loans (1)

TYPES OF LOANS

• PURE DISCOUNT

• INTEREST ONLY

• CONSTANT PAYMENT

TYPES OF LOANSPURE 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 theloan with 9% annual interest (compounded annually) fiveyears from today. What is your loan balance in five years?

FV = $10,000 (1 + 0.09)5 = $15,386.24

TYPES OF LOANSPURE 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 at7% interest is:

Par value $10,000.00Present value $ 9,345.79Interest paid $ 654.21

The interest rate quoted is $654.21/$10,000 = 6.54%(even though the interest rate is 7%).

TYPES OF LOANSINTEREST ONLY LOANS

INTEREST ONLY LOAN: the borrower receives the moneytoday and agrees to pay the lender interest periodically overthe loan term and the principal (the original loan amount) atthe end of the loan term.

Example: you borrow $10,000 today and agree to pay interestannually at the annual rate of 9% and repay the principal atthe end of five years. What is your annual interest payment?

Interest = 0.09 x $10,000 = $900

TYPES OF LOANSCONSTANT PAYMENT LOANS

• FIXED RATE OF INTEREST

• FIXED LOAN TERM

• FULLY AMORTIZING

• FIXED PERIODIC PAYMENTS


Computing the equal periodic payment for amortized loans:

PMT = Loan Amount

whereCR = the annual contract rate of interest n = the number of years in the loan term k = the number of payments per yearPMT = the equal periodic payment necessary to fully

amortize the Loan Amount with nk payments.

11

11( ) CR

ktt

nk


Compute the monthly payment necessary to fully amortize a30 year, 8% annual interest (compounded monthly), $100,000loan.

PMT = = $ 733.76

Annual debt service (DS) = 12 x PMT= $8,805.12

$100,

(.

)

0001

100812

1

360

tt


For a fixed rate, fixed term, fixed payment, fully amortizingloan, the mortgage balance (book value of the loan) is simplythe present value of the remaining stream of paymentsdiscounted at the periodic contract rate.

Let MBs = mortgage balance at the end of period s

= PMT 1

11 ( )

CRk

tt

nk s


What is the mortgage balance in five years for a $100,000, 30year, 8% annual interest rate, monthly payment loan?

The mortgage balance in five years is the present value of the300 (360-60) remaining monthly payments discounted at themonthly rate of 0.08/12.

MB60 = $733.76 = $ 95,069.261

100812

1

300

(.

)

tt


A l t e r n a t i v e l y , t h e m o r t g a g e b a l a n c e i s t h e f u t u r e v a l u e ( F V )i n :

P V P M TC Rk

M BC Rk

tt

ss

s

1

1 11 ( ) ( )

$ 1 0 0 , $ 7 3 3 .(

.) (

.)

0 0 0 7 61

10 0 8

1 21

0 0 81 2

1

6 0

6 0

tt

sM B


Amortization schedules separate the periodic payment intointerest and principal:

Periodic interest payment = beginning balance x periodic rate

orIs =MBs-1

Periodic principal = periodic payment - periodic interest

or Ps =PMT - Is

CR

k


Separate the $733.76 monthly payment into interest andprincipal 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 = $666.67Principal = $733.76 - $666.67 = $ 67.09MB1 = $100,000.00 - $67.09 = $99,932.91

Month 2:Interest = $99,932.91 x 0.0066667 = $666.22Principal = $733.76 - $666.22 = $ 67.54MB2 = $99,932.91 - $ 67.54 = $99,865.37


How would you calculate the amount of interest you paidduring the fifth year of a conventional mortgage?

You could separate the monthly payments into interest andprincipal for the 12 months of the fifth year and add themonthly interest payments.

Fortunately, there’s an easier way:

Principal paid between months s and t = MBs - MBt

Interest paid = PMT (t - s) - Principal paid


Compute the principal and interest paid during the fifth year ofa $100,000, 30 year, 8% annual rate, monthly paymentmortgage.

MB48 = $733.76 = $96,218.44

MB60 = $733.76 = $95,069.26

Year 5:Principal paid: $96,218.44 - $95,069.26 = $1,149.17Interest paid: $733.76 x 12 - $1,149.17 = $7,655.95

1

100812

1

312

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)

tt

1

100812

1

300

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)

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I n w h a t m o n t h i s o n e h a l f o f t h e l o a n r e p a i d ?

s = 2 6 9 ( t h e 5 t h m o n t h o f y e a r 2 2 )

$ 1 0 0 , $ 7 3 3 .(

.)

$ 5 0 ,

(.

)0 0 0 7 6

1

10 0 8

1 2

0 0 0

10 8

1 21

tt

s

s

Constant Payment Mortgages:Yields

The lender’s expected yield or borrower’s true borrowingcost 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 - Pointsy = the discount rate -- the lender’s yield, the

borrower’s borrowing cost.


