Problem $1$

$1\mathrm{.}1$ Suppose you received R$1000$ from your grandparents on your $18$th birthday, which was $17$ years ago. You put it in a savings account at a measly fixed rate of $4,5\%$ per year and forgot all about it until now. What should the account balance be now?

$1\mathrm{.}2$ You want to buy a new car which costs R$675000$ today. However, your plan is to buy the car cash in four years' time. How much should you save per month if you can get a rate of $10,5\%$ per year and the car's price increases at $8,75\%$ per year?

$1\mathrm{.}3$ Suppose you are the proud owner of a Private Bank platinum credit card which has all sort of wonderful benefits. However, you notice that the annual interest on outstanding balance is $15,75\%.$ What is the effective annual rate for this card if it is compounded

$1\mathrm{.}3\mathrm{.}1$ monthly

$1\mathrm{.}3\mathrm{.}2$ daily

$1\mathrm{.}3\mathrm{.}3$ continuously

$1\mathrm{.}3\mathrm{.}4$ what is the final balance after $3$ years for each case if the starting amount was R$50000$

$1\mathrm{.}4$ Given your excellent performance over the past year, your company has decided to pay you a bonus of R$100000.$ However, just as you start thinking of several ways to spend that money, your line manager informs you that the company is experiencing some cash flow problems and that they will only be able to pay the bonus in five annual installments of R$20000$ each, starting a year from now. Using a rate of $9,5\%$ per year, what is the bonus worth to you today?

$1\mathrm{.}5$ You have taken out a home loan of R$2359000$ at $10,55\%$ per year for $20$ years. How much will your monthly payment be$?$ How much will you pay for the house in total? How much will you save if you pay an extra R$3000$ per month?

$1\mathrm{.}6$ You seriously want to buy the latest $4$x$4$ vehicle but it costs a great deal of money: R$1,560$m$.$ The dealer explains his financing scheme in the following way: he needs a deposit of $10\%$ and then you must pay him a fixed monthly amount such that the present value of those payments over a period of $5$ years will cover the remaining cost of the car. How much should you pay per month if the offered interest rate is $16,5\%$ per year?

$1\mathrm{.}7$ Use the Rule of $72$ to estimate the time required for an amount to double with an annual interest rate of $9,5\%.$ Calculate the actual time required for an amount to double with that interest rate. What do you conclude?

Problem $2$

Mr Roll Thedice is a venture capitalist and has come across two business opportunities presented to him by two student teams that have recently completed the MEM programme. Each of the business ideas centre around an innovative new product that will require substantial funding and time to develop and introduce to the market. For purposes of confidentially Mr Thedice has assigned code names to the two teams: Flint and Steel.

Both teams have developed extensive business plans, including cash flow projections. After several meetings, Mr Thedice was able to produce the following table of cash flows for the two opportunities:

$2\mathrm{.}1$ Mr Thedice has to choose between the two opportunities as he has only limited capital available. He has indicated that he uses a WACC rate of $9,35\%.$ Based on NPV and IRR, what will you recommend? Explain your recommendation.

$2\mathrm{.}2$ Breakeven time is the time for a project or investment to reach positive cash flow. Determine the breakeven time for each opportunity. Does this help with the choice between Flint and Steel?

Problem $3$

A manufacturing company is considering the acquisition of a heavy press to be used for stamping and forming of sheet metal parts. The board of directors has determined that all capital investments should be evaluated with a $10$ year horizon and that the appropriate WACC to be used for this case is $8,75\%.$ The cost of the press is R$12560000.$ The head of the manufacturing plant has estimated that the income $($positive cash flow$)$ generated from the new press will be as follows $($in millions$)$:

Yr $1$ Yr $2$ Yr $3$ Yr $4$ Yr $5$ Yr $6$ Yr $7$ Yr $8$ Yr $9$ Yr $10$

R$0,750$ R$1,125$ R$1,500$ R$1,875$ R$2,220$ R$2,400$ R$2,700$ R$3,150$ R$3,300$ R$3,300$

$3\mathrm{.}1$ Determine the NPV and IRR. Does the investment seem viable?

$3\mathrm{.}2$ The CFO has looked at the numbers and suggests that the WACC should rather be treated as a normal random variable with mean $9,5\%$ and standard deviation of $1\%.$ Produce a histogram and CDF for the NPV with this new information and answer the following questions:

$3\mathrm{.}2\mathrm{.}1$ what is your conclusion about the viability of the capital investment now?

$3\mathrm{.}2\mathrm{.}2$ what is the probability that the NPV will be negative?

$3\mathrm{.}2\mathrm{.}3$ what is the probability that the NPV calculated in $3\mathrm{.}1$ will not be achieved?

$3\mathrm{.}2\mathrm{.}4$ what is the IRR?

$3\mathrm{.}3$ Explain how a real options approach can be applied to this decision.

Submit your Excel worksheet for this problem.

Problem $4$

$4\mathrm{.}1$ Consider again Problem $3($Real Estate Developer$)$ of Homework No $7$ and discuss it in the context of real options. Are there real options and what are they worth?

$4\mathrm{.}2$ Briefly mention and describe two examples of real options from your work environment or pe