Reading 7 Discounted Cash Flow Applications
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2. NET PRESENT VALUE AND INTERNAL RATE OF RETURN
Capital Budgeting refers to an investment decision-making process used by an organization to evaluate and select long-term investment projects.
Capital Structure is the mix of debt and equity used to finance investments and projects.
Working capital management refers to the
management of the company’s short-term assets (i.e. inventory) and short-term liabilities (i.e. accounts payable).
Capital budgeting usually uses the following assumptions:
1. Decisions are based on cash flows; not on accounting profits (i.e. net income):
• In addition, intangible costs and benefits are often ignored because it is assumed that if these benefits or costs are real, they will eventually be reflected in cash flows.
• The relevant cash flows need to be considered are incremental cash flows. Sunk costs should be ignored in the analysis.
2. Timing of cash flows is critical i.e. cash flows that are received earlier are more valuable than cash flows that are received later.
3. Cash flows are based on opportunity costs:
Opportunity costs should be included in project costs. These costs refer to the cash flows that could be generated from an asset if it was not used in the project.
4. Cash flows are analyzed on an after-tax basis. Cash flows on after-tax basis should be incorporated in the analysis.
5. Financing costs are ignored. Financing costs are reflected in the required rate of return which is used to discount after-tax cash flows and investment outlays to estimate net present value (NPV) i.e. only projects with expected return > cost of the capital (required return) will increase the value of the firm.
• Financing costs are not included in the cash flows; because when financing costs are included in both cash flows and in the discount rate, it results in double-counting the financing costs.
6. Capital budgeting cash flows are not accounting net income.
For details, refer to Reading 35, Capital Budgeting.s
Independent projects are projects whose cash flows are independent of each other. Since projects are
unrelated, each project is evaluated on the basis of its own profitability.
Mutually exclusive projects compete directly with each other e.g. if Projects A and B are mutually exclusive, you can choose A or B, but you cannot choose both.
2.1 Net Present Value and the Net Present Value Rule
NPV = Present value of cash inflows − initial investment
=
Independent projects: All projects with positive NPV are accepted.
Mutually exclusive projects: A project with the highest NPV is accepted.
• Positive NPV investments increase shareholders wealth.
• NPV is inversely related to opportunity cost of capital i.e. the higher the opportunity cost of capital, the smaller the NPV.
Advantages:
1) NPV directly measures the increase in value to the firm.
2) NPV assumes that cash flows are reinvested at r (opportunity cost of capital).
2.2 The Internal Rate of Return and the Internal Rate of Return Rule
IRR is the discount rate that makesPresent value of future cash inflows = initial investment
• In simple words, IRR is the discount rate where NPV = 0.
• IRR is calculated using trial and error method or by using a financial calculator.
As the name implies, internal rate of return (IRR) depends only on the cash flows of the investment i.e. no external data is needed to calculate it.
Example:
IRR is found by solving the following:
10,000 = 2000
Important to Note:
In the equation of calculating IRR, the IRR must be
compatible with the timing of cash flows i.e. if cash flows
are semi-annual (quarterly), the IRR will be semi-annual (quarterly).
When project’s cash flows are a perpetuity, IRR can be estimated as follows:
= − + • When IRR> opportunity cost of capital NPV> 0. • When IRR< opportunity cost of capital NPV< 0. •If projects are independent, accept both if IRR of
both projects ≥ Cost of Capital.
•If projects are mutually exclusive and project A IRR > project B IRR and both IRR ≥ Cost of Capital, accept Project A because IRRA>IRRB .
Advantages of IRR:
1) IRR considers time value of money. 2) IRR considers all cash flows. 3) IRR involves less subjectivity. 4) It is easy to understand. 5) It is widely accepted.
Limitations of IRR:
1) IRR is based on the assumption that cash flows are reinvested at the IRR; however, this may not always be realistic.
2) IRR provides result in percentages; however,
percentages can be misleading and involves difficulty in ranking projects i.e. a firm rather earn 100% on a $100 investment, or 10% on a $10,000 investment. 3) In case of non-conventional cash flow pattern, there
are or can be multiple IRRs or No IRR at all.
2.3 Problems with the IRR Rule
No conflict exists between the decision rules for NPV and IRR when:
1) Projects are independent.
2) Projects have conventional cash flow pattern.
Conflict exists between the decision rules for NPV and IRR when:
1) Projects are mutually exclusive.
2) Projects have non-conventional cash flow pattern.
NPV and IRR rank projects differently due to the following reasons:
1)Differences in cash flow patterns.