Computing Lender’s Yield (or Borrower’s Borrowing Cost)

There are 3 cases to consider:

(1) The loan is held to maturity;

(2) the loan is repaid prior to maturity withoutpenalty;

(3) the loan is repaid prior to maturity with aprepayment penalty.


C o m p u t i n g L e n d e r ’ s Y i e l d ( o r B o r r o w e r ’ s B o r r o w i n g C o s t )

1 ) I f t h e l o a n i s h e l d t o m a t u r i t y , s o l v e f o r y i n :

N L A P M Ty t

t

n k

1

1 1 21 ( / )


E x a m p l e : c o m p u t e t h e l e n d e r ’ s e x p e c t e d y i e l d ( o r t h eb o r r o w e r ’ s b o r r o w i n g c o s t ) f o r a $ 1 0 0 , 0 0 0 , 3 0 y e a r ,m o n t h l y p a y m e n t m o r t g a g e t h a t h a s a 7 . 5 % a n n u a l c o n t r a c tr a t e o f i n t e r e s t i f t h e l e n d e r c h a r g e s a $ 1 , 0 0 0 l o a no r i g i n a t i o n f e e , 2 d i s c o u n t p o i n t s , a n d e x p e c t s t h e b o r r o w e rt o h o l d t h e l o a n t o m a t u r i t y .

N L A = $ 1 0 0 , 0 0 0 - $ 1 , 0 0 0 - $ 2 , 0 0 0 = $ 9 7 , 0 0 0 . 0 0

P M T = = $ 6 9 9 . 2 1$ 1 0 0 , /(

.)

0 0 01

10 0 7 5

1 21

3 6 0

tt


E x a m p l e ( c o n t i n u e d ) : t h e l e n d e r ’ s e x p e c t e d y i e l d ( o r t h eb o r r o w e r ’ s t r u e b o r r o w i n g c o s t ) i s t h e I R R ( o r d i s c o u n t r a t e y )i n t h e f o l l o w i n g :

y = 7 . 8 1 %

$ 9 7 , $ 6 9 9 .( )

0 0 0 2 11

11 2

1

3 6 0

y tt



( 2 ) I f t h e l o a n i s r e p a i d p r i o r t o m a t u r i t y w i t h o u t p e n a l t y ,s o l v e f o r y i n :

N L A P M Ty

M Bytt

sS

s

1

11 2

11 2

1 ( ) ( )


E x a m p l e : c o m p u t e t h e l e n d e r ’ s e x p e c t e d y i e l d ( o r b o r r o w e r ’ sb o r r o w i n g c o s t ) i n t h e p r e v i o u s e x a m p l e i f t h e l e n d e r e x p e c t st h e b o r r o w e r t o r e p a y t h e l o a n , w i t h o u t p e n a l t y , a t t h e e n d o ff o u r y e a r s .

S o l v e f o r y = 8 . 4 0 % i n :

M Btt

4 81

3 1 2

2 11

10 0 7 5

1 2

8 6 0 0 0

$ 6 9 9 .(

.)

$ 9 5 , .

$ 9 7 , $ 6 9 9 .( )

$ 9 5 , .

( )0 0 0 2 1

1

11 2

8 6 0 0 0

11 2

1

4 8

4 8

y ytt



( 3 ) I f t h e l o a n i s r e p a i d p r i o r t o m a t u r i t y w i t h a p r e p a y m e n tp e n a l t y , s o l v e f o r y i n :

P r e p a y m e n t p e n a l t i e s a r e c o m p u t e d a s a p e r c e n t o f t h eo u t s t a n d i n g m o r t g a g e b a l a n c e .

N L A P M Ty

M B P Pytt

sS S

s

1

11 2

11 2

1 ( ) ( )


E x a m p l e : c o m p u t e t h e l e n d e r ’ s e x p e c t e d y i e l d ( o r b o r r o w e r ’ sb o r r o w i n g c o s t ) i n t h e p r e v i o u s e x a m p l e i f t h e l e n d e r e x p e c t st h e b o r r o w e r t o r e p a y t h e l o a n , w i t h a 2 % p r e p a y m e n t p e n a l t y ,a t t h e e n d o f f o u r y e a r s .

F V = $ 9 7 , 7 7 7 . 2 0 a n d y = 8 . 8 2 %

$ 9 7 , $ 6 9 9 .( )

$ 9 5 , . $ 1 , .

( )0 0 0 2 1

1

11 2

8 6 0 0 0 9 1 7 2 0

11 2

1

4 8

4 8

y ytt


Relationship between mortgage yields and prepayment (with noprepayment penalty) for a 7.5%, 30 year, constant paymentmortgage with a $1,000 loan fee and 2 discount points.

