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Formula Sheet Interest rate calculations Simple interest When money is put on deposit for a relatively short period of time, simple interest may be applied. The relevant parameters are i the annual rate of interest, d the number of days the cash will be on deposit, and a the amount of cash on deposit. We denote the interest received by r. This can we worked out as follows: aid r= (1.1) 365 Here we express the interest rate i as 0.05 for say 5%. Often it is more useful to know how much cash will be returned at the end of the deposit period and this is obviously: d FV = a(1 + i ) (1.2) 365 Where FV stands for Future Value. Compound Interest Interest is paid on deposits at set intervals of time, perhaps monthly, quarterly or annually. The interest is added to the amount on deposit and so increases the interest payable in future periods. This is known as compound interest. If interest is paid on an annual basis at rate i, and an amount a is on deposit for y years then the ﬁnal value is given by: FV = a(1 + i)y (1.3) If interest is paid monthly on a savings account and the interest rate applied is i% per annum, it is important to be able to compare this arrangement to what is known as the effective rate. That is the rate which would give the equivalent interest if paid once only at the end on the year. The general formula for this is: i n ie = 1 + −1 (1.4) n 1 Where ie is the effective rate and n is the number of payments per year. For example my savings account applies an interest rate of 0.045 per annum every month. The effective rate is just below 4.6% per annum rather than the 4.5% applied. Clearly a daily accrual of interest would be more favourable still and this comes out as a effective rate just over 4.6%. This formula can of course be rearranged to calculate the nominal rate given the effective rate. 1 i = n((1 + ie ) n − 1) (1.5) Continuous Interest There is no reason to make a day the shortest period over which interest is applied. Why not an hour, a minute, or a microsecond? The limit is the continuous rate where interest is applied at vanishingly small increments of time so as to be in effect continuous. The growth is of course exponential and given by the following equation: FV = aeiy (1.6) where y is again the number of years which might be expressed with decimal places. Bond Pricing Bonds issued by stable governments are regarded as risk-free and the current coupon offered is a good guide to a value that can be used in calculating the risk-free discount rate in ﬁnding the present value of future cash ﬂows. For a given expected rate of return i, a cash ﬂow C years in the future has a present value (PV )of: C PV = (1.7) (1 + i)N where N is the number of years ahead the cash ﬂow occurs. The expected yield i is can be taken as the current risk-free rate for government bonds. A more accurate valuation might use a future proﬁle of i rather than a single value. The value of a bond is simply the sum of all the future cash 2 ﬂows expressed in present value terms. The price is expressed per 100 units of the bond. This gives: Ck PV = ∑ ndk (1.8) k i 365 1+ n where Ck is the k’th expected cash ﬂow, dk is the number of days until Ck , and n is the number of payments per year. If inﬂation increases the yield will grow and the bond will become less valuable. Conversely if inﬂation falls then bond prices rise. There is a thriving market in bonds and the prices can be used to calculate an accurate value of future inﬂation as anticipated by the market. Bond coupons are paid at equal intervals, say every 0.5 years. That is to say a bond with a coupon of 10% will pay exactly 5% half yearly. At some point arbitrary point in time it is convenient to express equation 1.8 in terms of the coupon rate R and W which is the fraction of the coupon period between the date of purchase and the next coupon being received. 1 − (1+1i )N 100 R n 1 P= i + i (1.9) (1 + n )W n 1 (1 + n )N−1 1− i (1+ n ) Returns In ﬁnance we are often more concerned with returns than prices. A return r at time t is deﬁned as: (pt − pt−1 ) rt = (1.10) pt−1 Provided rt 1.0 this can be expressed in the more convenient form: rt = ln pt − ln pt−1 (1.11) 3 Risk The standard deviation (σ) of a time series is equated with risk and is given by the square root of the variance. In order to convert the standard deviation to annualised volatility it is necessary to divide by one √ over the square root of the sampling period. This is the same as multiplying by 252 for daily √ √ data, 52 for weekly data and 12 for monthly. Thus the annualised volatility for a daily time series is given