Investing is real assets – such as productive machinery, new production facilities, research or a new product line – creates new cash flows that did not previously exist, and thus new wealth that was not there before.
A good investment will produce more wealth than it consumes; a bad investment will do the opposite.
The future value of a present amount invested at the rate i t for t periods is the invested amount multiplied by (1 + i t ) t .
The present value of a future amount CF t is: PV = CF t / (1 + i t ) t .
The value of an asset that generates a stream of future cash flows is the sum of the present values of each of the future cash flows associated with the asset.
A constant annuity is a set of cash flows that are the same amounts across future time points. The present value of $1 per period for t periods implies a constant cash flow per period, and is therefore an annuity.
PV of $1 for t periods = [(1 + i ) t – 1] / i (1 + i ) t
Future value of $1 for t periods = [(1 + i ) t – 1] / i
A perpetuity is a cash flow stream that is assumed to continue forever. The formula for the present value of a perpetuity is: PV = CF/ i. For example, $100 per period forever at a discount rate of 10 per cent = $100/0.10 = $1,000.
If cash flows continue forever, but grow or decline at a constant rate g , the formula becomes: PV = CF/ ( i – g ). This might provide a reasonable approximation for a stream from a long-lived asset.
For example, a cash flow stream that starts at $100 at the end of this period and grows at 5 per cent per period, with a 10 per cent discount: PV = $100/ (0.10 – 0.05) = $2,000.
Interest rates that begin at the present and run to some future time point are called spot interest rates .
If interest rates are 5% between t 0 and t 1 , 6% between t 0 and t 2 and 7% between t 0 and t 3 , then the one-period spot rate is 5%, the two-period spot rate is 6% and the three-period spot rate is 7%.
The set of all spot rates is called the term structure of interest rates.
The yield-to-maturity is the rate that discounts a bond’s promised cash flows to equal its market price, analogous to IRR.
A difference in yields between bonds that have been subjected to the same discount rates is the coupon effect on the yield to maturity . It reflects that a greater or lesser portion of the bond’s value derives from the interest payments – because the interest is valued based on the spot discount rate for that period.
The YTM’s are expressing not only the earning rates but also the amounts invested in the bonds across time, i.e., higher interim cash interest payments means that relatively less is invested in the later periods.
The YTM, being a constant per-period average, cannot form the basis of comparison unless the cash flows are identical. Bonds of equal risk must earn the same rates during the same periods.
A forward interest rate is usually noted with the letter f surrounded by a left subscript indicating the rate’s beginning time point and a right subscript indicating the rate’s ending time point, e.g., the interest rate between t 1 and t 2 is noted as 1 f 2 .
The implied forward rate can be calculated by extracting the invested amount and comparing to the payment on that amount.
The present value of a t 2 cash flow can be arrived at:
Discounting with the spot rate for two periods
Discounting for the forward rates for one period each
Therefore: (1 + i 2 ) 2 = (1 + 0 f 1 )(1 + 1 f 2 )
If the forward rates are known, the spot rate of interest can be found by multiplying together 1 plus each of the intervening forward rates, taking the n th root of that product, and subtracting 1.
If the spot rates are known, the forward rates can be found by solving first for the forward rate nearest the present, and successively working to rates further in the future.
At t 0 , a market participant may have certain expectations about the rates at t 1 . In the interim, new information may appear that causes the market to revise its cash flow expectations, opportunity costs or both.
The possibility of a rate change can be hedged using interest rate futures.
The value of the futures contract depends on the implicit rate at the time that it is bought or sold. If rates increase, the value of the futures contract decreases.
Duration is a measure of the number of periods into the future where a bond’s value, on average, is generated.
The greater the duration of a bond, the further into the future its average value is generated, and the more its value will react to changes in interest rates.
Sum of: proportions of the present value of the bond represented by the cash flows, each multiplied by the number of years until the cash flow is received.
( i x CF i )/( r i i x PV), where r = interest rate
This is a particularly valuable concept for coupon bonds where, depending on their coupon rates, the duration of a nine-year bond could be longer than the duration of a ten-year bond.
