- (b) Prove that the fair price at time n of a European contingent claim h is given by WnEW (h/WN |Fn) .
- (a) Prove that the probability measure PW introduced in Question 4 is unique, and satises dPW/dP¤ =WN/S0N.
- 6. Assume that the market is viable and complete, and denote by P¤ the unique equivalent probability measure with respect to which ˜ S is a martingale.
- 5. Prove that, in a viable market, there is at most one deterministic numéraire.
- 4. Prove that the market is viable if and only if there exists a probability PW, equivalent to P, such that SW is a PW-martingale.
- 3. Prove that for any predictable sequence Á=(Án)n=0,...,N , with values in Rd+1 and any real number V0, there exists a self-nancing strategy ˆÁsuch thatˆÁn.SW n =V0 +Xn j=1Áj .¢SW j , n=0,
- 2. Prove that for n=1, . . . ,N, Pn j=1 µj .¢SW j =0.
- 1. Prove that a predictable sequence Á=(Án)n=0,...,N , with values in Rd+1, is a self-nancing strategy if and only if we have V W n (Á)=V0(Á) +Xn j=1Áj .¢SW j , n=1, . . . ,N, with the
- 2. Show that if F achieves a minimum at µ¤ 2Rd, we have EXeµ¤.X =0.
- 1. Show that if P(X =0)
- 7. We can now use the model to price a call or a put with maturity T on a single stock. In order to do that, we divide the interval [0, T] into N subintervals, so that we can refer to the N-period
- (1 + r)n +HnSn−1u=c(n, Sn−1u).Subtracting one from the other, it turns out that¢(n, x)=c(n, xu) − c(n, xd)x(u − d).
- (1 + r)n +HnSn−1d=c(n, Sn−1d)and
- (1 + r)n +HnSn =c(n, Sn).Since H0n and Hn are Fn−1-measurable, they are functions of S1, . . . , Sn−1 only and, since Sn is equal to Sn−1d or Sn−1u, the previous equality implies H0n
- 6. Show that the replicating strategy of a call is characterized by a quantity Hn =¢(n, Sn−1) at time n, where ¢ will be expressed in terms of the function c.We denote by H0n the number of
- By conditioning with respect to (Tn+2, . . . , TN) in (1.4.1), we have c(n, x)(1 + r)−(N−n) = pE¤xd NY i=n+2 Ti −K!++(1 − p)E¤xu NY i=n+2 Ti −K!+.
- (c) Prove that the function c satises the recursive equations c(n, x)=(pc(n + 1, xd) + (1 − p)c(n + 1, xu))1 + r, n=0, . . . ,N − 1.
- If the Ti's are independent and satisfy P(Ti =d)=p=1 − P(Ti =u), we have E(Tn+1|Fn)=E(Tn+1)=pd + (1 − p)u=1 + r, and thus ( ˜ Sn) is a P-martingale, according to Question 1.
- 4. From now on, we assume that r 2 (d − 1, u − 1) and we write p=(u − 1 − r)/(u − d). Show that ( ˜ Sn) is a P-martingale if and only if the random variables T1, T2, . . . , TN are
- 3. Give examples of arbitrage strategies if the no-arbitrage condition derived in Question 2 is not satised.Assume for instance that r ·u − 1. By borrowing an amount S0 at time 0, we can purchase
- 2. Deduce that r must belong to (d − 1, u − 1) for the market to be arbitrage-free.If the market is viable, there exists a probability P¤ equivalent to P, under which ( ˜ Sn) is a martingale.
- 1. Show that the discounted price ( ˜ Sn) is a martingale under P if and only if E(Tn+1|Fn)=1 + r, 8n 2 {0, 1, . . . ,N − 1}.The equality E( ˜ Sn+1|Fn)= ˜ Sn is equivalent to E( ˜ Sn+1/ ˜
- List and explain three limitations of ratio analysis.
- What is liquidity and how can it be measured?
- What is ROCE?
- What are the five key measures for a business?
- Having done all of the analysis what is your assessment of the organisation’s current financial position and performance? Remember to couple your analysis with the director’s report and the
- If you haven’t done so already follow the investment ratios and apply them to your company. What do your results tell you?
- Are cash and profit the same thing?
- What does the balance sheet show?
- What is the BTA’s overall financial health?
- What is the BTA’s financial position?
- What is the BTA’s financial performance?
- What is Net Profit?
- What is Gross Profit and how is it calculated?
- A business paid its annual rent of £12,000 in August. The business has an accounting period that ends in March. Explain the accounting entries that would be needed for the rent.
