# TYPES OF LOANS - Leeds School of Business TYPES OF LOANS PURE DISCOUNT INTEREST ONLY CONSTANT PAYMENT TYPES OF LOANS PURE DISCOUNT LOANS PURE DISCOUNT LOANS: the borrower receives the money today and repays the loan in one lump sum at some time in the future. Example: you borrow \$10,000 today and agree to repay the loan with 9% annual interest (compounded annually) five years from today. What is your loan balance in five years? FV = \$10,000 (1 + 0.09)5 = \$15,386.24 TYPES OF LOANS PURE DISCOUNT LOANS Another example: Treasury Bills -- for historical reasons, the interest rate on a T-Bill is quoted as a discount: interest rate quoted = interest paid/par value. For example, the quoted rate on a one year \$10,000 T-Bill at 7% interest is: Par value \$10,000.00 Present value \$ 9,345.79 Interest paid \$ 654.21 The interest rate quoted is \$654.21/\$10,000 = 6.54% (even though the interest rate is 7%). TYPES OF LOANS INTEREST ONLY LOANS

INTEREST ONLY LOAN: the borrower receives the money today and agrees to pay the lender interest periodically over the loan term and the principal (the original loan amount) at the end of the loan term. Example: you borrow \$10,000 today and agree to pay interest annually at the annual rate of 9% and repay the principal at the end of five years. What is your annual interest payment? Interest = 0.09 x \$10,000 = \$900 TYPES OF LOANS CONSTANT PAYMENT LOANS FIXED RATE OF INTEREST FIXED LOAN TERM FULLY AMORTIZING FIXED PERIODIC PAYMENTS TYPES OF LOANS CONSTANT PAYMENT LOANS Computingtheequal periodicpayment for amortizedloans: PM T = LoanAmount where CR

n k PM T = = = = 1 nk 1 CR t t 1 (1 ) k theannual contract rateof interest thenumber of yearsintheloanterm thenumber of paymentsper year theequal periodicpayment necessarytofully amortizetheLoanAmount withnkpayments. TYPES OF LOANS CONSTANT PAYMENT LOANS onthlypaym entnecessarytofullyam Com putethem ortizea 30year, 8%annual interest (com

poundedm onthly), \$100,000 loan. PM T = \$100,000 = 360 1 00 . 8t t 1 (1 ) 12 Annual debtservice(DS) \$733.76 = 12xPM T= \$8,805.12 TYPES OF LOANS CONSTANT PAYMENT LOANS For a fixed rate, fixed term, fixed payment, fully amortizing loan, the mortgage balance (book value of the loan) is simply the present value of the remaining stream of payments discounted at the periodic contract rate. Let MBs

= mortgage balance at the end of period s nk s = PMT 1 CR t t 1 (1 ) k TYPES OF LOANS CONSTANT PAYMENT LOANS What is the mortgage balance in five years for a \$100,000, 30 year, 8% annual interest rate, monthly payment loan? The mortgage balance in five years is the present value of the 300 (360-60) remaining monthly payments discounted at the monthly rate of 0.08/12. 300 MB60 1 = \$733.76 = \$ 95,069.26 008 . t t 1 (1 ) 12 TYPES OF LOANS

CONSTANT PAYMENT LOANS A lte rn a tiv e ly , th e m o r tg a g e b a la n c e is th e fu tu re v a lu e (F V ) in : s PV PM T t1 1 M Bs C R t C R s (1 ) (1 ) k k 60 \$ 1 0 0 , 0 0 0 \$ 7 3 3 .7 6 t1 1 M Bs 0 .0 8 t

0 .0 8 6 0 (1 ) (1 ) 12 12 TYPES OF LOANS CONSTANT PAYMENT LOANS A m o r t i z a t i o n s c h e d u l e ss e p a

r a t e t h e p e r i o d i c p a y m e n ti n t o i n t e r e s ta n

d p r i n c i p a l : i n t e r e s tp P e r i o d i c a y m e n t= b

e g i n n i n g b a l a n c e x p e r i o d i c r a t e C R o rI =M

B s s 1 k p r i n c i p a l = P e r i o d i c p e r i o d i c p a y

m e n t-p e r i o d i c i n t e r e s t o rP M T -I s =P s TYPES OF LOANS CONSTANT PAYMENT LOANS Separate the \$733.76 monthly payment into interest and principal for the first two months of the \$100,000, 30 year, 8% annual interest rate loan. Month 1: Interest

