Loans (1)

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
Computing the equal periodic payment for amortized loans:
PMT = Loan Amount

where
CR
n
k
PMT

=
=
=
=

1
nk

1
∑1 CR t
t = (1 +
)
k

the annual contract rate of interest
the number of years in the loan term
the number of payments per year
the equal periodic payment necessary to fully
amortize the Loan Amount with nk payments.

TYPES OF LOANS
Compute the monthly payment necessary to fully amortize a
30 year, 8% annual interest (compounded monthly), $100,000
loan.
PMT =

$100,000
=
360
1
∑ 0.08 t
t = (1 +
1
)
12

Annual debt service (DS)

$ 733.76

= 12 x PMT = $8,805.12

TYPES OF 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
= PMT

nk −s

1
∑ CR t
t = (1 +
1
)
k

TYPES OF 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
0.08 t
t = (1 +
1
)
12

TYPES OF LOANS
Alternatively, the mortgage balance is the future value (FV)
in:
s

1
MBs
PV = PMT ∑
+
CR t
CR s
t =1 (1 +
) (1 +
)
k
k
60

1
MBs
$100,000 = $733.76∑
+
0.08 t
0.08 60
t =1 (1 +
) (1 +
)
12
12

TYPES OF LOANS
Amortization schedules separate the periodic payment into
interest and principal:
Periodic interest payment = beginning balance x periodic rate
or Is = MBs-1
Periodic principal

CR
k
= periodic payment - periodic interest

or Ps = PMT - Is

TYPES OF 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
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, there’s an easier way:
Principal paid between months s and t = MBs - MBt
Interest paid = PMT (t - s) - Principal paid

TYPES OF 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
In what month is one half of the loan repaid?

s

1
$50,000
$100,000 = $733.76∑
+
0.08 t
.08 s
t = 1 (1 +
) (1 +
)
12
12
s = 269 (the 5th month of year 22)

Constant Payment Mortgages:
Yields
The lender’s expected yield or borrower’s 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 lender’s yield, the
borrower’s borrowing cost.

Yields
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 without
penalty;

(3)

the loan is repaid prior to maturity with a
prepayment penalty.

Yields

1)

If the loan is held to maturity, solve for y in:

nk

1
NLA = PMT ∑
t
t =1 (1 + y / 12 )

Yields
Example: compute the lender’s expected yield (or the
borrower’s borrowing cost) for a $100,000, 30 year,
monthly payment mortgage that has a 7.5% annual contract
rate of interest if the lender charges a $1,000 loan
origination fee, 2 discount points, and expects the borrower
to hold the loan to maturity.
NLA = $100,000 - $1,000 - $2,000
PMT =

360

= $97,000.00

1
=
$100,000 / ∑
0.075 t
t = (1 +
1
)
12

$699.21

Yields
Example (continued): the lender’s expected yield (or the
borrower’s true borrowing cost) is the IRR (or discount rate y)
in the following:
360

1

t =1

y t
(1 + )
12

$97,000 = $699.21∑

y = 7.81%

Yields
(2)

If the loan is repaid prior to maturity without penalty,
solve for y in:
s

1

MBS
NLA = PMT ∑
+
y t
y s
t =1 (1 +
) (1 + )
12
12

Yields
Example: compute the lender’s expected yield (or borrower’s
borrowing cost) in the previous example if the lender expects
the borrower to repay the loan, without penalty, at the end of
four years.
312

1
MB48 = $699.21∑
= $95,860.00
0.075 t
t =1 (1 +
)
12
Solve for y = 8.40% in:
48

1

$95,860.00
$97,000 = $699.21∑
+
y t
y 48
t =1 (1 +
)
(1 + )
12
12

Yields
(3)

If the loan is repaid prior to maturity with a prepayment
penalty, solve for y in:

MBS + PPS
NLA = PMT ∑
+
y t
y s
t = 1 (1 +
)
(1 + )
12
12
s

1

Prepayment penalties are computed as a percent of the
outstanding mortgage balance.

Yields
Example: compute the lender’s expected yield (or borrower’s
borrowing cost) in the previous example if the lender expects
the borrower to repay the loan, with a 2% prepayment penalty,
at the end of four years.

$95,860.00 + $1,917.20
$97,000 = $699.21∑
+
y t
y 48
t = 1 (1 +
)
(1 + )
12
12
48

1

FV = $97,777.20 and y = 8.82%

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%

Yields

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

The APR for the loan in the previous example is

7

3
.
4

Yields
Charging Points to Achieve a Desired Yield
If a lender has a required yield of y, then the points the lender
must charge to obtain the required yield are computed by
solving for ‘Points’ in:

MB s + PPs
1
Loan Amount − Points − Fee = PMT∑
+
y
y
t =1
(1 + ) t
(1 + ) s
12
12
s

Yields
Example: compute the points a lender must charge to earn a
9% required yield on a $100,000, 30 year, 7.5% annual
interest rate, monthly payment mortgage if the lender
charges a $1,000 loan origination fee and expects the
borrower to repay the loan, without penalty, at the end of
four years.
48

1
$95,860.00
$100,000 − Points − $1,000 = $699.21∑
+
0.09 t
0.09 48
t= 1
(1 +
) (1 +
)
12
12

$99,000 - Points = $95,066.75;

Points = $3,933.25

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
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

Price Level Adjusted Mortgage
For a fixed payment mortgage, the contract rate of interest, CR, is:
CR = rf + ρ
+ inf
where rf = risk free rate
=
ρ risk premium
inf = expected inflation rate
With a Price Level Adjusted Mortgage (PLAM),
CR = rf + ρ
and the outstanding mortgage balance is indexed to the price level
to compensate the lender for inflation.

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

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
•

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%
2/5—200bp 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

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

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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