by: √ υ= 252 σ (1.12) Time Series Analysis The mean of a time series pi for i = 1 . . . M is given by: 1 M p= ∑ pi (1.13) M i=1 and the variance by: 1 M σ2 = ∑ (pi − pi)(pi − pi) (1.14) M − 1 i=1 Portfolio Theory Let r to denote the actual return on a asset and r denote the average return. In ﬁnance theory the expected return is almost always a synonym for the average return. Where we have a portfolio of N assets we will denote their returns by ri where i = 1 . . . N. The return of the portfolio as a whole will be denoted using the subscript p that is r p . The standard deviation of return i will be denoted by σi and the fraction by value of a portfolio represented by asset i will be denoted by wi which we describe as the weighting. 4 The expected rate of return of a portfolio is the weighted sum of the expected rate of return of its components. N r p = ∑ wi ∗ ri (1.15) i=1 The variance of the portfolio is given by: N N σ2 = ∑ p ∑ wiw j σi, j (1.16) i=1 j=1 The CAPM There is a relationship between return and risk, which may be assumed to be linear. We know two points on this line, the risk and return of the market, for example the FTSE 100 index and the risk-free rate of return. This allows us to deﬁne expected return as a function of risk: (rM − r f ) E(σ) = σ+rf (1.17) σM Where rM is the market rate of return and r f is the risk-free rate. The slope of this line is clearly: (rM − r f ) (1.18) σM It is useful to be able to see whether an equity is more or less risky that the market as a whole and this is termed ins β which can be calculated as follows: σi,M βi = (1.19) σ2M In other words it is the ratio of the cross-variance of the share and the market to the variance of the market. These terms can be found in the covariance matrix found from the two time series of returns. 5 Option pricing The Black-Scholes formula for calculating the value of European call and put options is given below. We ﬁrst deﬁne: • K is the exercise or strike price in pence • S is the current or spot price in pence • T is the time to maturity in years • σ is the annualised volatility. • r is the risk-free annual interest rate, say on one year gilts Then calculate d1 and d2 and then use the cdf function in a formula for call or put option value. S 2 ln( K ) + (r + σ )T 2 d1 = √ σ T √ d2 = d1 − σ T C(S, T ) = SΦ(d1 ) − Ke−rT Φ(d2 ) P(S, T ) = Ke−rT Φ(−d2 ) − SΦ(−d1 ) Φ() is the standard normal cumulative distribution function. Company Valuation Valuing a company is a difﬁcult and contentious process. It could be argued that this is the job of a stock market, but it is useful to be able to estimate a share value using simple rules of thumb to give an indication of the assumptions the market is making. The ﬁrst part of such a valuation is the “liquidation value” of the company which is essentially the value of its assets minus its liabilities and should be available from a balance sheet showing recent accounts. The book value usually excludes intangible assets such as goodwill and often intellectual property, but this is a debatable point. If there are S shares issued and the book value is B, then the value per share due to this component is B/S. Companies often pay a dividend to shareholders. This is usually rather less that the amount of surplus cash generated after tax and interest payments and the ration between that amount and 6 the dividend is known as the “coverage ratio.” It is the total cash generated that adds shareholder value and so if the company gives a dividend of D units for each share and the coverage ration is R, then this is considered to be a cash ﬂow per share of RD. This can be brought back to present value in exactly the same way as a bond coupon. If the future free cash ﬂows per share are summed at their present value and the book value per share added to that then a valuation of the company has been made. However this neglects two issues. One is the expected rate of return i which will be higher than that for government bonds because the risk is greater. However a precise ﬁgure is rather difﬁcult to estimate. One could use the CAPM to estimate this or make a comparison with similar ﬁrms. The other factor that was neglected is growth. If the free cash ﬂows are expected to increase each year as the company grows then this will have a large effect on the valuation of the company. Anything in the future is problematic to predict and so a company valuation can never be a highly precise calculation, it will always involve the use of “reasonable assumptions.” 7

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posted: | 11/3/2009 |

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