= \$100,000.00 x 0.0066667 = \$666.67 Principal = \$733.76 - \$666.67 = \$ 67.09 MB1 = \$100,000.00 - \$67.09 = \$99,932.91 Month 2: Interest = \$99,932.91 x 0.0066667 = \$666.22 Principal = \$733.76 - \$666.22 = \$ 67.54 MB2 = \$99,932.91 - \$ 67.54 = \$99,865.37 TYPES OF LOANS CONSTANT PAYMENT LOANS How would you calculate the amount of interest you paid during the fifth year of a conventional mortgage? You could separate the monthly payments into interest and principal for the 12 months of the fifth year and add the monthly interest payments. Fortunately, theres an easier way: Principal paid between months s and t = MBs - MBt Interest paid = PMT (t - s) - Principal paid TYPES OF LOANS CONSTANT PAYMENT LOANS Compute the principal and interest paid during the fifth year of a \$100,000, 30 year, 8% annual rate, monthly payment mortgage. 312 MB48

1 = \$733.76 0.08 t t 1 (1 ) 12 = \$96,218.44 300 MB60 = \$733.76 1 0.08 t t 1 (1 ) 12 = \$95,069.26 Year 5: Principal paid: \$96,218.44 - \$95,069.26 = \$1,149.17 Interest paid: \$733.76 x 12 - \$1,149.17 = \$7,655.95 TYPES OF LOANS CONSTANT PAYMENT LOANS In w h a t m o n th is o n e h a lf o f th e lo a n re p a id ? s \$ 1 0 0 , 0 0 0 \$ 7 3 3 .7 6

t1 s = 2 6 9 (th e 5 th 1 \$ 5 0 ,0 0 0 0 .0 8 t .0 8 s (1 ) (1 ) 12 12 m o n th o f y e a r 2 2 ) Constant Payment Mortgages: Yields The lenders expected yield or borrowers true borrowing cost is the IRR on the expected mortgage cash flows. Let Fee = loan origination fee, Points = discount points in dollars (points are usually expressed as a percent of the loan amount), S = month that the loan is repaid,

PP = the dollar amount of the prepayment penalty (a percent of the mortgage balance), NLA = net loan amount = Loan Amount - Fee - Points y = the discount rate -- the lenders yield, the borrowers borrowing cost. Constant Payment Mortgages: Yields Computing Lenders Yield (or Borrowers Borrowing Cost) There are 3 cases to consider: (1) The loan is held to maturity; (2) the loan is repaid prior to maturity without penalty; (3) the loan is repaid prior to maturity with a prepayment penalty. Constant Payment Mortgages: Yields C o m p u tin g L e n d e rs Y ie ld (o r B o rro w e rs B o rro w in g C o s t) 1) If th e lo a n is h e ld to m a tu rity , s o lv e fo r y in :

nk NLA PM T t 1 1 (1 y / 1 2 ) t Constant Payment Mortgages: Yields E x a m p le : c o m p u te th e le n d e r s e x p e c te d y ie ld ( o r th e b o r r o w e r s b o r r o w in g c o s t) f o r a \$ 1 0 0 ,0 0 0 , 3 0 y e a r , m o n th ly p a y m e n t m o r tg a g e th a t h a s a 7 .5 % a n n u a l c o n tr a c t r a te o f in te r e s t if th e le n d e r c h a r g e s a \$ 1 ,0 0 0 lo a n o r ig in a tio n f e e , 2 d is c o u n t p o i n t s , a n d e x p e c ts th e b o r r o w e r to h o ld t h e lo a n to m a tu r i ty . N LA PM T = = \$ 1 0 0 ,0 0 0 - \$ 1 ,0 0 0 - \$ 2 ,0 0 0 360 \$ 1 0 0 ,0 0 0 /

t 1 1 = 0 .0 7 5 t (1 ) 12 = \$ 9 7 ,0 0 0 .0 0 \$ 6 9 9 .2 1 Constant Payment Mortgages: Yields E x a m p le (c o n tin u e d ): th e le n d e rs e x p e c te d y ie ld (o r th e b o rro w e rs tru e b o rro w in g c o s t) is th e IR R (o r d is c o u n t ra te y ) in th e fo llo w in g : 360 \$ 9 7 , 0 0 0 \$ 6 9 9 .2 1 t 1 y = 7 .8 1 % 1 y t

