Risk, Return, and the Time Value of Money

Chapter 14

Relationship Between Risk and Return

• Risk

– Uncertainty about the actual rate of return over the holding period

• Required rate of return

• Risk-free rate

Types of Risk

• Business risk (Changing Economy)

• Financial risk (Loan Default)

• Purchasing power risk (Inflation)

• Liquidity risk (Converting to Cash)

The Time Value of Money

• Money received today is worth more than money to be received in the future

• Interest Rates– Nominal Rates = Real Rates + Inflation

• Interest Rates are the rental cost of borrowing or the rental price charged for lending money

• Simple Interest – Interest on the initial value only (Not commonly used except for some construction loans)

• Compound Interest – Interest charged on Interest (Typical in Lending and Savings)

The Time Value of Money

• Present Value (PV) - a lump sum amount of money today

• Future Value (FV) - a lump sum amount of money in the future

• Payment (PMT) or Annuity - multiple sums of money paid/received on a regularly scheduled basis

The Six Financial Functions

• Future value of a lump sum invested today

• Compound Growth

• FV = PV(1+i)n

• PV=value today, i= interest rate, & n= time periods

• Example: where PV= $1, n=3, & i=10%

– FV = $1 x (1+.10) x (1+.10) x (1+.10)

– FV = $1 x 1.331

– FV = $1.331

The Six Financial Functions

• Present Value of a Lump Sum

• Discounting

– Process of finding present values from a future lump sum

• PV = FV [1/(1+i)n ]

• Example: where FV= $1, n=3, & i=10%

– PV = $1 x [1/(1+.10) x (1+.10) x (1+.10)]

– PV = $1 x [1/1.331]

– PV = $1 x 0.7513

– PV = $0.7513

The Six Financial Functions

• Future Value of an Annuity– FVA = PMT[((1+i)n -1))/ i]

– The future value of a stream of payments

The Six Financial Functions

• Present Value of an Annuity– PVA = PMT[(1-(1/(1+i)n ))/ i]

– The present worth of a stream of payments

The Six Financial Functions

• Sinking Fund– SF PMT = FVA [ i / ((1+i)n -1))]

– The payment necessary to accumulate a specific future value

The Six Financial Functions

• Mortgage Payments (Mortgage Constant)– MTG PMT = PVA [ i / ((1 - (1/(1+i)n )))]

– The payment necessary to amortize (retire) a specific present value

Effect of Changing the Compounding Frequency

– Interest Rates are quoted on an annual basis

– Increasing the frequency of compounding increases the amount of interest earned

– Increasing the frequency of payments for an amortizing loan decreases the amount of interest paid

Examples

• A Future Value Example:– You have just purchased a piece of residential land

for $10,000. Based upon current and projected market conditions similar lots appreciate at 10% per year (annually). How much will your investment be worth in 10 years? How about 20 years. Is the effect of compounding 2 times greater?

Examples

• A Present Value Example:– You have been offered the option of purchasing a

condo which will be sold for $150,000 at the end of 15 years. You need to make a reasonable offer for the investment so that you can purchase it today. You expect that similar investments would provide an 8% return per year (annually). How much should you be willing to pay (in one lump sum) today for this investment?

Examples

• Future Value of an Annuity Example:– You wish to save $2,000 per year over the 10 years

you operate an apartment property. You can invest your savings at 8% per year (annually). How much money will you have in the account when you sell the investment?

Examples

• Present Value of an Annuity Example :– You will receive $5,000 per year over the next 30

years as equity income from a ground lease you wish to purchase. Investors require an 8% return for similar investments If you wish to buy this property, how much should you offer (in one lump sum) for the investment today?

Examples

• Sinking Fund Payment Example:– You wish to buy a house in 5 years. The down

payment on a house, like you hope to purchase, will be $7,500. How much must you save every year to afford this down payment, given that you can invest the savings with the bank at 8%?

Examples

• Mortgage Payment Example:– You have negotiated the purchase of a condominium

for $70,000. You will need a loan of $60,000, which the local bank has offered based on a 30 year term at 6% interest (annually). How much will your annual payment be for the condo?

– Since nearly all mortgages are calculated on a monthly basis what is the monthly payment for the loan?

Net Present Value (NPV)

• Difference between how much an investment costs and how much it is worth to an investor

• NPV Decision Rule– If the NPV is equal to or greater than zero,

we choose to invest

Net Present Value (NPV)

• PV inflows – PV outflows

– NPV Formula:

0)1(CF

i

FVn

Internal Rate of Return (IRR)

• The discount rate that makes the NPV equal to zero - the rate of return on the investment

• IRR Decision Rule– If the IRR is greater than or equal to our

required rate of return, we choose to invest

Calculating Uneven Cash Flows

• Initial Cash Flow is the Cost of the Investment– Initial Cash Flow is Zero (0) if solving for PV

• Use the Nj Key for Repeating Sequential Cash Flows