Time Value of Money &

Introduction to Risk and Introduction to Risk and ReturnReturn

The value of money a firm has in its possession today is more valuable than money in the future because the money can be invested and earn positive returns.

Basic Concepts Used:• Time Line• Present Value (Discounting)• Future Value (Compounding)• Single or series of cash flows (annuity &

perpetuity)

niPVFV )1(

ni

FVPV

)1( • Discounting (Present

Value)• Compounding (Future Value)

Basic Models

• Annuities are equally-spaced cash flows of equal size.

• Annuities can be either inflows or outflows.

• An ordinary (deferred) annuity has cash flows that occur at the end of each period.

• An annuity due has cash flows that occur at the beginning of each period.

• An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

Annuities

Perpetuity

A perpetuity is a special kind of annuity.With a perpetuity, the periodic annuity or

cash flow stream continues forever.

PVPerp = Payment/Interest Rate

Compounding Interest more frequently than AnnuallyCompounding more frequently than once

a year results in a higher effective interest rate because you are earning on interest on interest more frequently.

As a result, the effective interest rate is greater than the nominal (annual) interest rate.

Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

Nominal and Effective Annual Rates of InterestThe nominal interest rate is the stated or

contractual rate of interest charged by a lender or promised by a borrower.

The effective interest rate is the rate actually paid or earned.

In general, the effective rate > nominal rate whenever compounding occurs more than once per year

Effective Rate: 1)1( m

m

iEFF

Using the Financial Calculator

[2nd] [PMT] [2nd] [ENTER] : changes mode to BGN

[2nd] [+/-] [ENTER] : resets the calculator

[2nd] [.] [6] [ENTER] : changes from 2 to 6 decimals

CF0 to enter uneven cash flowsNPV to find Present Value for Uneven Cash

Flows

You are planning on receiving $150,000 20 years from now. What is it worth today, if the required rate of return is 12%?

Sample Problems

$15,550.02

N= 20.00

I=12.00%

PV=Solve for

PMT= 0.00

FV = -150,000

Instead of receiving $150,000 in 20 years, you want to know what the equivalent installment amount would be if you received an equal amount each year from next year (time 1) to year 20. What $ amount would that be, assuming the same discount rate as in problem 1?

Sample Problems

N=20.00 I=12.00%PV=0PMT= Solve for $2,081.82FV=-150,000

You have $45,000 today and your aunt, who is a member of an investment club, says that she can turn it into $500,000 within 10 years. What annual rate of return is she implying that her club could earn with your money?

Sample Problems

N=10.00 I=Solve for 27.23%PV=-45,000PMT=0.00FV=500,000

*note the opposite signs for the PV and FV

You think your aunt’s investment club could earn 15% per year. How much would be in the account if you let her invest it for 35 years. For this problem, assume monthly compounding.

Sample Problems

N=420.00 35 X 12I=1.25% 15/12PV=-45,000PMT=0.00FV=Solve for $8,300,913.84

Harte Systems Inc: cash inflows $30,000 (year 1), $25,000 (year 2), $15,000 (years 3-9), $10,000 (year 10)

Required rate of Return: 12%

A second company offers $100,000 on an one-time payment

0 1 2 3 4 5 6 7 8 9 10

$30,000 $25,000 $15,000 $15,000 $15,000 $15,000 $15,000 $15,000 $15,000 $10,000

Use Financial Calculator, Inputs: CFo=0, C01=30,000, F01=1, C02=25,000, F02=1, C03=15,000, F03=7, C04=10,000, F04=1

NPV function: I = 12, Compute NPV = $104,508.28

$104,508.28 > $100,000 Accept the series of payments

Assigned Problem 4-31: Value of Mixed Stream

Loan Amount: $15,000Annual Rate of Interest: 14%Repaid Period: 3 years (end of year payments)

Assigned Problem 4-46: Loan Amortization Schedule

N = 3 , I/Y = 14, PV = 15,000, FV = 0, Solve for PMT = $ 6,459.97

End of Year

Beginning of Year Principal

Loan Payment

Annual Interest Paid

Annual Principal Paid

End of Year Principal

1 $15,000.00 $6,460.97 $2,100.00 $4,360.97 $10,639.032 $10,639.03 $6,460.97 $1,489.46 $4,971.51 $5,667.523 $5,667.52 $6,460.97 $793.45 $5,667.52 $0.00

Definition of Risk

In the context of business and finance, risk is defined as the chance of suffering a financial loss.

Assets (real or financial) which have a greater chance of loss are considered more risky than those with a lower chance of loss.

Risk may be used interchangeably with the term uncertainty to refer to the variability of returns associated with a given asset.

Sources of RiskFirm – Specific Risk

Business Risk

Financial Risk

Shareholder – Specific Risk

Interest Rate Risk

Liquidity Risk

Market Risk

Firm and Shareholder Risk

Event Risk

Exchange Rate Risk

Purchasing Power Risk

Tax

Return represents the total gain or loss on an investment.

The most basic way to calculate return is as follows:

Returns

Risk Preferences

An investment portfolio is any collection or combination of financial assets.

If we assume all investors are rational and therefore risk averse, that investor will ALWAYS choose to invest in portfolios rather than in single assets.

Investors will hold portfolios because he or she will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.”

If an investor holds a single asset, he or she will fully suffer the consequences of poor performance.

This is not the case for an investor who owns a diversified portfolio of assets.

Portfolio Risk and Return

The return of a portfolio is a weighted average of the returns on the individual assets from which it is formed and can be calculated as shown in the following equation:

Returns of a Portfolio

0 # of Stocks

Systematic (non-diversifiable) Risk

Unsystematic (diversifiable) Risk

Portfolio Risk (SD)

σM

Risk of a Portfolio

Derived using principles of diversification with simplified assumptions.

rRF :The rate of return on Treasury bills

rM :The average rate of return in the market

i :Correlation/ Coefficient of the risk of the market

compared to the returns

iRFMRFi rrrr )(

Capital Asset Pricing Model (CAPM)

Individual investors are price takers.Single-period investment horizon.Investments are limited to traded financial assets.No taxes and transaction costs.Information is costless and available to all investors.Investors are rational mean-variance optimizers.There are homogeneous expectations.

Assumptions of the CAPM

Currently under consideration is a project with a beta “b” of 1.5. At this time the risk free rate of return rf is 7% and the return on the market Rm is 10%. The project is expected to earn an annual rate of return of 11%.

a) If the return on the market portfolio was to increase by 10% what do you expect the return on the project’s required return? What if the market return were to decline by 10%?

b) Use the CAPM to find the required return on this investmentc) On the basis of the calculation in part “b” would you recommend this

investment and why.d) Assume that as a result of investors becoming less risk averse the

market return drops by 1% to 9%. What impact would this change have on your responses in part b and part c?

Problem on the CAPM

a) Since Beta is 1.5 the required return would change by

1.5 x (+/-) Rate

ri will increase by 15% if rm increases by 10%

ri will decrease by 15% if rm decreases by 10%

b) ri=0.07+(1.5)x(0.10-0.07) = 0.07+0.045=0.115 = 11.5%

c) Project’s expected return is 11% (0.5% lower than the required return) Reject the Project

d) ri=0.07+(1.5)x(0.09-0.07) = 0.07+0.03=0.10 = 10.0%

Go ahead with the project

Solution