Banking Law notes recently updated for exams at top-tier British Universities. These notes, written at King's College London, cover all the LLB banking law cases and so are perfect for anyone doing an LLB in the UK or a great supplement for those doing LLBs abroad, whether that be in Ireland, Hong Kong or Malaysia (University of London). These were the best Banking Law notes the director of Oxbridge Notes (an Oxford law graduate) could find after combing through over a hundred LLB samples from ou...
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Capital Budgeting
Methods to decide which projects to invest in and which to reject:
The Net Present Value Rules (which is the topic for today);
The Internal Rate of Return; and
Others (See I Welch, Corporate Finance, Chapter 4).
Net Present Value
The present value of all future cash flows of a project minus the present value of its costs.
I.e. The sum of the present value of all future positive and negative cash flows.
NPV = F_{0} + F_{1}/(1 + r_{0,1}) + F_{2}/( 1+ r_{1,2}) + … + F_{T}/(1 + r_{T-1,T})
Example:
You can buy a project today for 100, next year it will generate a return of 20, the following year of 50, and in year three, when the project comes to an end, of 75. The constant interest rate is 10%. What is the NPV?
NPV = -100 + 20/(1 + 0.1) + 50/(1 + 0.1)^{2} + 75/(1 + 0.1)^{3}
The NPV Rule:
Accept only projects with NPV > 0.
Accepting projects with a positive NPV increases firm value.
Rejecting projecting with a negative NPV that will decrease firm value.
Present (market) value of the future cash flow - cost = the profit (or loss) form the project.
Positive NPV means “free” money.
Application:
As the CFO of your company you are faced with the following investment options (with the cost of capital being constant at 8%):
Exploitation of a mine:
Cost today = 200;
Return in year 1 = 0;
Return in year 2 = 100
Return in year 3 = 300
Clean up costs in year 4 = 100.
Investment in a 4-year note with annual coupons:
Price today = 200;
Coupon in year 1 = 20
Coupon in year 2 = 20
Coupon in year 3 = 20
Coupon + Principal in year 4 = 220
Which project should we invest in?
NPV_{mine} = -200 + 0 + 100/1.08^{2} + 300/1.08^{3} - 100/1.08^{4} = 50.38
NPV_{note} = -200 + 20/1.08 + 20/1.08^{2} + 20/1.08^{3} + 220/1.08^{4} = 13.25
Conclusion = Invest in both if possible, but, if not enough cash, then only in the former, because it has a greater NPV.
But what if the interest rate is 16% instead of 8%? The bond is no longer feasible.
NPV_{mine} = 11.28
NPV_{note} = -33.56
The Internal Rate of Return
IRR = the rate-of-return-like number of NPV = 0.
The Internal Rate of Return:
0 = F_{0} + F_{1}/(1 + r) + F_{2}/(1 + r)^{2} + … + F_{T}/(1 + r)^{T}
Solve for r. But this is not possible to do by hand.
Invest if IRR > Required rate of return (that is the discount rate or the opportunity cost).
E.g. In the previous example, the required rate of return is the cost of capital of 8%.
Advantages:
Single number that is easy to understand; and
All you need to know is the cash flows emanating from the project.
Disadvantages:
There can sometimes be multiple IRRs;
When this is the case, the IRR method is then no longer accurate.
The IRR is not defined sometimes;
Comparison problems as it does not adjust for project scale.
Overall, NPV is the most reliable.
Valuing Risk
The NPV formula is easy.
But, in the presence of uncertainty, the inputs become difficult.
Future cash flows become “expected” cash flows, [E(F_{t})].
Rate of return becomes “expected risk adjusted rate” [E(r_{t})]. It is this concept of credit risk or risk of default that is being reflected in the expected cash flows and expected risk adjusted rate.
Expected value:
The probability-weighted average outcome of a random variable.
Example:
A bond for 200 promises to pay 210 the next year. The issuer’s business is risky. The bondholder may receive only a fraction of the promised cash flow at the end of the year. The probability distribution of payoffs (our random variable) is as follows:
Payoff | Probability |
210 | 50% |
150 | 10% |
90 | 10% |
50 | 10% |
20 | 10% |
0 | 10% |
What is the expected future value of the promised cash flow?
E(F) = p_{1}F_{1} + p_{2}F_{2} + p_{3}F_{3} + … p_{n}F_{n} = ^{n}_{j=1}p_{j}F_{j}
E(F) = (0.5 x 210) + (0.1 x 150) + (0.1 x 90) + (0.1 x 50) + (0.1 x 20) + (0.1 x 0) = 136
Variance and Standard Deviation:
Variance = The expected value of the squared deviations from the mean.
Payoff | Probability | Deviation (from the expected future value of 136) |
210 | 50% | 74 |
150 | 10% | 14 |
90 | 10% | -46 |
50 | 10% | -86 |
20 | 10% | -116 |
0 | 10% | -136 |
Var = (0.5 x 74^{2}) + (0.1 x 14^{2}) + (0.1 x (-46)^{2}) + (0.1 x (-86)^{2}) + (0.1 x (-116)^{2}) + (0.1 x )-136)^{2}) = 6,904
Standard Deviation = Square root of variance; that is, the expected squared deviation from the mean.
SD = var = 6,904 = 83.01
We can expect to receive 136 plus/minus 83.01 on this bond. This is very risky. The bond promises a rate of return of 5%. What is the expected rate of return?
E(r) = (E(F)-C)/C = (136 - 200)/200 = -0.32
Empirical Mean:
Expected value requires knowledge of the “true” probability distribution of the future cash flows.
This, in practice, means reliance on historical observations.
The realisation of cash flows (share prices, dividends, etc.);
For K periods;
This means that, the longer the period, the more accurate the approximation will be.
With empirical mean as an approximation of the expected value.
The NPV formula with credit risk accounted for:
NPV = F_{0} + E(F_{1})/(1 + r_{0,1}) + E(F_{2})/( 1+ r_{1,2}) + … + E(F_{T})/(1 + r_{T-1,T})
But what is the appropriate r (discount rate or cost of capital)?
Covariance and Linear Regression
Covariance: Measures the extent to which two random variables (x and y) move in conjunction with one another.
Say, two securities or a security and the market (FTSE 100).
Expected value of the product of the difference between each variable’s possible outcome and its expected value.
cov(y,x) = (^{n}_{j=1} (y_{outcome} - y_{expected})(x_{outcome} - x_{expected}))/n
Linear Regression: the line that “best” predicts the realisation of y given the realisation of x.
Y = a + Bx
Example:
The following table depicts the rate of return on a particular security and the market portfolio (say FTSE 100).
Market | Security |
3.6 | 9.8 |
2.7 | 5.6 |
7.3 | 7.8 |
4.6 | 6.5 |
2.6 | 2.4 |
7.3 | 8.6 |
5.4 | 5.9 |
2.4 | 1.8 |
4.6 | 7.0 |
Linear Regression:
The rate of return for a particular security and the market. The straight line is the closest prediction of the rate of return on our security (y) given...
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Banking Law notes recently updated for exams at top-tier British Universities. These notes, written at King's College London, cover all the LLB banking law cases and so are perfect for anyone doing an LLB in the UK or a great supplement for those doing LLBs abroad, whether that be in Ireland, Hong Kong or Malaysia (University of London). These were the best Banking Law notes the director of Oxbridge Notes (an Oxford law graduate) could find after combing through over a hundred LLB samples from ou...
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