Year of Prepayment Mortgage Yield 1 10.69%

2 9.16% 3 8.65% 4 8.40% 5 8.25%10 7.96%20 7.83%30 7.81%


The Annual Percentage Rate (APR) on a loan is the lender’syield (or borrower’s borrowing cost) computed assuming theloan is held to maturity rounded to the nearest one-eighth.

The APR for the loan in the previous example is 73

4.


C h a r g i n g P o i n t s t o A c h i e v e a D e s i r e d Y i e l d

I f a l e n d e r h a s a r e q u i r e d y i e l d o f y , t h e n t h e p o i n t s t h e l e n d e rm u s t c h a r g e t o o b t a i n t h e r e q u i r e d y i e l d a r e c o m p u t e d b ys o l v i n g f o r ‘ P o i n t s ’ i n :

L o a n A m o u n t P o i n t s F e e P M T1

( 1y

1 2)

M B P P

( 1y

1 2)tt 1

ss s

s


E x a m p l e : c o m p u t e t h e p o i n t s a l e n d e r m u s t c h a r g e t o e a r n a9 % r e q u i r e d y i e l d o n a $ 1 0 0 , 0 0 0 , 3 0 y e a r , 7 . 5 % a n n u a li n t e r e s t r a t e , m o n t h l y p a y m e n t m o r t g a g e i f t h e l e n d e rc h a r g e s a $ 1 , 0 0 0 l o a n o r i g i n a t i o n f e e a n d e x p e c t s t h eb o r r o w e r t o r e p a y t h e l o a n , w i t h o u t p e n a l t y , a t t h e e n d o ff o u r y e a r s .

$ 9 9 , 0 0 0 - P o i n t s = $ 9 5 , 0 6 6 . 7 5 ; P o i n t s = $ 3 , 9 3 3 . 2 5

$ 1 0 0 , 0 0 0 P o i n t s $ 1 , 0 0 0 $ 6 9 9 . 2 11

( 10 . 0 9

1 2)

$ 9 5 , 8 6 0 . 0 0

( 10 . 0 9

1 2)t 4 8t 1

4 8

Alternative Mortgage Instruments

Graduated Payment Mortgages (GPMs)

Price Level Adjusted Mortgages (PLAMs)

Adjustable Rate Mortgages (ARMs)

Reverse Annuity Mortgages (RAMs)

Shared Appreciation Mortgages (SAMs)


Graduated Payment Mortgage

Fixed Contract Rate

Fixed Loan Term

Payments Increase During First Few Years

Payments Known in Advance

Permit Negative Amortization


Graduated Payment Mortgage

Loan = $100,000Rate = 12%Term = 30 years with monthly payments;

payment increases 7.5% per year for first five years

Year Monthly Monthly EndingPayment Interest Balance

1 $ 791.38 $ 1,000.00 $ 102,645.822 850.73 1,026.46 104,874.523 914.54 1,048.75 106,576.644 983.13 1,065.77 107,624.725 1,056.86 1,076.25 107,870.63

6-30 1,136.13

Alternative Mortgage InstrumentsPrice Level Adjusted Mortgage

For a fixed payment mortgage, the contract rate of interest, CR, is:

CR = rf + + inf

where rf = risk free rate = risk premiuminf = expected inflation rate

With a Price Level Adjusted Mortgage (PLAM),

CR = rf +

and the outstanding mortgage balance is indexed to the price levelto compensate the lender for inflation.


Price Level Adjusted Mortgage

Loan = $100,000Rate = 5%Term = 30 years with monthly paymentsInf = 5%, 6%, and 4%

Year Beginning Monthly Ending Balance Balance Payment Before After

1 $ 100,000.00 $ 536.82 $ 98,524.34 $ 103,450.552 103,450.55 563.66 101,821.99 107,931.313 107,931.31 597.48 106,116.98 110,361.66


Adjustable Rate Mortgages

Contract Rate Indexed to Lender’s 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 InstrumentsAdjustable Rate Mortgages

Loan = $100,000Initial Rate = 9%Term = 30 years with monthly paymentsIndex = Yields on 1-Year Treasury Securities ( 8%, 9%, 7%)Margin = 2.5%Caps = 2/5—200bp annual cap and 500bp lifetime cap

Year Beginning Interest Rates Monthly Balance Market Contract Payment

1 $ 100,000.00 9.0% 9.0% $ 804.622 99,316.84 11.5% 11.0% 950.093 98,815.85 9.5% 9.5% 841.79

Alternative Mortgage InstrumentsReverse 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


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.

loans (1)

Business

borrowers

constant payment

lenders expected

equal periodic

outstanding

monthly payment

lenders yield

mortgage balance