FINM1001 equations compilation PDF

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A compilation of equations used in finm1001...

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FINM1001 equations FV= future value PV, F= present value F0= principal rs, r= interest rate n= number of terms N= total number of years P0= price of shares D= dividend re= required rate of return rd= required rate of return on debt F= face value of coupon Cr= coupon rate Rt = Cash revenues generated by the firm in period t; Et = Cash expenses paid by the firm in period t (Does NOT include depreciation); It = Capital expenditure paid in period t; and, Tt = Taxes paid in period t. t = The corporate tax rate; and, Dt= Depreciation charge for period t. Note that depreciation is not included as a cash expense in Et. This is because it is not a cash flow. However, depreciation gives rise to a tax deduction, which is a cash flow, and is therefore included in tax calculations. SLD= straight line depreciation of asset IC= initial cost of asset ESV= estimated salvage value of asset EUL= estimated useful life of asset

Simple Interest Calculations Future value FV= F0 + F0rsn = F0 (1 + rsn) Present value FV PV= (1+r s n)

Compound Interest Calculations Future value FV= F0 (1 + r)n Present value n 1+r ¿ ¿ PV= FV ¿

Multiple cash flows that are uneven in amount Future value

FV =P1 (1 + r) N- n1 + P2 (1 + r )N- n2 + P3 (1 + r)N- n3 +... n1, n2, n3 refer to year 1, 2 or 3. e.g. calculate the total value of deposits in exactly three years’ time as follows of cash flows of $100, $200 and $500: FV = $100(1.1)2 + $200(1.1) + $500 = $841 Present value

P3 P2 + PV = P 1 + +... 2 1+ r (1 + r ) (1 + r )3 300 500 580 290 PV = + + + 2 3 (1.10) (1.10) (1.10) (1.10) 4 =$1,284.20 e.g. Pn = whatever the amount is that comes in (e.g. $300, $290) r is the interest rate and is to the power of what ever year the cash flow comes in Ordinary annuity Future value é (1 + r) n - 1 ù FV =F ê ú r ë û Present value é 1- (1 + r)- n ù =F ê ú r ë û PV Annuity due Future value é (1 + r) n - 1ù FV =F ê ú (1 + r ) r ë û Present value é 1- (1 + r)- (n- 1) ù =F + F ê ú r ë û PV

Deferred annuity Present value é 1- (1 + r) - nù Fê ú r ë û PV = m- 1 (1 + r)

Ordinary perpetuity Present value F PV = r Perpetuity due Present value F PV =F + r

Deferred perpetuity Present value éFù ê ú ë rû PV = (1 + r)m- 1

Interest rate

Valuing shares 1. Identify all the cash flows generated by the asset as well as when they occur; 2. Find the present value of each cash flow that was identified in 1; and, 3. Sum the present values of all cash flows calculated in 2.

Constant dividends (with no growth in dividends)(e.g. preference shares)

P0 =

D re

Dividends with constant growth

P0 = =

D0 (1 + g ) re - g D1 re - g

D0 is the dividend just paid D1 is the dividend paid next Bond valuation Coupon paying bonds é 1- (1+ r )- n ù F d B =C ê ú+ rd úû (1+ rd )n êë Cr C =F ( ) n In this equation n is number of times paid per year (so usually equal to 2)  n for finding c is the number of payments in a year, so usually 2. But the n in finding B is number of overall periods. So if payment twice a year and over 5 years then n is 10.  Calculate C using this equation for the bond price equation.  Make sure than frequency of n and rd is the same. Normally compounded semi-annually so n is usually times 2. E.g. if over 3 years then n is 6.  Assume semi-annual compounding interest rate and coupon payment unless otherwise stated. However compounding frequency is the same as payment frequency, so if stated payment is annual then compounding also annual. – Face value amounts: Bond prices are quoted as if the face value was $100. Therefore, if a bond with a face value of $100,000 has a reported price of 95.00, its actual value is 95% of $100,000 or $95,000; and, – Coupon payments: The majority of coupon-paying bonds pay coupons semi-annually. Further, coupon rates and yields are quoted as annual nominal rates compounded semi-annually. Note: in an exam/quiz question: – unless told otherwise , assume coupons are paid semi-annually and face value is $100. – If told that coupons are other than semi-annual (e.g. quarterly), coupon rates and yields are quoted as annual nominal rates with compounding frequency to match the frequency of the coupon payments (implication: for coupon bonds can always convert straight from rn to rp).

We can determine whether a bond’s price will exceed its face value simply by comparing its percentage yield to it’s percentage coupon rate. More specifically: – Cr > rd if and only if B > F; – Cr = rd if and only if B = F; and, – Cr < rd if and only if B < F.