Those having claims upon a company’s assets include those having contracts (both formal and informal), government (for taxes) and those holding securities (common shares).
The quid pro quo of limited shareholder liability is that equity is a residual claim: all other claimants will be satisfied before a distribution to shareholders.
Shareholder wealth (market cap) is the market value of the common shares. To maximize the former is to maximize the latter.
The value of a share is related to the discounted value of the stream of future dividends. Different bids and offers result from the different perceptions of value and of the discount rate.
Wealth of existing shareholders is increased by the NPV of the investment, regardless of the source of investment capital.
If an investment will generate $X and costs $Y, the new capital suppliers will get $Y of the $X value increase, thus leaving $X - $Y = NPV for the equity holders of the company.
The P/E ratio is a complex function of the size, timing and risk of cash flows, usually based on the preceding period’s earnings (as a proxy for earnings at the end of the current period).
The nexus between the discount rate (which applies only to dividends) and the P/E ratio (which applies only to earnings) is the payout ratio .
Price per share 0 = dividend per share 1 / (re – g)
g = re – (dividend per share 1 / price per share 0 )
(Price per share 0 / earnings per share 1 ) = [payout ratio / (re – g)]
Free (as in ‘unfettered’) Cash Flow is interpreted to be the amount that could be withdrawn by capital suppliers without impairing the operating expectations. If the amount is negative, it is to be contributed.
Customers : includes sales revenue actually received, but also later capital dispositions
Operations : (1) paid in the current period, (2) tax deductible in the current period, and (3) not a payment to a capital supplier. These ‘expenses’ do not include interest or depreciation.
Assets : are cash flows at the time listed, but must be capitalized and deducted in future periods.
Government : A negative tax amount for a project will reduce overall tax for the corporation in the current period or allow recovery of taxes through a “carry-back”.
The common practice in financial analysis of corporate investment is that, when estimating the cash flows of a project, its interest tax shields are not included in the cash flows.
The flows are calculated as if the project will be financed with equity, even if the plan is to finance it with debt.
The Adjusted Present Value (APV) allows the analyst to incorporate various adjustments to cash flows, but this method requires that their dimensions (rather than proportions) be estimated.
APV will find the same NPV as WACC by: (1) discounting the unleveraged cash flows based on financing completely with equity, (2) then adding the discounted (by the pre-tax rate of interest) interest tax shields.
Issue #1: Valuations should consider economic opportunity costs (e.g., alternate use of equipment)
Issue #2: Changes may result from interactions with other activities (e.g., the new product results in declining sales of another product, unless this would have happened anyway.)
Issue #3: Cash outlays that have already been made – sunk costs – are to be ignored.
Issue #4: Accounting numbers may relate neither to cash flow nor to the changes caused by the project.
Sales revenue can be converted to cash receipts by adjusting for the change in accounts receivable, but it is probably preferable to simply adjust for revenues and expenses by adjusting for the change in working capital.
Working capital = current assets – current liabilities
Interest and capital repayment are not cash flows, but merely claims on the cash flows created.
If calculating NPV or IRR, the tax calculation should not include the deduction for interest, since this is already included in the discount rate for NPV, and IRR assumes equity capital only. APV requires an estimate of the tax shields, then discounts at the pre-tax rate.
Overhead allocations are a costing device, and unrelated to the change in cash flows.
Payback Period is the number of periods until a project’s cash flows recoup the investment outlay.
It ignores all cash flows (the profit!) beyond the payback period.
The cash flows are not typically discounted, so that a dollar received at t 1 is valued the same as one in t 3 so long as both are within the payback period.
The Profitability Index (PI) is very similar to CBR. It has the NPV of all future cash flows in the numerator and the initial cash flow in the denominator. It can only be used when the t 0 cash flow is an outlay.
It will produce the same ratio as the CBR when the only negative outlay in in the t 0 period.
It calculates the NPV of the project per dollar of initial outlay, but is unsuitable for ranking projects because the criterion should be the quantum of wealth increase and not the ratio.
