1 n=1 n (32) N [ ( k )] 1 = i a P P (k) (1 + ) n (33) N = i a P P (k) = i a = i a = i a k=1 { N k=1 { n=1 P P (k) LB(0) 1 (1 + ) k k=1 } P P (k) (1 + ) k k=1 } P P (k) (1 + ) k (34) (35) (36) LoanP incipalv alue. (37)

9 Obsevations: 1. When the discount ate is chosen equal to i a, then the Loan Value Fomula (25) immediately shows that the LoanV alue = 0. You can veify this fo the afte-tax cash flow steams of Examples 1, 2 and The T axshieldv alue := τ n=1 IP (n) (1 + ) n (38) is the pesent value of the tax shield cash flow steam. The T axshieldv alue is an impotant component of the Adjusted Pesent Value (APV) appoach to value copoate pojects financed in pat by debt. By the definitions τ T axshieldv alue = ( ) AfteT axinteestv alue (39) 1 τ τ = ( 1 τ ) (1 τ)i LoanP incipalv alue using (28) (40) = τ i LoanP incipalv alue. (41) 3. An appopiate discount ate fo the pupose of copoate valuation is the loan inteest ate, i.e., = i. In this special case, and so (1 i a i (1 τ)i ) = (1 ) = τ (42) i LoanV alue = τ LoanP incipalv alue = T axshieldv alue. (43) The second identity follows fom (41). That is, when = i, then the LoanV alue is the T axshieldv alue. Fo this special case identity (43) is easily obtained if one ecalls that the pesent value of the total payment cash flow steam at the loan inteest ate always equals the initial loan balance. (Can you pove this?) Example 5. What ae the LoanV alue and T axshieldv alue associated with the loan of Example 1 when the discount ate equals the loan inteest ate of 10%? A diect calculation shows that [ LoanV alue = ] = T axshieldv alue = = Of couse, we have aleady shown in (43) that these two values must be the same when = i. We can also use the Loan Value Fomula (25) to calculate eithe quantity: LoanV alue = [ ( )] = 0.40( ) =

10 Why use the Loan Value Fomula? Both calculations of the LoanV alue o T axshieldv alue use the numbes obtained fom the Table. Fo this type of loan no table is equied when = i! Hee is why. We have shown that the pincipal payments gow exponentially at the loan inteest ate. Since the pincipal payment cash flow steam is being discounted at this same ate, it follows that = = (1.1) (1.1) (1.1) (1.1) = 1 [ ] (1.1) (1.1) (1.1) (1.1) (1.1) = 5( ). 1.1 Fo this type of loan the geneal ule fo calculating the LoanV alue o T axshieldv alue when = i is this: [ LoanV alue = T axshieldv alue = τ LB(0) N P P (1) ] (44) 1 + i [ ] N (T P ilb(0)) = τ LB(0). (45) 1 + i You only need to calculate the value of T P = ilb(0)/(1 (1 + i) N ). Example 6. Conside a conventional 30-y fixed-ate loan fo 100,000 at 6%. What ae the LoanV alue and T axshieldv alue when the discount ate equals the loan inteest ate of 6%? The tax ate is 40%. The calculations ae: T P = 0.005(100, 000) = (1.005) 360 ilb(0) = 0.005(100, 000) = P P (1) = = LoanV alue = T axshieldv alue = 0.40[100, (99.55) ] = 25, Example 7. Suppose the loan of Example 6 is paid off at the end of yea 10. (This is typical of a eal estate loan.) What ae the LoanV alue and T axshieldv alue associated with this epayment schedule when the discount ate equals the loan inteest ate of 6%? To apply the geneal fomula, keep in mind that (i) thee will be an additional balloon pincipal payment at the end of peiod 120 in the amount of LB(120) whose pesent value must be accounted fo, and (ii) only multiply P P (1) by the numbe of peiods n = 120. The calculations ae: LB(120) = 100, (1.005) 240 = 83, (1.005) 360 LoanV alue = T axshieldv alue = 0.40[100, (99.95) , ] = 16,

