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Financial Mathematics Exam for Business Administration Students, Ejercicios de Matemática Financiera

Universitat de València (UV)Matemática Financiera

A business administration exam focused on financial mathematics. It includes questions on defining financial transactions, calculating interest rates, determining bond prices, and understanding concepts such as true effective rate of cost and outstanding balance. Students are required to use formulas for present and future value of constant and varying payments, as well as annuities.

Tipo: Ejercicios

2017/2018

Subido el 25/02/2018

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Degree: Business Administration (ADE)
FINANCIAL MATHEMATICS (35804) 15th January 2015
Group………………….
Surname......................................................………..................................
Name....................................
EXAM RULES
• Please write your answers inside the boxes.
• Do not use pencil.
• Only the stapled exam will be handed in.
FIRST PART: (Marks: 70%)
1. Define and explain the concept of “financial transaction”.
2. Obtain the annual nominal interest rate payable monthly and the annual effective interest rate which are
equivalent to a 0.5% monthly effective rate.
3. What would be the price paid (market value) the 1st of March 2015 for a zero-coupon bond with a
nominal value of 1,000 Euros, issued the 1st of January 2015, at a 3% annual effective rate, with maturity
date 1st of January 2016, if the (annual effective) market interest rate in the purchase date was 2%?
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Degree: Business Administration (ADE)

FINANCIAL MATHEMATICS (35804) 15th January 2015

Group………………….

Surname......................................................………..................................

Name....................................

EXAM RULES

  • Please write your answers inside the boxes.
  • Do not use pencil.
  • Only the stapled exam will be handed in.

FIRST PART: (Marks: 70%)

  1. Define and explain the concept of “financial transaction”.
  2. Obtain the annual nominal interest rate payable monthly and the annual effective interest rate which are equivalent to a 0.5% monthly effective rate.
  3. What would be the price paid (market value) the 1 st^ of March 2015 for a zero-coupon bond with a nominal value of 1,000 Euros, issued the 1 st^ of January 2015, at a 3% annual effective rate, with maturity date 1st^ of January 2016, if the (annual effective) market interest rate in the purchase date was 2%?
  1. An investor buys, through a broker, a financial asset that gives the right to receive a nominal amount of 5,000 Euros after 120 days. The price paid for this asset is calculated with the simple interest rule, using a 5% annual interest rate. The investor must pay to the broker a brokerage fee of 0.15% on the asset’s nominal, at the purchase date, and another fee of 0.10% on the asset’s nominal at the maturity date. Calculate the true effective rate of return for the investor.
  2. Assume that you sign a savings plan where you compromise to pay constant amounts of 1,000 Euros during 15 years, and you pay the first amount on the contracting date, the 15 th^ of January 2015. Obtain the final accumulated value the 15 th^ of January 2030, if you use a 3% annual effective rate during the five first years and a 4.5% annual effective rate during the last ten years.
  1. Represent graphically on a time line the following annuity whose financial value at time τ is given by: 𝑉𝜏 = 50𝑎 5 �|𝑖 (1 + 𝑖)^5 Assume that 𝑖 stands for an annual effective rate and that the first amount is paid the 1 st^ of February 2015. Indicate also the exact date for τ.
  2. Explain one similarity and one difference between the concepts “Interest” and “Discount”.
  3. Explain one similarity and one difference between the concepts “outstanding balance” and “market value” of a bond.

SECOND PART: (Marks: 10%)

  1. Proof, mathematically, step by step, the decomposition of the total periodic payment (as) of a loan into

interest payment (Is ) and principal repayment (A (^) s ). In order to do so, you will have to use the outstanding balance calculated using the recursive method.

  1. Assume that, in a loan, there is a bilateral initial expense of 120 Euros paid by the borrower and a unilateral initial expense of 60 Euros also paid by the borrower. Sort from lowest to highest the effective rate of the pure financial transaction, the effective rate of cost and the effective rate of return. Explain your answer.
  2. Indicate whether the following statement is true or false. Explain carefully your answer. “In a bullet loan with a constant interest rate, the total periodic payments are decreasing in time, because the outstanding balance is also decreasing in time.”

THIRD PART: (Marks: 20%)

The 15 th^ of January 2008 a person signed a loan contract with a bank with the following characteristics:

  • Loan amount: € 168,000.
  • Loan length: 12 years.
  • Type of loan: Adjustable-rate loan with monthly total periodic payments recalculated at the end of each interest adjustment period.
  • Annual interest adjustment periods.
  • Nominal interest rate for the 1 st^ year: 4.8%.
  • Remaining years: Reference rate plus 0.5%.
  • Bilateral initial expenses paid by the borrower: 1% of the initial amount (C (^) o ).
  • Unilateral initial expenses, also paid by the borrower: € 1,200.
  • Early-cancellation fee: 0,5 % of the outstanding balance (C (^) s).

The 15 th^ of July 2008, 18 months after the contracting date, the debtor decided to cancel the original loan in order to benefit from another bank loan with the following characteristics:

  • Loan amount: Maximum € 200,000.
  • Maximum length: 12 years.
  • Monthly total periodic payments.
  • Type of loan: The first two years the monthly payments only correspond to the interest due (bullet loan) and the remaining years the loan is repaid with monthly level (constant) payments.
  • Nominal interest rate (fixed): 4.75%

Obtain:

  1. Total periodic payments of the initial loan for the first and second years, taking into account that the

reference rate for the second year was i (^) r2 = 5.45%.

  1. Total amount to be paid in order to cancel the initial loan on the 15 th^ of July 2008.

F ORMULAS FOR F INANCIAL MATHEMATICS

Present value formulas

PV of n constant payments of 1 per period:

i

i

a

n ni

1  ( 1 )^ 

|

PV of a perpetuity:

i

a i

f|

PV of n payments varying in geometric progression:

z 





C n i if q i

if q i

i q

q i

C

AC q

n n

ni

1

|

PV of an annuity with payments varying in geometric progression, payable m thly:

ni n i

m ACq

jm

i

A (^ ) Cq | ( , ) |

Future value formulas

FV of n constant payments of 1 per period:

i

i

S

n ni

|

FV of n payments varying in geometric progression:

z 

C n i ^ if q i

if q i

i q

i q

C

SCq

n

n n

ni

1

|

FV of an annuity with payments varying in geometric progression, payable m thly:

ni n i

m SCq

jm

i

S (^ ) C q | ( , ) |