Both IRR and NPV are discounted cash flow methods: IRR calculating a rate (and assuming reinvestment) to get an NPV of 0. The NPV uses the hurdle rate to discount, and assumes reinvestment at that rate, to calculate a change to shareholder wealth.
The IRR is unable to contend with each cash flow having a unique discount rate.
If the sign on the returns changes more than once, the IRR method will calculate a return rate for each additional change in sign.
When considering an investment in different assets that will be replaced (renewed), the comparison can be facilitated by applying the concept of an annuity .
Divide the NPV of a single cycle for each investment by the annuity present-value factor for the number of years in the asset replacement cycle at the appropriate discount rate.
The result is a constant annuity outlay per period that has the same NPV as the asset.
Inflation is an increase in the price unaccompanied by any other changes (such as quantity or quality).
The “inflation-free” return is the real rate, and the return that compensates for both that and inflation is the nominal rate.
(1 + Nominal return) = (1 + Real return) x (1 + Inflation)
We typically quote the nominal rate. The real rate is the difference between the nominal rate and the effects of expected inflation (including uncertainty about it).
The most common error in dealing with inflation is in estimating cash flows. The estimates should include the effect of inflation on the cash flows, and then be discounted with nominal rates.
Inflation has the effect of reducing the value of depreciation deductibility for tax. An optimal method might be determined treating these expenses as generating ‘tax shields’, then comparing the present values from the competing methods.
The lease may allow the lessor to capture a tax advantage when it may not otherwise be able to benefit from the deductibility of interest and depreciation.
Leasing can lower the cost of ‘information asymmetries’ that exist for some assets.
There are ‘economies of scale’ in the management of specialized asset leasing.
An analysis of the whether to purchase or lease should subject the cash flows to discounting at the comparable after-tax rate.
The application of ‘economic income’ concepts has become a popular approach to measuring economic performance.
A division may be ‘charged’ for the capital invested, and the period’s income (positive or negative) indicates whether the performance has compensated the company for its opportunity costs.
Activities that produce positive economic profit have positive effects on share value.
The greater difficulty is in implementation, since (1) it is a period by period measure of a longer-term operation, and (2) accounting performance is not the equivalent of economic performance.
The Security Market Line ( SML ) describes the relationship between risk and return: the higher the risk, the higher the required return. It depicts the set of risk-adjusted returns available in the market (i.e., opportunity costs).
An average expected return (a ‘mean’) on a project can be derived from a probability distribution of returns.
To a capital supplier, risk is best measured by the standard deviation of rates of return for a portfolio of assets (and not those of the assets themselves).
A portfolio will have less risk than the average of the securities in it due to the interactions between them. This is the value of diversification .
To the extent that the returns from securities are correlated (unless perfectly so), their risks will cancel each other out. The correlation coefficient runs in a continuum from +1.0 (perfectly correlated) to 0.0 (independent) to –1.0 (perfectly inversely correlated).
The covariance between a pair of assets is given by the product of the standard deviations and their correlation coefficient.
If market participants understand the benefits of diversification, only the risk that cannot be diversified away is relevant: the systematic risk of the asset.
There appears to be a lower limit to risk to be obtained by diversification, due to a common correlation among all securities, called the market factor.
The beta ( ) of an asset is a ratio of its standard deviation of returns x its correlation with the market, over the standard deviation of market return.
i = i i / m
This is the same as the covariance over the variance of the market return, or im / 2 m ; the variance – covariance model
This is sometimes called the regression coefficient, as it provides the slope of the regression line between the asset and the market.
The beta is adjusted for revenue volatility by multiplying the ungeared beta by the ratio of project revenue volatility over company revenue volatility.
Revenue adjusted = u x (Project revenue volatility / Company revenue volatility)
There is a further adjustment for the ratio of fixed prices in the project and company, because financial results will always be more variable with high fixed costs ( operational gearing ).
Ungeared beta for the project is obtained by multiplying the revenue-adjusted beta by the ratio of fixed costs in the project and in the company.
Project u = r.a. u (1 + Project fixed cost %/ 1 + Company fixed cost %)
The WACC of any company is in fact the average of the risk-adjusted rates of return of the company’s various endeavors, including its asset types and associated future cash flow expectations.
