Lecture 6: Fixed rate mortgages

Question 1

Annual Compounding formula?

Answer 1

Annual Compounding:

FV = PV * (1 + R)^T

Question 2

Monthly Compounding formula?

Answer 2

Monthly Compounding:

FV = PV * (1 + R/12)^(T*12)

Question 3

Effective Annual Yield (Annualised Interest Rate):

EAY = ????

Answer 3

Effective Annual Yield (Annualised Interest Rate):

EAY = (FV - PV) / PV

Question 4

? means repaying a loan

Answer 4

Amortisation: repaying a loan

Question 5

??: how much t’ principal rises as interest on t’ loan rises.

Answer 5

Accrual rate: how much t’ principal rises as interest on t’ loan rises.

Question 6

Types of ??? (CPM) Loans | Pay rate | Balance at Maturity:

1. ?? (FA) | ? than accrual rate | ? repaid
2. ? Amortising (PA) | ? than accrual rate | ? repaid
3. ?? (IO) | ? to accrual rate | ? to amount borrowed
4. ? Amortising (NA) | ? than accrual rate | ? than amount borrowed.

Answer 6

Types of Constant Payment Mortgage (CPM) Loans | Pay rate | Balance at Maturity:

1. Fully Amortising (FA) | greater than accrual rate | Fully repaid
2. Partially Amortising (PA) | greater than accrual rate | Partially repaid
3. Interest Only (IO) | equal to accrual rate | equal to amount borrowed
4. Negative Amortising (NA) | less than accrual rate | greater than amount borrowed.

Question 7

3. Interest Only (IO) Mortgage:
Repayment amount of each period:
PMT = ???
(PV = present value = amount borrowed)
 => Balance at maturity = FV = ? ( (i.e. still owe lender the ?)

Answer 7

3. Interest Only (IO) Mortgage:
Repayment amount of each period:
PMT = r * PV 
(PV = present value = amount borrowed)
 => Balance at maturity = FV = PV (i.e. still owe lender the principal)

Question 8

1. Fully Amortising mortgage:
   PMT = ?????
   FV = ?: i.e. Loan is fully repaid.

Answer 8

1. Fully Amortising mortgage:
   PMT = { r / [1 - 1/ (1+r)^n] } * PV
   FV = 0: i.e. Loan is fully repaid.

Question 9

2. Partially amortising mortgage:
   { r / [1 - 1/ (1+r)^n] } * PV > ? > ??
   Some balance left at maturity: ? < FV < ?

Answer 9

2. Partially amortising mortgage:
   { r / [1 - 1/ (1+r)^n] } * PV > PMT > r * PV
   Some balance left at maturity: 0 < FV < PV

Question 10

4. Negatively amortising loans:
   PMT < ??
   FV ? PV

Answer 10

4. Negatively amortising loans:
   PMT < r * PV
   FV > PV

Question 11

Equation holding for all CPM:
PV = ................?
or PV = .....?
r: interest rate of each period
n: number of periods

Answer 11

Equation holding for all CPM:
PV = [PMT / (1+r)^1] + [PMT / (1+r)^2] + … + [PMT / (1+r)^n] + [FV / (1+r)^n] (Equation 1)
or PV = PMT * { [1 - 1/(1+r)^n] / r} + [FV/(1+r)^n]

r: interest rate of each period
n: number of periods

Question 12

For FA mortgages:
FV = ?
PMT = ????

Answer 12

For FA mortgages:
FV = 0
PMT = { r / [1 - (1+r)^n] } * PV
(derived from equation 1 & 2)

Question 13

For IO loans, PMT = r*PV & FV = PV, substitute to equation 1, we get:
??? = [1 / (1+r)] + [1 / (1+r)^2] +…+ [1/ (1+r)^n] (Equation 2)

Answer 13

For IO loans, PMT = r*PV & FV = PV, substitute to equation 1, we get:
{ 1 - [1 / (1+r)^n] } / r = [1 / (1+r)] + [1 / (1+r)^2] +…+ [1/ (1+r)^n]
(Equation 2)

Question 14

Functions in Excel:
'?': future value: outstanding loan balance
'?': present value (amount borrowed)
'?': fixed regular repayments per period
'?': number of payment periods
'?': interest rate per period

Answer 14

Functions in Excel:
‘FV’: future value: outstanding loan balance
‘PV’: present value (amount borrowed)
‘PMT’: fixed regular repayments per period
‘nper’: number of payment periods
‘rate’: interest rate per period

Question 15

Reverse Annuity Mortgages (RAM): aka:

• ? mortgage (US)
• ? mortgage / ?? mortgage (UK)

Answer 15

Reverse Annuity Mortgages (RAM): aka:

• Reverse mortgage (US)
• Lifetime mortgage / Equity release mortgage (UK)

Question 16

UK Equity Release mortgage:
Use the house as a collateral to borrow money, ? a constant regular payments from lender & pay back interest & principal at loan maturity date.
=> PV = ?
- if borrower > 65 yrs, cash is ??.
- borrower continues to live in the house
But:
- interest rate is quite ?
- reduce ? => nothing/ little left for heirs.

Answer 16

RAM / (UK) Equity Release mortgage:
Use the house as a collateral to borrow money, RECEIVE a constant regular payments from lender & pay back interest & principal at loan maturity date.
=> PV = 0
- if borrower > 65 yrs, cash is tax-free.
- borrower continues to live in the house
But:
- interest rate is quite high
- reduce estate => nothing/ little left for heirs.

Question 17

Formula needed to calculate number of repayment periods:

a^n = b => n = ??

Answer 17

Formula needed to calculate number of repayment periods:

a^n = b => n = log_a (b)