B = B(T ) = P e r T = P (0)e r T :

Transcription

1 ath 165 Revenue Steams and otgages This le is http// jlewis/math165/165motgage.pdf I. Income and Investment Steams Simple odel If a pesent value P is invested at time t = with continuous compounding CC at ate, the futue value at time t = T is B = BT = P e T = P e T If we wish to have futue value B at time t = T, we should invest a pesent value by P = P = Be T = BT e T P given If we wish to withdaw amounts B 1 ; B 2 ; ; B N, at times T 1 ; T 2 ; ; T N, we must have pesent value P = P 1 + P P N = B 1 e T 1 + B 2 e T B N e T N = B i e T i Continuous odel { Income Steam We wish to withdaw a continuous income steam { to withdaw continuously at a ate R [dollas/yea] fo T [yeas]. At a typical time t, ove a peiod t, we will withdaw B Rt, a futue value at time t. Thus we need a pesent value investment P e t Rt. The total pesent value needed is P = P e t Rt Re t dt t fom to T Figue 1. Pesent Value P of an Income Steam R, t T. P e t Rt. B Rt T t t P = P! Re t dt 165motgage.pdf page 1/7

2 Similaly, if we invest continuously at ate R [dollas/yea] fo T [yeas], the futue value B at time T will be given by B = R e T t dt Figue 2. Futue Value B of an Investment Steam R, t T. P = Rt. B e T t Rt T t t B = B! Re T t dt Ination Adjustments Simila aguments can be made if the ate, R = Rt, depends on t. Fo example, the ate of contibution investment o income evenue might be continuously adjusted fo ination. In this case the fomulas become If we wish to withdaw a continuous income steam { to withdaw continuously at a ate Rt [dollas/yea] fo T [yeas], we need a pesent value P = Rt e t dt In paticula if we make a cost of living adjustment COLA, of 1 % annually, Rt = R e 1t ; P = R e 1 t dt If we invest continuously at a ate Rt [dollas/yea] fo T [yeas], the futue value B at time T will be given by B = Picing Annuities Rt e T t dt Fo moe on cost of living adjustments COLA, see an execise fom ath 165 at UIC ath 165 Sping 29 SA3 Picing Annuities http// jlewis/math165/165sa39.pdf 165motgage.pdf page 2/7

3 II. otgage Payments A motgage of P fo T yeas with an annual inteest ate is taditionally paid o at a xed [annual] ate R. If P t is the pincipal emaining at time t, in a [small] time peiod t, the payment is R t, the inteest accued is o is P t t so that the change in pincipal is P R t + P t Using the language of dieentials, dp = R dt + P dt Using the magic of dieentials, we divide by dt to obtain the dieential equation dp dt = R + P Thee ae many ways to nd all solutions. Let P t = ute t. Then P P = u e t ; u e t = R; u = Re t ; ut = R e t + C; P t = R + Ce t In addition thee ae two bounday conditions satised by P t P = P ; P T = In ou case, means that the eo, which depends on t and t, satises eot; t lim t! t = divide by dt means eplace dt by t 6=, divide by t, and let t!. 165motgage.pdf page 3/7

4 Using the initial condition, P = P, R C = P ; P t = R + P R e t If T is given, solve the equation P T = fo R R = P e T e T 1 Since R is the annual ate of payment, the monthly payment is = R=12. If R is given, solve the equation P T = fo T = R + P R e T R = R P T = 1 R ln R P e T N.B. In a pactical application, R P >. The case R P = coesponds to paying inteest only. 165motgage.pdf page 4/7

5 Investigations 1. Vaiable Rate otgages. At a time t = T 1, the ate changes fom to 1. Investigate If the monthly payment is held constant, how does the tem expiation date of the loan change? If the tem expiation date of the loan is unchanged, how does the monthly payment change? 2. Acceleated Payments. At time t = T 1, make an exta payment R 1 and continue making the monthly payment. How is the expiation date of the loan changed? 3. Rules of Thumb? Detemine the veacity of the statements Doubling the monthly payment halves the peiod T of the motgage. Doubling the peiod T of the motgage halves the monthly payment. 4. Compae. Seveal motgage calculatos ae available on the web. otgage Calculato - otgage-calc.com http// Take seveal common loans fo example $1,, 3 yeas at 5.75%,. Compae the monthly payment you have calculated using continuous compounding CC and the monthly payment calculated by you favoite web motgage calculato. Ae the esults dieent? 5. onthly Compounding. In pactice, inteest is not compounded continuouslycc. A discete calculation using monthly compounding is used. If an annual ate is compounded equally spaced times in a yea, the annual pecentage ate APR is APR = The actual annual inteest paid is the same as if the APR ate wee simply compounded. On most consume loans, the APR is ecoded. 165motgage.pdf page 5/7

6 Discete Payments otgages and othe loans ae usually paid on a monthly basis. We will assume that thee is a nominal annual inteest ate, and that payments of $R ae made times pe yea so that in a time peiod of 1= yeas, fo a loan of $P, $ 1 + P is due. Stat with a loan of P, and let P n denote the pinciple emaining afte the nth payment is made. The nth payment of R [dollas] coves the accued inteest P n 1 and educes the pincipal by R P n 1. Thee is a fomula fo P n in tems of P ;, and. It is convenient to let Then = 1 + P n = P n 1 R = P n 2 R R = 2 P n R = 2 P n 3 R + 1 R = 3 P n R = = n P n R = n P n 1 1 R If the loan is paid o in T yeas with N = T payments, we have the equation fo R. N P = N = N 1 1 R; R = R N N 1 P Note that 1 = ; 165motgage.pdf page 6/7

7 N = 1 + T 1 + T = [ = 1 + ] T Let E = 1 + Note that is small and lim! = e, so that E e. We obtain E T R = E T 1 P ; E = 1 + e N.B. Since we have thee ae payments in a yea, the total yealy payment, Y, is E T Y = E T 1 P y If Y and P ae given, solve the equation y fo T E T Y = E T P E T 1 ; 1 Y P = E T ; E T = Y Y P ; T = 1 ln E 1 ln Y Y P 165motgage.pdf page 7/7