Financial Mathematics
0 0 0 0 0 0 1 1 0 3 4 4 3 Mid Term Exam 12/04/2017
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The exam is made of two parts: (A) Multiple choice questions, (B) Exercises. Each part will be graded from 0 to 30; the final grade is made 40% by the grade obtained in part (A) and 60% by the grade obtained in part (B). A minimum grade of 15/30 in part (A) and an overall grade of 15/30 are required to pass the exam.
Part A. Multiple choice questions.
Answer the multiple choice questions choosing among one of the four possible answers. For every correct answer you will get 3/30, for every wrong answer you will get -1/30; if a question has no answer you will get 0/30. This point system delivers an expected grade of 0 to random answers.
1. Given a financial law, and t < s, which of these relationships is true:
◻ m(t, s) = 1 + v(t, s).
◻ m(t, s) = 1 − v(t, s).
◻ m(t,s)v(t,s) =1.
◻ m(t, s) =v(t,s)1 .
2. The linear law is
◻ decomposable but not time uniform.
◻ the only time uniform law.
◻ time uniform but not decomposable.
◻ time uniform and decomposable.
3. A financial contract, whose value in time is described by a value function W (t), has value W (0) = 100 and W (2) = 110. The instantaneous interest rate (force of interest) δ of the underlying financial law
◻ cannot be determined given the above information.
◻ is equal to -0.05
◻ is equal to 0.048809.
◻ is equal to 0.05
4. Consider a financial transaction x/t whose net present value according to a certain financial law F is W (0, x) = K > 0. The transaction y/t that adds to x/t the additional cash flow −K at time 0 is:
◻ fair at time 0 only if financial law F is an exponential law.
◻ not fair according to financial law F.
◻ is fair at time 0 according to any financial law.
◻ fair at time 0 according to financial law F.
5. The transaction x/t has Internal Rate of Return (IRR) i∗. The IRR of −x/t
◻ cannot be determined given the above information.
◻ is 0.
◻ is i∗.
◻ is −i∗.
6. An annuity with duration n and unitary installments, deferred mperiods, has present value equal to
◻ (1 + i)man i.
◻ (1 + i)−nam i.
◻ (1 + i)nam i.
◻ (1 + i)−man i.
7. The IRR of a transaction in which a debt of 100 is reimbursed with a payment of 100 after n years, together with a payment of 6 at the end of each of the n years is
◻ 0.6%.
◻ greater than 6%.
◻ lower than 6%.
◻ 6%.
8. In an amortization plan in which a loan is amortized in n periods, is the last amortization quota Cn equal to the next to last debt value Dn−1?
◻ Only in a French amortization.
◻ Yes, always.
◻ Only in an Italian amortization.
◻ No, never.
9. During the pre-amortization period
◻ the interest quotas are decreasing.
◻ the amortization quotas are positive.
◻ the residual debt stays constant and equal to the initial debt value.
◻ the residual debt decreases.
10. In an amortization plan, where an initial debt S is reimbursed in n periods and the periodic interest rate is i > 0, the succession of installments Rk with k = 1, ...n paid at the end of each period can be chosen freely, provided that
◻ S = ∑nk=1Rk(1 + i)−k.
◻ ∑nk=1Rk<S.
◻ ∑nk=1Rk= ∑nk=1Ck.
◻ ∑nk=1Rk=S.
Part B. Exercises.
Answer the questions using the given spaces. The decimal mark or the negative sign occupy one box. Please do not use additional sheets, write clearly the essential steps in the solution procedure in the blank spaces.
Exercise 1.
Mr. Smith wants to invest an amount S = 31000 euros. He has two investment options:
(a) an account provided by Bank Red, that returns an annual compound interest rate ia=3%;
(b) an account provided by Bank Brown, that returns a semi-annual simple interest rate ib=1.51.61.71.81.92%.
Determine the IRR of the two alternatives, i∗a and i∗b respectively, expressed in percentage terms and on an annual basis, if he invests his money for 1 year and 8 months:
i∗a=3% i∗b =2.97068%
i∗a is indeed the annual compound interest rate of the exponential law; i∗b is obtained as:
i∗b = (1 + ib∗2 ∗ 20/12)12/20−1.
Which of the two alternatives should he choose?
Bank Red Bank Brown The two alternatives are equivalent The above information is insufficient to answer the question.
The first alternative (Bank Red) should be chosen, since the IRR i∗a is greater than i∗b. Provide an explanation in the space below.
For personal reasons, Mr.Smith ultimately decides to split his investment in the two options, investing 40% of the initial amount in option (a) and 60% in option (b). Determine the interest he earns if he keeps the position for 2 years.
I = 1871.16euros
I = 40% ∗ 31000 ∗ (1 + 3%)2+60% ∗ 31000 ∗ (1 + 2 ∗ 1.5% ∗ 2) − 31000 = 1871.16.
The IRR i∗ of such an investment, expressed as a percentage and on an annual basis, is:
i∗=2.973783%
i∗= [
31000 + 1871.16
31000 ]
1/2
−1.
Exercise 2.
An investor buys at time 0 the following financial instruments, for a price P : 1. an annuity due, paying 15 installments R1=200euros, paid every 3 months;
2. a perpetuity, with monthly installments R2=1euros.
Knowing that the IRR of the transaction is i∗=2.1%, determine the price P the investor has paid.
P = 3470.4888euros The price is equal to the sum of the values of the two instruments:
P1=R1. . . a15i4=200 . . . a150.5209%=2893.58.
P2=R2ainfi12=1ainf0.1733%=576.91.
The accumulated value M and the residual value V at time t = 2/12 of the transaction, including the payment of the price P , evaluated at i∗ are respectively:
M = −3279.83euros V = 3279.83euros
Detail the reasoning that lead you to the above answer.
A time t = 2/12, the accumulated value is the sum of the appropriately capitalized values of:
1. the price paid at t = 0;
2. the instalment of the first annuity paid at t = 0;
3. the instalments of the perpertuity paid at t = 1/12, 2/12.
Hence the accumulated value M is equal to
M = (−3470.49 + 200)(1 + 2.1%)2/12+1(1 + 2.1%)1/12+1 = −3279.83.
V, by the fairness property, is equal to -M: V = 3279.83.
Exercise 3.
An entrepreneur receives a loan S = 110 000 euros from a bank, that he will need to reimburse in 4 periods at an interest rate i=3.1%. Fill the amortization plan, knowing that:
• the first period is a pre-amortization period;
• the amortization quotas paid in the following three periods are increasing by 5000 euros per period.
Justify the amounts inserted in the cells.
Period Residual Debt Instalment Amortization quota Interest Quota
0 110 000 0 0 0
1 110000 0 3410 3410
2 78333.33 31666.67 35076.67 3410
3 41666.67 36666.67 39095 2428.33
4 0 41666.67 42958.33 1291.67
Being the first period a pre-amortization one, the Amortization quota is 0, while the interest quota is I1=3.1% ∗ 110000 = 3410.
This is equal the instalment as well, while the residual debt is still 110000 euros. Knowing that the amortization quotas have to sum to the initial debt, it follows that:
C2+C3+C4=110000,
C2+ (C2+5000) + (C2+10000) = 110000 3C2=95000 Ô⇒ C2=95000/3 = 31667.67.
Hence:
C3=31667.67 + 5000 = 36667.67 C4=36667.67 + 50000 = 41667.67 All the other values in the plan follow from the usual relations:
In=3.1%Dn−1; Dn=Dn−1−Cn; Rn=In+Cn.