Zero coupon bonds

B=

F (1 + rd )n

Make sure than frequency of n and rd is the same. Normally compounded semiannually so n is usually times 2. E.g. if over 3 years then n is 6. Assume semi-annual compounding interest rate unless otherwise stated. However compounding frequency is the same as payment frequency, so if stated payment is annual then compounding also annual. – Face Value: As was the case with coupon-paying bonds, zero coupon bond prices are quoted as if the face value is $100; and, – Bank Accepted Bill Yields: Bank Accepted Bill Yields are quoted on a nominal basis, with the price able to be computed simply by discounting the face value of the bill at the periodic rate. Note: in an exam/quiz question, unless told otherwise, assume a face value of $100. Note: in a question, unless told otherwise, for zero-coupon bonds with maturities ≥ 1 year assume the yield is quoted as an annual effective rate. Elsewhere, assume an annual nominal rate with a compounding frequency that matches the time until bond maturity. E.g. the price of a 90-day bank accepted bill quoted as having a yield of 8% p.a. is calculated as:

$100 90 x0.08)) (1+ ( 365 =$98.07

B=

Note: this formula can also be applied to Treasury Notes.

Investment decisions Net cash flows X t =CashInflowst- CashOutflowst

=Rt - Et - I t - Tt

X t =(1- t )(Rt - Et) + t Dt - It (When substituting income tax equation) Income tax for corporations

Tt =t (Rt - Et - Dt ) Depreciation The cost of an asset = Purchase price + delivery and installation charges Depreciation = Process of allocating costs over the expected useful life of the asset. Use straight-line depreciation (equal allocation of depreciation over the life of the asset such as a machine). Depreciation is tax deductible. IC - ESV SLD = EUL

Depreciated value=IC − Depreciation ×Years

Risk and return Expected returns

N

E (r ) = ri P (ri ) i =1

So multiply each possible outcome by their probability of occurring, then add the result of each to find the expected return.

Diversification and risk Correlation coefficient:

 X,Y =

 X,Y  X Y

Top is covariance of X and Y Bottom is standard deviation of X multiplied by standard deviation of Y Expected return of a two-asset portfolio:

E ( R p ) =w1E ( R1 ) + w2 E ( R2 ) Where: w1 =Proportion invested in Asset 1; w2 =Proportion invested in Asset 2; E(R1) =Expected return on Asset 1; and, E(R2) =Expected return on Asset 2. The variance of expected return of a two-asset portfolio is:

2 2 2 2 2  p =w1  1 + w2  2 + 2w1w2 12

Or if the formula for covariance is substituted

 p =w1  1 + w2  2 + 2w1w2 12 1 2 2

2

2

2

2

σ12 is covariance of 1 and 2 Standard deviation is just the square root of the variance The less positively correlated assets are, the less overall risk there is for the portfolio, this is because if one asset does badly the other may not follow by doing badly. So there are no diversification benefits from having a two-asset portfolio with two perfectly correlated assets. In order to determine the weight of each asset in two-asset portfolio that will allow the least amount of risk:

-    =  +  - 2   2

w1

1,2 1

2

2

2

1

2

1,2

2

1

2

and w2 =1 - w1 CAPM - Review the 7 steps to deriving CAPM - The Beta measures the extent to which security movies with the market. So how sensitive and risky an asset is to a market. - The Capital Asset Pricing Model (CAPM) determines the appropriate discount rate to use in computing net present value (NPV) and valuing shares, using CAPM we can calculate the required return of an asset using the equations:

-

-

é E (rm ) - r f ) ù E (ri ) =rf +  im ê ú 2 ë m û

E (ri ) =rf + i éë E ( rm ) - rf ) ùû E(rm) is the return on the market E(ri) is the required return on asset i rf is the risk-free rate (is the absolute minimum required rate of return for any investment) Beta is the covariance of the market

  i = im2 m

mm is covariance of the stock with the market mis the variance of the market Example: Beta of a Portfolio You invest in two stocks, A and B. These stocks have betas of 1.5 and 1.2 respectively. If you hold an equally weighted portfolio of the two stocks, the beta of your portfolio is simply the weighted average of the betas for each asset, or:

 p =(0.5 x1.5) + (0.5 x1.2) =1.35 Hence, this portfolio is approximately 35% more risky (more variable) than the market portfolio.

Future/forward contracts - To value futures/forward contracts: F = S0(1 + rf + q)T F is the price you pay after at the end of the contract (so the current price locked) S0 is the spot price now rf is the interest rate you earn by placing money in a bank account q is the cost of carry (storage costs) T is time - If rf and q are per annum, then T can be per annum too. Even if T is only 6 months, it can still be expressed as T=0.5, while rf and q are expressed as their normal per annum rate (e.g. 5% p.a.= 0.05). - Sometimes there are no storage costs but there are cash inflows from holding an asset, such as dividends for shares, so q is replaced with d for the equation: F = S0(1 + rf – d)T...