MATH3090/7039: Financial mathematics Lecture 3
Discounted cashflow (DCF) valuation
Share valuation
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Investment project evaluation
DCF valuation and risky cashflows
The basic philosophy behind the PV approach is to determine the future expected cashflows arising from holding an asset and then discount these cashflows back to the present value by an application of a discount factor (or required rate of return) that compensates the investor for the time value of money, expected inflation and the risk of the asset.
Investments: Concepts and Applications in Australia
Recommended resources for this week
ASX Shares and the booklet Getting Stated in Shares.
Brealey, Myers and Allen, Principles of Corporate Finance, McGraw-Hill/Irwin.
Discounted cashflow (DCF) valuation
Share valuation
Investment project evaluation
DCF valuation and risky cashflows
Discounted cashflow (DCF) valuation
Share valuation
Investment project evaluation
DCF valuation and risky cashflows
Discounted cashflow (DCF) valuation
Recall the basic DCF principal that we covered in Lecture 2. • The value of an asset is
◦ the present value of its expected future cashflows
◦ discounted at an appropriate required rate of return.
The DFC principle is typically represented by the discrete formula:
CCTC PV=1+…+T=t.
(1+y) (1+y)T (1+y)t t=1
where y is the required rate of return.
Note the continuous version of this formula on L2.30.
DCF valuation
We now extend the DCF valuation methodology to:
• Share valuation.
• Investment project evaluation.
• Simple DCF model for uncertain/risky cashflows.
Dividend discount model (DDM)
A share represents an investment in a company in return for receiving a proportion of the company’s profits in the form of dividends:
D1, D2, . . . received in 1, 2, . . . years from now. Here, we assume that the dividends are known constants.
For simplicity, we have assumed annual dividends.
Hence, if we expect the share to keep paying dividends forever, then
P = ∞ D t = D 1 + D 2 + . . .
(1+y)t 1+y (1+y)2
is the today’s price or value of the share.
This is called the dividend discount model (DDM).
Dividend discount model (DDM)
Now consider the scenario in which you will sell the share in T years. You will also receive the selling price at time T :
PT =∞ DT+i = DT+1 + DT+2 +…
(1+y)i (1+y)1 (1+y)2 i=1
Thus, we have the following relation
P = D1 + . . . + DT + PT .
The dividend discount model needs to be adjusted to incorporate e.g. semiannual or quarterly dividends.
(1+y) (1+y)T
Dividend discount model (DDM): example
A share pays a dividend of $0.50, and y = 0.12. What is the price today of the share?
Solution: This is a perpetuity, so
P =∞ Dt = D = 0.50 =$4.167.
(1+k)t k 0.12
(Here we use the preference share formula on L2.42.)
Consider the following case: you expect D1 = $0.30, D2 = $0.40, D3 = $0.40 and P3 = $4.40. Then
P=D1 + D2 +D3+P3=$4.28. 1+y (1+y)2 (1+y)3
Constant growth DDM model
Under this model, starting with the current dividend D0, let the constantgrowthrateisg,i.e.D1 =D0(1+g),D2 =D0(1+g)2),…, Dt =D0(1+g)t.
P=∞ D0(1+g)t
=D0 1+y+(1+y)2+…
1+g (1+g)2
= D0(1+g). y−g
which is derived using the idea of a geometric sequence Proof: a tutorial question next week.
You may have to derive this formula in the exam.
Constant growth DDM: example
Assume D0 = 0.23, g = 5%, and k = 0.11. Then
P = D0(1+g) = 0.23(1+0.05) = $4.03. k − g 0.11 − 0.05
The search for value
Companies invest in a variety of real assets. These include tangible assets such as plant and machinery, and intangible assets such as management contracts and patents. The object of the investment, or capital budgeting, decision is to find real assets that are worth more than they cost.
. Brealey and Stewart C. Myers
Principles of Corporate Finance
Investment decisions of corporations
Purchase of plant, machinery and equipment.
Creation and development of new product lines.
Purchase and/or building of new factories and premises. Property construction and development projects.
Massive mining projects.
Mergers and acquisitions: purchase of other companies.
Corporations don’t take these huge financial decisions lightly . . ..
Investment decisions of corporations
The question is: How do companies decide on whether to go ahead and undertake these huge investment projects.
In other words, how do companies decide if these investment projects are financially viable or profitable investment decisions?
DCF valuation of investment projects: net present value
One way to answer these questions is to apply the discounted cashflow (DCF) valuation framework:
• View an investment project as a set of future cashflows.
• Determine the present value of this cashflow stream by DCF. • Compare this valuation to the outlay or cost of the project.
• Invest if the project’s valuation is larger than the project cost.
Makes sense yes?
Net present value (NPV) project evaluation
Some notation:
IO: the initial outlay or project cost.
k: the required rate of return or cost of capital.
T : the forecast timespan of the project in years.
C1,…,CT is the discrete net cashflow stream at times 1,…,T. C(t): the continuously received net cashflow stream.
Initial outlay and cost of capital
The initial outlay (IO) is the financial or capital investment the company needs to make in the project or investment.
Where does the company get the money from?
