Financial Math Answers

Project R delegates all the development work to outside companies. The estimated cash flows for project R are: (last column indicates expenditure) Beginning of year 1 150,000 dollars Contractor fees Beginning of year 2 250,000 dollars Contractor fees Beginning of year 3 250,000 dollars Contractor fees End of year 3 1,000,000 dollars Sales Table 1 Project S carries out all the development work in-house by purchasing the necessary equipment and using the company’s own staff. The cash flows for the project S are: (last column indicates expenditure) Beginning of year 1 325,000 dollars New equipment Throughout year 1 75,000 dollars Staff cost Throughout year 2 90,000 dollars Staff cost Throughout year 3 120,000 dollars Staff cost End of year 3 1,000,000 dollars Sales Table 2 The staff cost can be estimated to be paid uniformly throughout the year.

b) Find the IRR for the project R 

An oil company has issued preferred stock with $10 annual dividend that will be paid in perpetuity.

a. If the discount rate is 14%. What price must the stock sale?

The following information is given with respect to the ratio's of two companies

Aman Ltd Roger Ltd

Current ratio 2:01 1.60:1

Quick Ratio 1.35:1 1:01

Return on investment 15% 13%

Debt Equity Ratio 2.5:1 1:01

a. Define the concepts of Current and Quick ratio’s and also, reflect on your understanding

towards the financial performance of the companies by looking to the above information.

b. Define the terms- Return on Investment and Debt equity ratio and also, reflect on

your understanding towards the financial performance of the companies

Alice is closing on a house on Aug 13t the buyer owens the property on the day of the closing. the selling price of the home is 350,700. Alice was accepted for a 20 year fixed-rate mortage for 324900 at 5.25% interest. the seller has paid 4478.51. In property taxes for the coming year. How much will Alice owe in prorated taxes and interest?

Question 2 [9 marks]. This question is about the Arbitrage Theorem.

(a) State the Arbitrage Theorem. [3]

(b) A 6-sided die is rolled. The 6 possible outcomes are 1, 2, 3, 4, 5, 6. You can bet on any outcome. If you bet £1 on i and the outcome is j 6= i, then you lose your pound. But if you bet £1 on i and the outcome is i, then you get back your pound and a reward of £u. For what value of u will this game be arbitrage-free? Justify your answer.

The price of a share follows the geometric Brownian motion with parameters µ = 0.2 and σ = 0.18. Presently, the share’s price is £38. Consider a call option having one year until its expiration time and having a strike price of £40. The continuously compounded interest rate is 5%.

(a) What is the risk-neutral price C of this call option?

(b) Suppose now that you are the seller of this option. At time t = 0 you get £C from the buyer of the option, where C is the risk-neutral price of the option. You then have to design a hedging strategy which would allow you to meet your financial obligation in one year’s time. Your portfolio should consist of two investments: you are allowed to buy the underlying shares and to deposit money in the bank.

(i) The price of the share evolves according to a geometric Brownian motion. State the formulae you will need to compute the number of shares in the portfolio and the capital deposited in the bank at any time t, 0 6 t 6 1.

Question 5 [9 marks]. The price S(t) of a share follows the geometric Brownian motion S(t) = Se µt+σW(t) . The price of the share at t = 0 is S = £25 and µ = 0.1. The continuously compounded interest rate is 8%. However, the volatility σ is not known.

(a) Explain the definition of implied volatility. [3]

(b) A European call option on the above share with the strike price K = £23 and expiration time of 6 months has the market price £3.1. Does the equation defining the implied volatility have a solution? [6] Hint. You are not supposed to solve the equation defining the volatility to answer this question.

Question 6 [26 marks]. Consider the extension of the Vasicek model for a variable interest rate rt , t > 0, which is described by the following stochastic differential equation: drt = −a(rt − µ)dt + σ(t)dWt , where a > 0, µ > 0 are constants and σ(t), t > 0, is a strictly positive function of t.

(a) Compute the differential of the function U(t) defined by U(t) = e at(rt − µ). [6]

(b) Solve the equation for rt with the initial value r(0) = r0 [7]

(c) State the distribution of rt and provide formulae for E(rt) and Var(rt). [6]

(d) Suppose that a = 2, µ = 0.05, and σ(t) = 0.1e t 4 . In other words, the interest rate is governed by the following stochastic differential equation: drt = −2(rt − 0.05)dt + 0.1e t 4 dWt .

Given that r0 = 0.05, what is the probability that after one year the interest rate will be less than 0.03? 

The following facts about a company are known: 1. At present, its total capital is £3 million. 2. It has just sold zero-coupon bonds with the total nominal value of £2 million which it promises to repay in 18 months from now. 3. The total capital F(t) of the company follows the geometric Brownian motion with parameters µ = 0.15 and σ = 0.2. The continuously compounded annual interest rate r = 6%. Within the framework of the Merton model, establish the following.

(a) What is the total value of the shares of this company? [5]

(b) How much money has the company raised from the sale of the bonds? [2]

(c)What is the probability that the company would default on its promise to bond holders? [5]

Mozumuhle takes out personal loan of R439200,00 to help finance the building of his holiday house,. The terms of his loan specify equal three monthly repayments over five years, with 13,2%interest per 5,compounded quarterly. The first payment is made three months after the loan was taken out. Find the size of the 3 monthly payment