11 Example 8. Let s evisit the efinancing example. We obtained a value of The incemental cash flow steam obtained in Table 4 was discounted at the afte-tax loan inteest ate of 4.8%. The incemental cash flow steam is the diffeence between the afte-tax cash flows steams associated with the loans at 8% and 10%, espectively. So, in effect, the value of equals the LoanV alue of the 8% loan minus the LoanV alue of the 10% loan both discounted at 4.8%. But the LoanV alue of the 8% loan discounted at 4.8% equals zeo, so the value of the efinancing equals minus the LoanV alue of the 10% loan discounted at 4.8%. When applying the Loan Value fomula to obtain this latte quantity, you can not use the simple expession (45). The eason is that the pincipal payment cash flow steam gows at 10% but the discount ate equals 4.8%. Hee, you must apply the gowing annuity fomula: [ LoanP incipalv alue = 10, ] [ = 10, (1.1) ] (1.1) (1.1) (1.1) [ ( )] 1 ( = 10, ) = 10, = Now, we ae in position to calculate the LoanV alue as (1 0.40)0.10 (1 ) ( ) = , (46) which confims ou ealie calculation. (Keep in mind the efinance value hee is minus the LoanV alue.) Example 9. Suppose you company is consideing puchasing a machines that costs 1 million. The manufactue offes to finance the puchase by lending you the puchase pice fo 5 yeas with annual inteest payments of 5%. The pincipal of 1 million is paid at the end of yea 5. (This is diffeent than the loans we have consideed up to now, but the geneal fomula still applies.) The local bank will chage you 15% fo such a loan. Assume that you will commit to epay the loan epayment schedule. You tax ate is 40%. What is the value of this loan? If you take this loan, the cash flows ae shown in Table 7. What is the appopiate discount ate? Fo easonably safe loans (i.e. vey low pobability of default), the coect discount ate is you company s afte-tax, unsubsidized boowing ate, which is 9% in this case. (You can use the same agument we used to justify the discount ate in the efinance example.) In this case, the pesent value of the afte-tax cash flow is 233, 379 = 1, 000, , 000(1 (1.09) 5 ) 0.09 We can also apply the Loan Value Fomula, as follows: LoanV alue = ( ( 0.09 ) 1, 000, 000 1, 000, (47) ) 1, 000, = 233, 379. (48)

12 Table 7: Afte-tax cash flows fo Example 9 when i = 5%. Loan Balance (LB) 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 0 Total Payment (TP) 50,000 50,000 50,000 50,000 1,050,000 Inteest Payment (IP) 50,000 50,000 50,000 50,000 50,000 Pincipal Payment (PP) ,000,000 Tax Shield (TS) 20,000 20,000 20,000 20,000 20,000 Afte-tax cash flow 1,000,000-30,000-30,000-30,000-30,000-1,030,000 In effect, the manufactue is subsidizing the puchase pice by the amount of 233,379. Fo puposes of any othe poject evaluation, the cost to the company to acquie the machine is 766,621. We can also detemine the value of the subsidy povided by the manufactue by examining the incemental afte-tax cash flows. The afte-tax cash flow associated with financing the machine puchase at the company s cost of funds is shown in Table 8. Compaing the afte-tax Table 8: Afte-tax cash flows fo Example 9 when i = 15%. Loan Balance (LB) 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 0 Total Payment (TP) 150, , , ,000 1,150,000 Inteest Payment (IP) 150, , , , ,000 Pincipal Payment (PP) ,000,000 Tax Shield (TS) 60,000 60,000 60,000 60,000 60,000 Afte-tax cash flow 1,000,000-90,000-90,000-90,000-90,000-1,090,000 cash flows shown in Tables 7 and 8, we see that the incemental afte-tax cash flow gain fo financing at the manufactue s ate nets 60,000 each yea fo 5 yeas. The PV of this cash flow steam at the company s afte-tax cost of funds of 9% is P V = 60, (1.09) = 233, 379. (49)