(1 ) 12 Constant Payment Mortgages: Yields C o m p u tin g L e n d e rs Y ie ld (o r B o rro w e rs B o rro w in g C o s t) (2 ) If th e lo a n is re p a id p r io r to m a tu rity w ith o u t p e n a lty , s o lv e fo r y in : s NLA PM T t 1 1 M BS y t y s (1 ) (1 ) 12 12 Constant Payment Mortgages: Yields

E x a m p le : c o m p u te th e le n d e rs e x p e c te d y ie ld (o r b o rro w e rs b o rro w in g c o s t) in th e p re v io u s e x a m p le if th e le n d e r e x p e c ts th e b o rro w e r to re p a y th e lo a n , w ith o u t p e n a lty , a t th e e n d o f fo u r y e a rs. 312 M B 48 \$ 6 9 9 .2 1 t 1 S o lv e fo r y 1 \$ 9 5 ,8 6 0 .0 0 0 .0 7 5 t (1 ) 12 = 8 .4 0 % in : 48 \$ 9 7 , 0 0 0 \$ 6 9 9 .2 1 t 1 1 \$ 9 5 ,8 6 0 .0 0

y t y 48 (1 ) (1 ) 12 12 Constant Payment Mortgages: Yields C o m p u tin g L e n d e rs Y ie ld (o r B o rro w e rs B o rro w in g C o s t) (3 ) If th e lo a n is re p a id p rio r to m a tu rity w ith a p re p a y m e n t p e n a lty , s o lv e fo r y in : s NLA PM T t 1 1 M B S P PS y t y s (1 ) (1 )

12 12 P re p a y m e n t p e n a ltie s a re c o m p u te d a s a p e rc e n t o f th e o u ts ta n d in g m o rtg a g e b a la n c e . Constant Payment Mortgages: Yields E x a m p le : c o m p u te th e le n d e rs e x p e c te d y ie ld (o r b o rro w e rs b o rro w in g c o s t) in th e p re v io u s e x a m p le if th e le n d e r e x p e c ts th e b o rro w e r to re p a y th e lo a n , w ith a 2 % p re p a y m e n t p e n a lty , a t th e e n d o f fo u r y e a rs . 48 \$ 9 7 , 0 0 0 \$ 6 9 9 .2 1 t1 FV = 1 \$ 9 5 ,8 6 0 .0 0 \$ 1 ,9 1 7 .2 0 y t y 48 (1 ) (1 ) 12 12 \$ 9 7 ,7 7 7 .2 0 a n d y = 8 .8 2 %

Constant Payment Mortgages: Yields Relationship between mortgage yields and prepayment (with no prepayment penalty) for a 7.5%, 30 year, constant payment mortgage with a \$1,000 loan fee and 2 discount points. Year of Prepayment 1 2 3 4 5 10 20 30 Mortgage Yield 10.69% 9.16% 8.65% 8.40% 8.25% 7.96% 7.83% 7.81% Constant Payment Mortgages: Yields A n n u a

lP e r c e n t a g e R a t e ( A P R )o T h e n a l o a n i s t h e l

e n d e r s y i e l d ( o rb o r r o w e r s b o r r o w i n g c

o s t )c o m p u t e d a s s u m i n g t h e l o a n i s h e l d t o

m a t u r i t y r o u n d e d t o t h e n e a r e s to n e e i g h t

h . 3 . 4 A P R T h e f o rt h e l o a n i n t h e p r e v i o u

s e x a m p l e i s7 Constant Payment Mortgages: Yields C h a rg in g P o in ts to A c h ie v e a D e s ire d Y ie ld If a le n d e r h a s a re q u ire d y ie ld o f y , th e n th e p o in ts th e le n d e r m u s t c h a rg e to o b ta in th e re q u ire d y ie ld a re c o m p u te d b y s o lv in g fo r P o in ts in : s L o a n A m o u n t P o in ts F e e P M T t1 1 (1 y 12 )

