3. Investment Planning. 7. Time Influence on Valuation Flashcards
In general, if an investor plans on holding an investment for a long time-period, he or she has the opportunity to take on a riskier investment. Take stocks, for example: Over the past 73 years, large-company stock prices have risen an average of 11.2 percent per year. However, it has not been a smooth ride. The problem with stocks is that “the average” is just that, it takes out all of the ups and downs encountered by investors. Some investors invest during a period of low returns do not earn the “average return,” and experience a loss of their investment.
If an individual needs money for his or her child’s college education, which begins next year, the stock market is probably not an appropriate investment. A rationale investor would not risk their child’s college education on whether or not this will be a good year in the stock market. Contrary to the behavior of the stocks, fixed income investments become riskier in the long term as opposed to the short-term. In addition to immediate exposures to interest rate risk and reinvestment risk, over the long run, fixed income securities have tremendous exposure to inflation (or purchasing power) risk. By looking at stocks and bonds, we can see that time can have both favorable and unfavorable effects on investments.
The Time Influence on Valuation module, which should take approximately three hours to complete, will explain the concept that time affects the value of money.
Upon completion of this module you should be able to:
* Define the time value equations in relation to the present and future value of an investment, and
* Discuss time influence on bond prices in terms of their exposure to interest rate risk.
Module Overview
To ensure that you have a solid understanding of the influence of time on security valuation, the following lessons will be covered in this module:
* Present Value and Future Value
* Convexity and Duration
Click here to view an equation sheet that will be needed for this Module’s Lesson Exercises and Module Quiz.
Section 1 – Present Value and Future Value
There are several ways to estimate the value of an investment. One way of valuing an investment is to use the discounted present value model, which assumes that money has time value. This assumption is relevant because borrowers pay interest to lenders to induce them to make loans. Interest is the rent on borrowed money. It causes money to have a terminal value in the future that differs from its present value. The discounted present value model can be used to help you estimate the value of securities like stocks, bonds, or even rental property.
Once you have an estimate of the investment’s value, you can compare its price with its value and decide whether you think it is underpriced, overpriced, or priced appropriately.
The concepts introduced in this lesson are fundamental to all forms of investing and are essential to wealth maximization. To ensure that you have a solid understanding of the present and future value of a security, the following topics will be covered in this lesson:
* Present Value
* Future Value
Upon completion of this lesson, you should be able to:
* Define and calculate the net present value of a security, and
* Define and calculate the future value of a security.
What is the formula for
Net Present Value (NPV)?
NPV = PV of Future Cash Flows - Purchase Price
What does a positive NPV mean?
What does a negative NPV mean?
A positive NPV means that the present value of all the expected cash inflows is greater than the cost of making the investment.
Conversely, a negative NPV means that the present value of all the expected cash inflows is less than the cost of making the investment.
What is the Equation for the One-Period Rate of Return?
The following shows how the equation for determining the one-period rate of return is derived by rearranging the equation so that it is equal to the time value model:
r = (Terminal value - Present value)/Present value
r = (Terminal value/Present value) - 1
(1 + r) = Terminal value/Present value
(Present value)(1 + r) = Terminal value
What is the Equation for Present Value?
Present value =
(Terminal value)/(1 + r)
How do you calculate for k using the CAPM model?
k = Rf + bi(Rm - Rf)
k = Market Capitalization Rate
Rf = Risk-Free Return
bi = Beta for security (how closely the security correlates with movements of the market)
(Rm - Rf) = Risk Premium: the difference between the market return and the risk free return.
Thus, suppose that the risk-free rate is .03, Beta = 1.5, and the risk premium on the market portfolio is .08.
What is k?
k = .03 + 1.5(.08) = .15 or 15%
What are the Values for K?
The following are created based on a combination of risk premiums to compute a security’s Required Rate of Return (Cost of Capital):
* For Treasury bills, k = 4.5%
* For Treasury notes, k = 5.5%
* For Treasury bonds, k = 5.9%
* For corporate bonds, k = 6.3%
* For large-cap stocks, k = 13.0%
* For small-cap stocks, k = 14.5%
What is the formula for Present value?
Present value, or PV = CF1/(1+k)1 + CF2/(1+k)2 + CFT/(1+k)T
This valuation model says that the value of a series of cash flows equals the discounted present value of all future cash flows.
* CF stands for cash flow (either inflows or outflows). The cash flows could be cash dividends from a common stock, coupon interest from a bond, rent from a piece of real estate, the asset’s selling price, or other cash flows.
* The subscripts and exponents are time period indicators. The terminal time period, when the cash flow occurs, is denoted T. These cash flows are expected to arrive at the end of successive time periods denoted t = 1, t = 2, t = 3,…., t = T.
* The term k represents the required rate of return that is appropriate for the investment.
What is the formula and keystrokes for PV of Common Stock?