This WACC should not be applied as an investment criterion unless the investment risk is the same as the average risk of the entire company.
Risk free returns can be obtained from the YTM on a government bond of comparable maturity.
Estimating E(r m ) is isolation is more problematic, but:
E(r m ) is a function of rf
The whole term – the difference between the market return and the risk free rate – is more stable over time than the market return itself: averaging 9.1% in UK and 8.8% in USA.
As an alternative to adding a risk premium to the risk free rate to produce a risk-adjusted discount rate, we could adjust the future cash flows downward according to their risk characteristics, and then discount the (now ‘certain’) cash flow at the risk free rate.
CF ce = CF – [Covariance(CF, r m ) / Variance (r m )] x (E(r m ) – rf)
The future cash flow is reduced by a portion of the spread over the risk free rate, according to its systematic risk.
This amount is still a future amount, and must be discounted by the risk free rate to present value.
If an investment’s cash flows do not have constant risk across time, and can be resolved as time passes, this must be accounted for in the analysis, or the risk may be over-estimated for future cash flows (beyond the resolution point.)
The dividend decision is, in mirror image, a cash retention and cash reinvestment decision.
If the were no frictions in transaction costs (taxes, brokerage fees or flotation costs), the shareholder would be indifferent to the form of wealth: cash or share value. These frictions may give shareholder ‘clienteles’ a preference for one form over another.
There is enough competition amongst capital users that there is little to be gained by catering to a clientele by changing dividend policy, and it may indeed prejudice existing shareholders.
For any company, its debt claims require lower returns than its equity, because equity, being residual, must bear greater risk than debt.
Their interaction means that the addition of new debt has the effect of raising the risk to equity. Thus, geared equity is riskier than ungeared equity.
Earnings before interest and taxes (i.e., operating income) is a measure of the total amount of money available to service debt, taxes (the priority dividends paid to an unwelcome equity claimant) and equity.
The more borrowing a company does, the steeper will be the line depicting the EBIT -EPS relationship (hence the term ‘leveraged’).
Franco Modigliani and Merton Miller ( M&M ), in ground-breaking analysis, argued that if shareholder wealth was the same regardless of capital structure, then the latter is irrelevant.
An efficient capital market adjusts the returns of debt and equity claimants according to the cost of acquiring the same future cash flow expectations.
Competitive markets abhor an arbitrage opportunity as nature abhors a vacuum. Participants seeking greater wealth create an elegant consistency among security prices.
The transaction might be regarded as the sale of a claim to assets, with an option to repurchase that claim by making interest payments.
The incremental return required on geared equity as the result of increasing debt is often called the implicit cost of borrowing.
As both the equity (re) rate and the debt rate (rd) increase with increases in D/V, the value weight [1 – D/V] on the higher equity rate steadily declines, exactly offsetting the higher proportion of lower-cost debt, so that their weighted average is unchanged.
The operating cash flows of a company are transformed by the taxation system before they can be claimed by capital suppliers. This transformation is different depending on the capital structure of the company.
We can think of a company’s value as the sum of its unleveraged value plus the value of its leveraging-based tax benefits.
Interest deductibility makes the company’s cost of capital lower, the more debt it uses.
The essence is a change in legal ownership of the company’s assets from the shareholders to the bondholders.
Bankruptcy costs are not the declines in value that precipitate the default – those are independent of capital structure. The true costs are those of the legal process and of the opportunity costs (when compared to equity financing).
One consideration for the lender is the value of the debt claim to the total value of the company. A market value of assets might be difficult to obtain, so practitioners often rely on book values . These are thought less vulnerable to upset when the company is in financial distress.
A financial planning exercise, using a simulation model, would compare the financing alternatives under a range of economic outcomes.
Maturity matching associates low risk-and-return assets with high risk-and-return financing (both short term), and high risk-and-return assets with low risk-and-return financing (both long term)
The result is a mixture of risks and returns that is both potentially profitable and survivable.
Some current assets have a long-term characteristic, that is, the company always requires some amount of cash, inventory, debtors regardless of immediate market position.