It has to raise the capital in financial markets:
• Debt capital markets: bonds, bank lending, others. See ASX Interest Rate Securities and
Macquarie Debt Capital Markets.
• Equity: ordinary shares, preference shares, others. See ASX Shares and Macquarie Equity Capital Markets.
Also see NAB Wholesale Funding, Westpac Debt Markets, Commonwealth Bank Capital Raising, and ANZ Markets.
Initial outlay and cost of capital
Hence the initial outlay IO typically needs to be funded by raising funds in capital markets.
The company will have to make bank loan payments, bond coupon payments, share dividends, etc, on these funds.
Important question: how could the required rate of return k (cost of capital) be determined from these funding sources?
We can compute k as the weighted average cost of these sources of finance.
An example will clarify this.
Initial outlay and cost of capital: example
Suppose a company raises $1, 000, 000 for a project: $200, 000 via bank lending at an interest rate of 7%, $300, 000 via issuing corporate bonds yielding 7.5%, $250, 000 via issuing preference shares at 11%, and $250, 000 via ordinary shares at 15%.
Solution: The cost of capital is calculated by
k = 0.2×0.07+0.3×0.075+0.25×0.11+0.25×0.15 = 0.1015.
For simplicity, this value of k will be used for discounting. In practice, we need to take into account other factors, such as inflation, risk-premium, etc (see L1.29).
Forecast discrete Ct and continuous C(t) cashflows
The value of an investment project is the present value of its expected future net cashflows Ct and/or C(t).
The forecast project cashflows are net cashflows:
net cashflows = cash inflows − cash outflows.
(Think of them as project revenues minus expenses/costs.)
In this course we won’t go into how to evaluate these cashflows for an investment project. Basic books on corporate finance do this.
Project evaluation: net present value (NPV)
The net present value rule is: NPV =−IO+PV
• Invest in the project if NPV ≥ 0.
• Don’t invest if NPV < 0.
T T Cte−kt +
C(t)e−ktdt.
Again, in practice, some people still invest when NPV < 0. Can you think of an example?
Project evaluation: simple NPV example
Suppose IO = $100, 000, C1 = $40, 000, C2 = $30, 000, C3 = $30, 000, and C4 = $35, 000, find the project NPV under:
(i) k = 7% and (ii) k = 14% (both discrete compounding).
(i)NPV = −100,000+40,000+30,000+30,000+35,000 = $14,477. 1.07 1.072 1.073 1.074
(ii)NPV = −100,000+ 40,000 + 30,000 + 30,000 + 35,000 = −$856. 1.14 1.142 1.143 1.144
⇒ Don’t invest.
Discounted cashflow (DCF) valuation
Share valuation
Investment project evaluation
DCF valuation and risky cashflows
DCF valuation: risky cashflows
Some notation
Mt: the number of possible discrete cashflows at time t.
Ctm: the mth possible discrete cashflow at time t, m = 1, . . . , Mt
ptm: the probability of the mth discrete cashflow at time t, m = 1,...,Mt.
C(t): still the instantaneous continuous cashflow at time t.
p(t): the probability density of the instantaneously received C(t). We assume that all of these are know.
DCF valuation: risky cashflows
Let Ct be the time t random variable presenting the cashflow received at time t. Note that Ct have Mt possible outcomes: Ct1, . . . , CtMt . (Don’t confuse Ct with C(t) which the continuous cashflow at time t.) The expected discrete cashflow at time time t is
E[Ct] = ptmCtm. m=1
To explicitly incorporate risk risky cashflows into DCF:
T T E[Ct]e−rt +
p(t)C(t)e−rtdt.
DCF valuation: example of risky cashflows
Consider pricing a zero-coupon bond with possible default p: the probability of default on the face value F .
α: the recovery rate, which is the percent recovered in default of the face value F.
Then the expected face value equals E[F]=pαF +(1−p)F.
The value is
P = E[F] or P =E[F]e−yT. (1+y)T
DCF valuation: example of risky cashflows
Consider pricing coupon-paying bond with possible default qt: the probability of default on the tth coupon payment.
βt: the recovery rate on the tth coupon.
p and α: same as above for a zero-coupon bond. The value of a risky coupon-paying bond is
B = T E t [ C ] + E [ F ]
(1+y)t (1+y)T
whereE[F]=pαF+(1−p)F andEt[C]=qtβtC+(1−qt)C. (Note: annual coupons are assumed in this example. Needs to be adjusted for a different paying frequency.)
DCF valuation: example of risky cashflows
Consider pricing shares with uncertain dividends
Mt: the number of possible dividends at time t.
Dtm: the mth possible dividend at time t, m = 1,...,Mt.
ptm: the probability the mth dividend at time t, m = 1, . . . , Mt.
The value of a share with uncertain dividends is
P=∞ Et[Dt] =E1[D1]+ E2[D2] + E3[D3] +...
Et[Dt] = ptmDtm.
(1+k)t 1+k (1+k)2 (1+k)3
(Note: annual dividend are assumed in this example. Needs to be
adjusted for a different paying frequency.)
Discounted cashflow (DCF) valuation
Share valuation
Investment project evaluation
DCF valuation and risky cashflows
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