2) Size (scale) differences: Required rate of return favors small projects because the higher the opportunity cost, the more valuable these funds are. Sometimes, the larger, low-rate-of-return project has the better NPV.
3) Timing differences: Project with shorter payback
period provides more CF in early years for
reinvestment. Therefore, when required rate of return is high, it favors project with early CFs.
NPV versus IRR:
• NPV rule is based on external market-determined discount rate because it assumes reinvestment at r (opportunity cost of capital).
• IRR assumes that cash flows are reinvested at IRR; thus, IRR and IRR rankings are not affected by any external interest rate or discount rate.
• It is more realistic to assume reinvestment at opportunity cost ‘r’; thus, NPV method is the best.
It implies that whenever there is a conflict between NPV and IRR decision rule and to choose between mutually exclusive projects, we should always use NPV rule.
3. PORTFOLIO RETURN MEASUREMENT
Holding Period Return (HPR): A holding period return refers to the return earned by an investor from holding an asset for a specified period of time e.g. 1 day, 1 week, 1 month, 5 years etc.
Total return = Capital gain (or loss) yield + Dividend yield
=−+
3.1 Money-Weighted Rate of Return
The money-weighted rate of return (MWR) measures the compound growth rate in the value of all funds invested in the account over the entire evaluation period. In U.S., it is known as “dollar-weighted return”. It represents an internal rate of return (IRR) of an investment. Like IRR,
• Amounts invested (initial market value of the portfolio) are cash outflows for the investor.
• All additions to the portfolio are cash outflows for the investor.
• Amounts returned (receipts) or withdrawn by the investor are cash inflows for the investor.
• The ending market value of the portfolio is a cash
inflow for the investor.
It is computed as follows:
1 + = 0
where,
IRR represents the MWR.
T = number of periods
CFt = cash flow at time t
• MWR is preferred to use to evaluate the
performance of the portfolio manager when the manager has discretion over the deposits and withdrawals made by clients.
Advantages of MWR: MWR requires an account to be valued only at the beginning and end of the evaluation period.
Disadvantages of MWR:
• MWR is highly affected by the size and timing of
external cash flows to an account.
• It is not appropriate to use when investment manager has little or no control over the external cash flows to an account.
Example:
Assume,
• Amount invested in a mutual fund at the beginning of 1st year = $100
• Amount invested in a mutual fund at the beginning of 2nd year = $950
3.2 Time-Weighted Rate of Return
The time-weighted rate of return (TWR) measures the compound rate of growth over a stated evaluation period of one unit of money initially invested in the account.
• In TWR, the account needs to be valued whenever an external cash flow occurs.
• TWR measures the actual rate of return earned by the portfolio manager.
• TWR is preferred to use to evaluate the performance of the portfolio manager when the manager has no control over the deposits and withdrawals made by clients.
When there are no external cash flows, TWR is computed as follows:
= r=
MV− MV MV
rtwr = (1+rt,1)×(1+rt,2) × … (1+rt,n) –1
• Note that unless the sub-periods represent a year, the time-weighted rate of return will not be expressed as an annual rate.
• Each sub-period return within the full evaluation period has a weight = (length of the sub-period / length of the full evaluation period).
If the investment is for more than one year, time-weighted return can be annualized by calculating geometric mean of n annual returns:
Time – weighted return = [(1+R1)(1+R2)…(1+Rn)]1/n – 1
Where,
Rit = return for year i
n = total number of annual returns
Method of computing Time-weighted Return for the Year:
i. Calculate holding period return for each day (i.e. 365 days daily returns) using the following formula:
=
where, ri = r1, r2, …r365
ii. Calculate annual return for the year by linking the daily holding period returns as follows:
Time – weighted return = [(1+R1)(1+R2)…(1+R365)] – 1
This annual return represents the precise time-weighted return for the year IF withdrawals and additions to the portfolio occur only at the end of day. Otherwise, it represents the approximate time-weighted return for the year.
Time-weighted return can be annualized by calculating geometric mean of n annual returns:
Time-weighted return = [(1+R1)(1+R2)…(1+Rn)]1/n –1
where,
Rit = return in period t
n = total number of periods
Advantage of TWR: TWR is not sensitive to any external cash flows to the account i.e. additions and withdrawals of funds.
Disadvantage of TWR:
• TWR requires determining a value for the account each time any cash flow occurs.
• Marking to market an account on daily basis is administratively more cumbersome, expensive and potentially more error-prone.