- How do accruals and prepayments affect the trial balance?
- Explain what is meant by (i) an accrued expense and (ii) a prepaid expense?
- Does a trial balance list all the transactions made by the business?
- years. It depreciates the van by£2,000 each year. What would be in the trial balance at the end of the second year for the van and its depreciation?
- A company has owned a van worth £10,000 for
- A company takes out a loan to pay for a new van. Is the loan an asset?
- What effect does profit have on capital?
- Should the cost of equipment we have purchased be shown in the income statement?
- Indicate where each of the following items will be shown (in the trading income statement or balance sheet):■ Trade Debtors■ Cash and Bank■ Bank Loans■ Rent■ Wages■ Loan interest paid■
- List the five circumstances in which a trial balance might balance even though some of the balances are incorrect.■■■■■
- Would ‘discounts allowed’ be shown as an expense?
- Explain the difference between ‘interest payments’ and ‘capital repayments’ on a loan.
- Would ‘interest receivable’ be shown as an expense?
- per cent per year, how much extra cash would leave the company in the second year of the machine’s usage?
- per cent per annum. If the depreciation rate was changed to
- A machine cost £50,000 and was being depreciated at
- What is the net book value of an asset?
- What are the accounting entries for question 4?
- the company decides to sell the machine for £3,000. What is the profit/loss for the machine?
- At the end of year
- What will be the NBV of the machine at the end of year 5?
- What will be the accumulated depreciation by the end of year 5?
- What will be the annual depreciation charge for the machine? (Assume the straight line method for depreciation.)
- The company returns goods to one of its suppliers and receives a credit note(i.e. does not obtain a cash refund)?
- Customers return goods and obtain a cash refund immediately?
- What impact will the discount have on the profit on the period?
- What accounting entries will be required to record the discounted payment(ignore the original purchase)?
- What is the amount of the discount?
- Will this be ‘discount allowed’ or ‘discount received’ in Kitlocker’s accounts?
- What will the net book value of the machine be at the end of year 6?
- What will be the balance on the Provision for Depreciation of Machinery account at the end of year 4?
- What will be the annual percentage rate of depreciation?
- How much is the depreciation charge per annum?
- per cent per annum.■ Calculate the interest payable in the third year of the loan.■ What is the balance on the loan account at the end of the third year.
- A business takes out a 5-year ‘interest only’ loan of £250,000, at an interest rate of
- What is the percentage rate of interest per annum on a loan of £25,000 where the interest payable for the year is £1,250?
- per cent?
- What is the amount of interest payable for the year on a loan of £50,000 at an annual rate of interest of
- Which of the following items are shown under the wrong headings?
- Complete the ‘T’ accounts for the following examples:a The owner of the firm introduces capital of £15,000 in cash.b Goods for re-sale are purchased, by cash, for £6,000.c The rent bill arrives
- Are these statements true or false?a Assets are items or resources that have a value to the business and things that are used by the business and for the business.b Liabilities are amounts owed to
- Identify three of the regulatory bodies that oversee accounting.
- For the following statements fill in the blanks.a The concept recognises that each transaction conducted by a business will affect two items within the business.b The concept dictates that
- List the accounting concepts and principles that you can remember from this chapter.
- Is your favourite sport team listed?
- Which sector, sport or leisure, is represented the most on the UK Stock Exchange?
- How many leisure organisations are listed?
- How many sport organisations are listed?
- True or False: The balance sheet helps us determine the profit of an organisation?
- What type of information does financial accounting examine?
- What are the four main company types?
- List three users of financial information (ignoring managers).
- Why do managers use financial information?
- The strategy to buy players was the Chairman’s idea but what other things could Leeds have done to improve on-the-field success?
- Do you think that there is a right level of borrowing and if so how do you think that it could be calculated?
- Do you think that the Leeds United model suggests that football clubs, or any sport organisation, should avoid borrowing?
- Appraise an organisation’s financial position and performance through the application of ratio analysis and interpretation of the Annual Report.
- Explain the purpose of analysing financial statements.
- Understand the reporting and measurement of financial performance and position. Understand and explain the accounts of non-trading organisations.
- Understand and explain the information needed to construct the financial statements.
- Make adjustments to the trial balance (cost of sales, mispostings, accruals and prepayments, and others).
- Construct an outline income statement and balance sheet.
- Determine balances on ‘T’ accounts and construct a trial balance.
- Understand and communicate the interrelationship between ‘T’ accounts and financial statements.