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UNIT 9: ADJUSTABLE-RATES LOANS
9.1. Adjustable-rate amortization transactions.
An adjustable (floating) rate loan allows the lender to adjust the interest rate so it reflects fluctuations in the cost of money more accurately. If interest rates rise, the borrower’s payments also go up - if the rates fall, the borrower’s periodic payments will drop along with the declining rates.
Thus, in the adjustable rate loans the rate is tied to an index. The future interest adjustments are then based on the upward or downward movements of this index. An index is a reliable statistical report that reflects the approximate change in the cost of money. Some examples of this would be the one-year EURIBOR (average rate of the 1 year transactions among financial institutions in Europe), or the CECA (Confederación Española de Cajas de Ahorro) reference rate. The rise and fall of the payments will fluctuate with the index.
To insure that the expenses of administration and profit are included in the payments to the lender when an interbank index (as Euribor) is used, it is necessary for the lender to add a margin to the index. Margins are added to the index to come up with the interest rate you pay (margin + index = nominal interest rate). Therefore, it is the fluctuation of the index rate that causes the borrowers interest rate to increase or decrease.
Typically, interest payments depend on the future dynamics of a reference rate. Notice that neither the future dynamics of the reference rate nor the future interest payments are known at the time the loan is agreed upon. That is why the true internal effective rate (and, obviously, the true lending and borrowing rates) will be only known at the end of the financial transaction, i.e., after the evolution of the reference rate is known.
In this kind of financial transactions the return on the investment for the lender and the financial cost for the borrower only can be known ex-post.
Elements of an adjustable-rate loan:
- Rate adjustment period: The interest rates on an adjustable-rate mortgage are allowed to be adjusted at certain intervals during the loan term. During the rate adjustment period the interest rate is kept constant. Depending on the type of adjustable loan you have, this interval could be six months, one year (the usual period in the Spanish case), three years or more.
- Index: It is a reliable statistical report that reflects the approximate change in the cost of money. In the Spanish mortgage market there are 7 official indexes. Examples of this would be the average rate of mortgages for three or more years offered by banks, the one-year EURIBOR, the internal rate of return in the secondary market for public debt securities with a remaining term between two and six years, or the CECA reference rate.
- Margin: The margin or spread is the difference between the index rate and the interest charged to the borrower. The margin constitutes a fixed number of percentage points, so it does not change throughout the loan term.
- Teaser rate: A teaser rate is a reduced, introductory interest rate designed to attract borrowers to adjustable rate loans. In US markets, to make the adjustable- rate loans attractive to borrowers, a low beginning interest rate is usually offered and through time these introductory rates became known as “teaser rates”. The interest rate would then rise at each rate adjustment period until the rate equalled the index rate + the margin. After the fully-indexed rate was reached, your loan would then fluctuate with the index on your loan. In the Spanish case, the teaser rates only are taken into account for the period prior to the first rate adjustment (usually one year).
- Interest rate cap or floor: Sometimes there are limits on just how much the payments can go up or down if you have an adjustable-rate loan. An interest rate cap determines an upper bound for the interest rate that the lender can charge every period to the borrower over the life of the loan regardless of what happens in the market whereas the interest rate floor determines the analogous lower bound.
- Conversion option: In the US market, a conversion option on an adjustable-rate loan gives the borrower the option to convert their adjustable-rate loan to a fixed-rate loan. These kind of loans normally have a higher initial interest rate (even the converted fixed rate will usually be higher). You will usually have a time frame in which to convert the loan to a fixed rate. For example, you might have to make your decision to convert the loan sometime after the first year and before the fifth year ends. In most cases, there is also a conversion fee imposed on the borrower (for instance 1% of the total loan amount).
How it works?
- Interest adjustment periods should be established.
- For the first interest period (it could be usually the first year, or the first six months) the nominal interest rate to be applied j 1 (m) is known. It is the so-called “teaser rate”.
- For the rest of interest periods, their respective interest rates are obtained from the following equation: j (^) s(m) = r (^) s ± spread
where: j (^) s(m) : nominal interest rate for the s-th period r (^) s : market rate, index rate, or reference rate at the beginning of the s-th period
Therefore, at the initial date, C 0 , j 1 (m) , spread , and adjustment periods are known.
First adjustment period:
Periodic payments:
nxm|i 1 (m)
0 1
a
C
a =
Outstanding balance at the end of the first interest adjustment period:
C 1 =a 1 a(nxm)−m|i 1 (m)
Second adjustment period:
Periodic payments:
nxm m|i 2 (m)
1
2 a
C
a −
Outstanding balance at the end of the second interest adjustment period:
C 2 =a 2 anxm− 2 m|i 2 (m)
n-th adjustment period:
Periodic payments:
m|in(m)
n 1 n
a
C
a = − → C (^) n = 0
Therefore, following this procedure, this financial transaction will have constant total periodic payments within each interest period (year), which will change each interest period (year), according to the evolution of the reference rate.