Example:
• Beginning portfolio value for period 1 = $10,000 • Ending portfolio value for period 1 = $10,050 • Dividends received before additional investment in
period 1 = $100
• Beginning portfolio value for period 2 = $10,350 • Ending portfolio value for period 2 = $10,850 • Dividends received in period 2 = $100
(ℎ ℎ)
=10,050 − 10,000 + 100
10,000 = 1.50%
(ℎ ℎ)
=10,850 − 10,350 + 100
10,350 = 5.80%
The annual return (based on the geometric average) over the entire period is
r = [(1.0150)(1.05800)] –1=0.0739 or 7.39%
TWR versus MWR:
• When funds are contributed to an account prior to a period of strong (positive) performance, MWR > TWR. • When funds are withdrawn from an account prior to
a period of strong (positive) performance, MWR < TWR.
• When funds are contributed to an account prior to a period of weak (negative) performance, MWR < TWR.
• When funds are withdrawn from an account prior to a period of weak (negative) performance, MWR > TWR.
• Under normal situations, both TWR and MWR provide similar results.
• When large external cash flows occur (i.e. > 10% of account) and during that evaluation period, account’s performance is highly volatile, then MWR and TWR will provide significantly different results.
4. MONEY MARKET YIELDS
Money market instruments are short-term debt instruments i.e. having maturities of one year or less. These instruments pay par value (face value) at maturity and are usually discount instruments i.e. they do not pay coupons, but instead are sold below (at discount from) their par (face) value. For example, T-bills are discount
instruments where,
• Investor buys the T-bill at (Face value – discount) and receives face value at maturity.
• Investor earns a dollar return equal to the discount when he/she holds the T-bill to maturity.
Other types of money-market instruments include commercial paper and bankers’ acceptances (which are discount instruments) and negotiable certificates of deposit (which are interest bearing instruments that pay coupons).
1)Bank Discount Basis: T-bills are quoted on a 360-day discount basis rather than price basis using the bank
discount rate (a 360-day year is commonly used in
pricing money market instruments). The bank discount rate is defined as:
!=
rBD= Annualized yield on a bank discount basis
n = Actual number of days remaining to maturity
Limitations of Yield on a bank discount basis: Bank discount yield is not a meaningful measure of investors’ return because:
1. It is based on the FV (par value) of the bond instead of its purchase price; but returns should be
evaluated relative to the amount invested (i.e. purchase price).
2. It is annualized based on a 360-day year rather than a 365-day year.
3. It is annualized based on simple interest; thus, it ignores the compound interest.
• The discount rate for the T-bill can be used to find PV of other cash flows with risk characteristics similar to those of the T-bill.
• However, when risk of cash flows is higher than that of T-bill, the T-bill's yield can be used as a base rate and a risk premium is added to it to represent higher risk of cash flows.
2)Holding period yield (HPY): HPY reflects the return earned by an investor by holding the instrument to maturity.
$=−+
where,
P0 = initial purchase price of the instrument
P1 = Price received for the instrument at its maturity
D1 = Cash distribution paid by the instrument at its
maturity (i.e. interest)
For interest-bearing instruments: The purchase and sale prices must include any accrued interest* when the bond is purchased/sold between interest payment dates.
*Coupon interest earned by the seller from the last coupon date but not received by the seller as the next coupon date occurs after the date of sale.
NOTE:
• When the price is quoted including accrued interest, it is called Full price.
• When the price is quoted without accrued interest, it is called Clean price.
3)Effective annual yield (EAY):
EAY = (I + HPY) 365/t - 1
Rule: The bank discount yield < effective annual yield.
4)Money market yield (or CD equivalent yield): Money market yield can be used to compare the quoted yield on a T-bill to quoted yield on interest-bearing money-market instruments that pay interest on a 360-day basis.
• Generally, the money market yield is equal to the annualized holding period yield (assuming a 360-day year) i.e.
Money market yield = rMM = (HPY) × (360/ t)
• Unlike bank discount yield, the money market yield is based on purchase price.
%""= (%#$) ×&'()* +(,-* ./ 01* 23*(4-35 67,,8-3)1(4* 837)* 9
• Thus, money market yield > bank discount yield.
Or
%"": 360
!
360 > !
5)Bond-equivalent yield: When a semi-annual yield is annualized by multiplying it by 2, it is referred to as the bond-equivalent yield. It ignores compounding of
interest. The bond equivalent yield is calculated as follows:
Bond Equivalent Yield = Semiannual Yield ? 2