‘ Accounting definition’ assets are presumably fixed assets.
Rather than considering the costs and benefits of short-term assets and financing, managers devise policies to govern the firm’s investment in each type.
Types include cash, marketable securities, accounts receivable and inventory.
Efficient management balances costs and benefits to produce the highest company value.
Typically benefits reach a point of diminishing returns, while costs steadily increase with usage.
Cash is required for transaction uses, precautionary and anticipatory reserves and compensating balances (left on deposit with bank).
For cash and securities, the balance is in minimizing the sum of transaction costs and foregone interest. When cash usage is steady , cash replenishment amounts are solved by:
$r = sqrt [ (2 x $ D x $T) / i ] , where D is the total annual cash spent by the firm, T is the transaction cost and i is the interest foregone.
The formula equalizes the transaction cost with the cost of foregone interest on the replenishment amount.
At best, a manager may be able to specify a probability distribution of potential cash balance changes.
The company may establish minimum and maximum cash balances to be held, and will calculate a ‘ return point ’ to which the balance will be reset when it hits one of those bounds.
$R = cube root [(3 x $T x s 2 ) / (4 x i )] + $M , where s 2 is the variance of the change in cash balances, i is a daily rate and $M is the minimum threshold.
The s 2 term is expected cash balance change, squared, times the number of times per day there will be changes: s 2 = $c 2 x t
This formula gives an upper threshold at three times the differential between the minimum and the return point, plus $M.
A higher level of receivables may boost credit sales, but at the costs of a longer collection period and of bad debt.
An attempt to discriminate between good and bad credit risks should continue until the incremental expenditure is equal to the expected gain.
To analyze the effects of a change, accumulate: (1) the NPV of the sales value when received, less the previous NPV, (2) less the change in costs, (3) plus the cost of increased working capital, (4) less the NPV of the recoupment of that capital.
If a company has monetary assets denominated in a foreign currency, it is exposed to the risk of exchange rate fluctuations.
Purchasing power parity is an application of the assumption that free markets do not allow arbitrage opportunities to exist for long.
Exchange rates portray relationships in wealth exchanges across national borders in much the same manner as interest rates portray wealth exchanges across time.
Even with its imperfections, there is an impressive elegance and consistency in exchange rates across currencies based on the general economics of international transactions.
By entering a forward exchange contract , a trader commits to purchase or sell an amount of currency at a fixed rate at a fixed time in the future.
The same result can be achieved by borrowing in the foreign currency, exchanging spot and investing the proceeds in the domestic currency.
Maintaining purchasing power across time requires that the forward exchange rates for any two currencies be consistent with the inflation expected in those currencies.
Nominal interest rate = Real interest rate + Effect of inflation
(1 + nominal rate) n = (1 + Real rate) n + (1 + inflation rate) n
The connection is that interest rate differentials are caused by inflation differentials, which are the root cause of the observed discount or premium on forward exchange.
Monetary assets are those whose returns are expected to be fixed in monetary or money terms in the future, regardless of the inflation rate in the economy.
Only these are serious candidates for the hedging of exchange risk.
Real assets are not fixed in foreign currency value, but will increase in value with increases in foreign inflation.
This applies to plant and equipment, but also other longer-term productive assets.
When an investment proposal carries with it an option to alter, curtail or extend a project’s cash flows at some future time, classic NPV is an inadequate valuation technique.
Project cash flows after the exercise, discounted back to the time of the exercise, represent the option payout. This payout, discounted to the present, represents the S 0 value.The exercise price is the premium.
An agent is an individual, group or organization to whom a principal has designated decision-making authority.
The primary engines driving agency situations are conflicts of interest . Typically, the interests of shareholders conflict with managers or bondholders.
Agency concepts can explain real market actions that at first seem irrationally complex or outside the scope of traditional financial economics.
A derivative is simply any financial security whose return or outcome set is derived from some other asset’s value or return outcome.
The well-publicized derivatives trading disasters tend to be manifestations of control failure, regardless of whether the sufferer had intended to hedge or speculate.
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