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I need help completing half of a Financial Mathematics quiz, this should only take about an hour and I have included an entire example quiz so that you have an idea of the type of questions in the actual quiz that I shall provide shortly.
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Practice Quiz
Question 1 [23 marks].
(a) Suppose you put £1500 in a bank account with nominal interest rate 1.2% and
make no withdrawals. How much money will be in the account 6 years later if
the interest is compounded monthly? State your answer to the nearest pence.
[3]
(b) Suppose a bank account has a nominal interest rate of 4.4% compounded
semi-annually. Find the effective interest rate reff to three significant figures.
[3]
(c) Suppose that for time t ≥ 0 the instantaneous interest rate of a bank account is
given by
2
r (t) = 0.016 + 0.01te−t .
(i) Determine the yield curve r (t).
[4]
(ii) Determine lim r (t).
[3]
t→∞
(d) Suppose that Bank A offers deposits and loans continuously compounded with
discount factor D A (t) and that Bank B offers deposits and loans continuously
compounded with discount factor DB (t). Moreover, suppose that
D A (1) D A (2) > D B (3).
Show that an arbitrage opportunity exists.
[10]
Question 2 [8 marks]. Consider the cash flow ( a1 , a2 , a3 ) = (−2, 1, −1), where ai is
the payment at the beginning of year i for i = 0, 1, 2.
(a) Show that this cash flow does not have an Internal Rate of Return.
[5]
(b) Why is this cash flow not subject to the theorem proved in lectures about the
existence of an Internal Rate of Return r satisfying −1 < r < ∞.
[3]
Question 3 [9 marks]. A 2-year bond has face value £700,000 semi-annual
coupons at rate 3% per annum, and is redeemable at half par. The current rate of
interest is 3% compounded continuously.
(a) Determine the coupon and redemption payments in pounds.
[4]
(b) Determine the no-arbitrage price of the bond to the nearest pound.
[5]
Continue to next page
Question 4 [10 marks]. Consider the three cash flow streams of the form
( a1 , a2 , a3 ) where ai is the amount of money in thousands of pounds received at the
end of year i for i = 0, 1, 2:
x = (2, 2, 2)
y = ( a, 2, 2)
z = (2, 2, a),
where a > 2. Interest is 3% compounded continuously. Order x, y, and z from
smallest effective duration to largest effective duration. Justify your answer.
[10]
Question 5 [14 marks]. Suppose that in the fixed interest rate model the interest
rate compounded yearly has the continuous distribution R ∼ Uniform(1.3%, 2.7%).
(a) Determine the probability that £200 accumulates to less than £210 after three
years? State your answer as a decimal to three significant figures.
[5]
(b) Find the expected present value of a payment of £10,000 received five years
from now. State your answer to the nearest pound.
[6]
(c) Find the present value of a payment of £10,000 received five years from now if
interest is not random any more, but is compounded yearly at rate E( R), where
E denotes expected value. State your answer to the nearest pound.
[3]
Question 6 [8 marks].
(a) Assume that Corner Bank quotes spot rate rate s8 = 1.5% and forward rate
f 8,10 = 1.9%. Find the spot rate s10 . State your answer as a percentage to three
significant figures.
[4]
(b) Suppose that the price of 100 6-year zero-coupon bonds each paying £1 is £96
and that the price of 120 8-year zero-coupon bonds each paying £1 is £105.
Assuming there is no-arbitrage, find the forward rate f 6,8 . State your answer as
a percentage to three significant figures.
[4]
Question 7 [8 marks].
A company issues new shares to fund a new manufacturing plant. Explain the
meaning of the Arbitrage Theorem with respect to the price of the new shares.
Continue to next page
[8]
Question 8 [20 marks]. A share price is modelled via a two-period binomial
model with initial stock price S = 250, up/down multiplication factors u = 1.2 and
d = 0.8, and interest rate 3.2% compounded continuously.
(a) Verify that the no-arbitrage assumption is valid in this model.
[3]
(b) Find the risk-neutral probabilities of up and down movements in the share
price. State your answers to three significant figures.
[4]
(c) Find the no-arbitrage price of a European call option on the share with strike
K = 200 and expiry date T = 2. State your answer to the nearest pence.
[7]
(d) Suppose that we let the strike price K vary and keep the other parameters the
same. What is the smallest value of K for which the call would has value zero?
Explain your answer.
[6]
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