Example 9.1:
Obtain the total periodic payments of a loan of 75,000€ and three years term, with the following conditions: Monthy constant payments over each adjustment period. Annual adjustment periods. Nominal interest rate for the 1st period : 6%. Remaining periods: index rate plus 0.75 percentage points. Assume that the index rates take the following values for each of the periods:
ir 2 = 0. 05 ; ir 3 = 0. 045
Solution:
A posteriori , which is the only way to solve this type of financial transactions, the periodic payments will be:
(^1124) | 24 | 0 , 005
12 | 36 | 0 , 005
0 1
( 12 ) 1 1
1 (^12 )
1 (^12 ) → = ⋅ = ⋅ =
a a
a a
i
nx i C a
C
a
Year j i
(^2212) | 12 | 0 , 00479 ...
24 | 24 | 0 , 00479 ...
1 2
( 12 ) 2 2 2
2 (^12 )
2 (^12 ) → = ⋅ = ⋅ =
a a
a a
i
i
r
C a
C
a
Year j i i
3 12 | 12 | 0 , 004375
2 3
( 12 ) 3 3 3
3 (^12 )
C
C
a
Year j i i
a i a
r
9.2.2. Loan with predetermined total periodic payments and varying term.
The parties agree on the payments amount, based on a given initial term and interest rate. These payments do not have to be necessarily constant, even though usually they are.
If the market (index) rate increases a lot, most part of the periodic payment will go to interest due and the principal repayment will be lower than expected, so that the term might increase indefinitely. To avoid this, usually a maximum term is set, and a balloon payment is done at the end of the maximum term in order to pay off the outstanding debt.
How it works:
For every period s :
- Set the as.
- I (^) s = Cs (^) − 1 ⋅ is
- As = as − Is
- Cs = Cs (^) − 1 − As
6
6
5 4 5
( 2 ) 5 43
( 2 ) 3 3 3
= − = − × =
C
A
C C A
A a Ci
Year j ir i
7 6 7
7
( 2 ) 4 4 4
= − × =
C C A
A
Year j ir i
As the maximum term is 4 years, the eighth payment will be the last one and, therefore, it must be enough to cancel the outstanding debt. This is,
a 8 =C 7 ( 1 +i( 42 ))= 19. 842 , 03 ( 1 + 0 , 0325 )= 20. 486 , 90 >a
If the maximum term had not arrived yet, the financial transaction could continue for another year and the last periodic payment would be smaller than the periodic payment initially agreed.
9.3. Other adjustable-rate loans with fixed term: known principal repayments.
Since (^0) 1
n h h
C A
=
= ∑ , knowing the principal repayments implies knowing the total term of
the loan.
It constitutes the simplest case. How it works?
- Cs = Cs (^) − 1 − As
- I (^) s = Cs (^) − 1 ⋅ is
- as = Is + As
Example 9.3:
Obtain the periodic payments of the following adjustable-rate loan:
- C 0 : 90,000€.
- n : 4 years.
- Constant annual principal repayments.
- Annual adjustment periods.
- Teaser rate: 4%
- Remaining periods: reference rate plus 0.25 percentage points. Assume that the index rates take the following values for each of the periods:
ir 2 = 0. 035 ; ir 3 = 0. 03 ; ir 4 = 0. 0275
Solution:
This is solved, a posteriori, as it is done with the corresponding fixed-interest loan:
- 500 4
A = = €.
Year 1 → i 1 = 0 , 04 → a 1 = C 0 i 1 + A = 90. 000 ⋅ 0 , 04 + 22. 500 = 26. 100
2 12 0 2
(^2 ) = + = − + = ⋅ + =
a Ci A C Ai A
Year i ir
3 23 0 3
(^3 ) = + = − + =
a Ci A C Ai A
Year i ir
4 i 0 , 0025 0 , 0275 0 , 0025 0 , 03
4 34 0 4
(^4 ) = + = − + =
a Ci A C Ai A
Year ir
Example 9.4:
Obtain the periodic payments for the first two years of the previous problem, if the interest was paid semi-annually.
Solution:
A = = €.
1 0
( 2 ) 1 01
( 2 ) 0 , 5 01
( 2 ) 1
C C A
a Ci A
a Ci
Year i
2 1
( 2 ) 2 12
( 2 ) 1 , 5 12
2 2 2
C C A
a Ci
a Ci
i Year i r