t M B PP y s (1 ) 12 s s Constant Payment Mortgages: Yields E x a m p le : c o m p u te th e p o in ts a le n d e r m u s t c h a rg e to e a rn a 9 % r e q u ir e d y ie ld o n a \$ 1 0 0 ,0 0 0 , 3 0 y e a r, 7 .5 % a n n u a l in te re s t ra te , m o n th ly p a y m e n t m o rtg a g e if th e le n d e r c h a r g e s a \$ 1 ,0 0 0 lo a n o r ig in a tio n f e e a n d e x p e c ts th e b o rro w e r to re p a y th e lo a n , w ith o u t p e n a lty , a t th e e n d o f fo u r y e a rs. 48 \$ 1 0 0 ,0 0 0 P o in ts \$ 1 ,0 0 0 \$ 6 9 9 .2 1 t1 \$ 9 9 ,0 0 0 - P o in ts = \$ 9 5 ,0 6 6 .7 5 ;

1 \$ 9 5 ,8 6 0 .0 0 0 .0 9 t 0 .0 9 48 (1 ) (1 ) 12 12 P o in t s = \$ 3 ,9 3 3 .2 5 Alternative Mortgage Instruments Graduated Payment Mortgages (GPMs) Price Level Adjusted Mortgages (PLAMs) Adjustable Rate Mortgages (ARMs) Reverse Annuity Mortgages (RAMs) Shared Appreciation Mortgages (SAMs) Alternative Mortgage Instruments Graduated Payment Mortgage Fixed Contract Rate Fixed Loan Term Payments Increase During First Few Years Payments Known in Advance Permit Negative Amortization Alternative Mortgage Instruments Graduated Payment Mortgage Loan Rate

Term = = = \$100,000 12% 30 years with monthly payments; payment increases 7.5% per year for first five years Year 1 2 3 4 5 6-30 Monthly Payment \$ 791.38 850.73 914.54 983.13 1,056.86 1,136.13 Monthly Interest \$ 1,000.00 1,026.46 1,048.75 1,065.77

1,076.25 Ending Balance \$ 102,645.82 104,874.52 106,576.64 107,624.72 107,870.63 Alternative Mortgage Instruments Price Level Adjusted Mortgage For a fixed payment mortgage, the contract rate of interest, CR, is: CR = rf + + inf where rf = = inf = risk free rate risk premium expected inflation rate With a Price Level Adjusted Mortgage (PLAM), CR = rf + and the outstanding mortgage balance is indexed to the price level to compensate the lender for inflation. Alternative Mortgage Instruments Price Level Adjusted Mortgage Loan Rate Term Inf

= = = = Year 1 2 3 \$100,000 5% 30 years with monthly payments 5%, 6%, and 4% Beginning Balance \$ 100,000.00 103,450.55 107,931.31 Monthly Payment \$ 536.82 563.66 597.48 Ending Balance Before After \$ 98,524.34 \$ 103,450.55 101,821.99 107,931.31 106,116.98 110,361.66

Alternative Mortgage Instruments Adjustable Rate Mortgages Contract Rate Indexed to Lenders Cost of Funds (plus a margin) Term May Adjust Monthly Payment May Adjust Negative Amortization May be Permitted Typically Have Periodic and Lifetime Interest Rate Caps Alternative Mortgage Instruments Adjustable Rate Mortgages Loan Initial Rate Term Index Margin Caps Year 1 2 3 = = = = = = \$100,000 9% 30 years with monthly payments Yields on 1-Year Treasury Securities ( 8%, 9%, 7%) 2.5%

2/5200bp annual cap and 500bp lifetime cap Beginning Balance \$ 100,000.00 99,316.84 98,815.85 Interest Rates Market Contract 9.0% 9.0% 11.5% 11.0% 9.5% 9.5% Monthly Payment \$ 804.62 950.09 841.79 Alternative Mortgage Instruments Reverse Annuity Mortgages The borrower: receives the loan in periodic installments repays the loan in one lump sum at the end of the term

The monthly RAM receipt on a 10 year, \$50,000, 8% annual interest rate RAM is \$273.30. The borrower will recieve 120 of these monthly payments. At the end of the loan term, the borrower will repay the lender \$50,000. Principal = 120 x \$ 273.30 = \$ 32,796.56 Interest = \$50,000 - 32,796.56 = \$ 17,203.44 Alternative Mortgage Instruments Shared Appreciation Mortgages The lender provides the borrower with: a below market rate of interest, or cash to pay a portion of the down payment, or both In exchange for a share of the property value appreciation during the hoding period.

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