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Grado: Administración y Dirección de Empresas
FINANCIAL MATHEMATICS June 2012
Group AR
Surname......................................................……….................................. Name....................................
Please write your answers inside the boxes.
1. Determine whether the following statements are true or false. Explain carefully your answers.
(2 marks)
1.1. With the compound interest rule, the equivalent rates relationships let us confirm that the nominal rate
will always be higher than its equivalent effective rate.
1.2. In a financial transaction with several inflows and several outflows, where the last amount of money is laid
out by the lender, the sign of the outstanding balance will change at some point during the financial
transaction.
1.3. The present value and the accumulated value of a constant annuity will increase if the interest rate
increases.
1.4. In a financial transaction with just bilateral terms and conditions, the lending effective rate and the
borrowing effective rate will be the same and they will be equal to the pure effective rate.
1.5. In a “level-payment fixed-rate loan” (equal payments and constant interest rate), over the term of the loan,
the interest portion of each payment increases.
1.6. In a loan, the principal amount C 0 is equal to the arithmetic sum of all principal repayments.
1.7. In a zero-coupon bond, the outstanding balance will never be equal to the principal (or face value), except
at the issuing date t 0.
3. Represent graphically the evolution of the outstanding balance of a loan. Represent the different important
variables, explain their meaning and explain the relationships among them.
(1,5 marks)
4. Due to his future retirement, one person decides to deposit 110.000€ in a bank so that he will be able to receive
an annuity during the future 12 years.
He is offered the following possibilities:
Monthly constant payments during the whole financial transaction.
Option A.
Monthly constant payments, constant during the year and increasing 5% annually.
Option B
In both cases, the deposit is made the 01.01.2013, the first payment will be received one month after the deposit has been made (02.01.2013) and the nominal rate compounded monthly is 2%.
During the first two years he will not receive any payment and, after that, he will receive monthly constant
payments. The deposit is made the 01.01.2013 and the first payment will be received the 02.01.2015.
Option C
In this case, the nominal rate would be 2,25% during the two first years and 2,75% the following ten years.
According to this information, obtain: (1,5 marks)
4.1 The monthly payments to be received in option A.
4.2. The monthly payments to be received during the first two years in option B.
5. Mr YYY, in order to buy a house, signed a level-payment fixed-rate loan with the following conditions:
C (^) o = 170.
Term: 10 years.
Monthly constant payments
Annual nominal rate: 5,25%
Early cancellation penalty: 1% of the outstanding balance
Consider, moreover, the following expenses paid by the debtor:
Unilateral initial expenses 500€.
Loan origination fee 1%
After two years, Mr. YYY decided to cancel the financial transaction. He decided to refinance the transaction with
an adjustable rate loan. The conditions of this new loan are as follows:
Amount: the amount needed to cancel the initial loan.
Interest rate:
First year: 4,25% (nominal).
Remaining: Reference rate plus 1 %.
Term: 12 years.
Annual constant principal repayments with quarterly interest payments.
Obtain:
(2,5 marks)
5.1. Monthly constant payments for the initial loan.
5.2. Total amount to be paid in order to cancel the initial loan.
5.3. Set the equation of equivalence that allows obtaining the true borrowing effective rate for the initial loan
taking into account the cancellation.
5.4. Instalments for the first and second years for the adjustable-rate loan. Suppose that the reference rate for
the second year was 3,15%.
6.3. Market price of a single bond at 06.30.10.
6.4. Equation of equivalence that allows obtaining the true lending effective rate of buying at the issue date and
afterwards selling in the secondary market.
6.5. Without any calculation, return will be higher or lower than 3%? Justify your answer.
1. Equivalent interest rates
( ) = + −
m m i i
( )^1 / = + −
m m i i
j m i m
m = ⋅
( ) ( )
m
jm i
( m ) ( )
2. Ordinary Annuities: Present value formulas
2.1. PV of n payments of 1 per period:
i
i a
n
ni
− − + =
|
2.2. PV of a perpetuity:
i
a i
|
∞
2.3. PV of n payments varying in geometric progression:
−
−
C n i if q i
if q i i q
q i C AC q
n n
ni
( 1 ) 1
1
|
2.4. PV of an annuity with payments varying in geometric progression, payable m thly:
ni n i
m ACq jm
i A Cq | |
( ) ( , ) ( )
3. Ordinary Annuities: Future value formulas
3.1. FV of n payments of 1 per period:
i
i S
n
ni
|
3.2. FV of n payments varying in geometric progression:
− C n i if q i
if q i i q
i q C SCq n
n n
ni
( 1 ) 1
1
|
3.3. FV of an annuity with payments varying in geometric progression, payable m thly:
ni n i
m SCq jm
i S C q | |
( ) ( , ) ( )