PV = [CF1/(1+k)1]+[CF2/(1+k)2]+[CFT/(1+k)T]
Keystrokes
0 g CF0 2 g CFj 43 g CFj 14.5 i f NPV
The calculator returns: 34.5455
Example PV of Common Stock:
Brenda is thinking of purchasing stock in a small corporation. She thinks the stock should earn a required rate of return of k = 14.5%. Brenda expects to sell the stock for $40 after collecting cash dividends of $2 per share at the end of the first year, and $3 per share at the end of the second year. The present value of this stock is $34.5455 per share.
PV = [CF1/(1+k)1]+[CF2/(1+k)2]+[CFT/(1+k)T]
= [$2/(1.145)1] + [$3/(1.145)2] + [$40/(1.145)2]
= $1.7467 + $2.2883 + $30.5105 = $34.5455
Brenda makes a wealth-maximizing decision to buy the stock if she can get it for less than $34.5455
Keystrokes
0 g CF0 2 g CFj 43 g CFj 14.5 i f NPV
The calculator returns: 34.5455
Example for Stock with Constant Growth Rate
Jana is wondering whether or not she should pay the market price of $51.50 for a stock issued by a large NYSE-listed corporation that is currently paying an annual cash dividend of $3 per share. Jana believes this dividend will grow at a rate of g = 3% per year for as long as she can see. Assume we use k = 13.0% as the risk-adjusted discount rate to use in valuing the stock.
What is the PV of the stock?
PV = DIV 0 (1+g) / (k−g)
= $3 (1.03) / (.13 – .03)
= $30.90
Based on these calculations Jana decides not to buy the stock because it is overpriced by $51.50 – $30.90 = $20.60 per share.
Example for Stock with No Growth Rate (also works for Preferred Stock)
Alex is considering paying the market price of $50 for a share of preferred stock that will pay an annual cash dividend rate equal to 4.5% of its $100 face value per share forever. This $4.50 annual cash dividend is fixed, g = 0. Alex plans to hold the preferred stock indefinitely. Some financial research leads Alex to conclude that k = 13.0% is an appropriate risk-adjusted discount rate to use in valuing this preferred stock.
What is the PV of the stock?
PV = DIV0 / k
= $4.50/.13
= $34.615
The stock’s perpetual stream of constant cash dividends is worth $34.615 per share.
Alex maximizes his wealth by deciding not to buy the stock, because it is overpriced by $50 - $34.62 = $15.38 per share.
What is the formula for Sustainable growth rate?
Sustainable growth rate =
(1 – Payout ratio)ROE
The following are sources of growth rates: historical average, industry average, and sustainable growth rate*.
What is the formula for Internal Rate of Return (IRR)?
P=∑∞t=1 Ct / (1+k∗)t
Again, since k* will be compounded over t periods, time has a significant influence over the internal rate of return.
Calculating the internal rate of return (IRR) associated with the investment is similar to the NPV method and offers an alternate method for making investment decisions.
The IRR for a given investment is the discount rate that makes the NPV of the investment equal to zero. To compute the IRR, the NPV is set equal to zero, and the discount rate, which is unknown, is then calculated.
The decision rule for IRR involves comparing the investment’s IRR (denoted by k) with the required rate of return for an investment of similar risk (denoted by k). Specifically, the investment is viewed favorably if k greater than k, and unfavorably if k* less than k. As with NPV, the same decision rule applies if either a real asset or a financial asset is being considered for possible investment.
PRACTITIONER ADVICE
In comparing the NPV and IRR methods, the NPV method is superior in that the underlying assumption in the calculation is the opportunity to reinvest future cash flows at the investors required return. IRR however, makes the unrealistic assumption that the investor has an opportunity to reinvest at the IRR! This difference is significant, and the ramifications are such that given a choice on these methods, the NPV should always be used.
What is the formula for Future Value?
FVn = PV(1+k)^n
FVn = the future value of the investment at the end of n years
n = the number of years during which the compounding occurs
k = the annual interest rate, and
PV = the present value, or the current value in today’s dollars.
The value of an investment at a future point in time is called future value. To derive the future value, we do not look at the present value of future cash flow through discounting. Instead, we look at the future value of investment through compounding.
The future value of an investment for any number of years can be calculated using the following equation, where:
Again, time influences future value through compounding. Time also allows for compounding of inflation rate to erode away the purchasing power of the investment. The inflation adjusted compounding rate or the real rate is equal to:
Real rate = [(1 + interest rate)/(1 + inflation rate)] – 1
For example, if a stock investment is expected to yield 11% on average over the next 10 years and the inflation rate is expected to average 4% during the same time, then the real rate = (1.11/1.04) - -1 = .0673 or 6.73%. The real rate can be used for the compounding rate to solve for the future value of the investment to show the return net of inflation.
