Part 1: The Time Value of Money

Question 1

Compound Interest

Answer 1

The growth in the value of the investment from period to period reflects not only the interest earned on the original principal amount, but also on the interest earned on the previous periods interest earnings (interest on interest).

Question 2

Future Value (FV)/Compound Value

Answer 2

Involves projecting the cash flows forward, on the basis of an appropriate compound interest rate, to the end of the investment’s life.

The amount to which a current deposit will grow overtime when it is placed in an account paying compound interest.

Question 3

Present Value (PV)

Answer 3

Involves bringing the cash flows from an investment back to the beginning of the investment’s life based on an appropriate compound rate of return.

Of single sum, is today’s value of cashflow that is to be received at some point in the future.

The amount of money that must be invested today, at a given rate of return over a given period of time, in order to end up with a specified FV.

Question 4

Purpose of PV and FV

Answer 4

Useful when comparing investment alternatives because the value of the investment cash flows must be measured at some common point in time, typically at the end of the investment horizon (FV), or at the beginning of the investment horizon (PV).

Question 5

TVM Problems using Financial Calculator

Answer 5

N = Number of compounding periods
I/Y = Interest rate per compounding period
PV = present value
FV = future value
PMT = Annuity payments, or constant periodic cashflow
CPT = Compute

Question 6

Time Lines

Answer 6

A diagram of cash flows associated with TVM problem, where a cash flow that occurs in the present (today) is put at time zero.

Cash outflows (payments) are given a negative sign, and cash inflows (receipts) are given a positive sign.

Discounting = when cash flows are assigned to a time line, they may be moved to the beginning of the investment period to calculate the PV through a process.

Compounding = when cash flows are assigned to a time line, they may be moved to the end of the investment period to calculate the FV through a process.

Question 7

Interest Rates

Answer 7

Measure of time value of money, although risk differences in financial securities lead to difference in their equilbrium interest rates.

Question 8

Equilbrium interest rates/discount rates

Answer 8

The required rate of return for a particular investment, in the sense that the market rate of return is the return that investors and savers require to get them to willingly lend their funds.

• If an individual borrows funds at an interest rate of 10%, then that individual should discount payments to be made in the future at that rate, to get equivalent value in current dollars or other currency.

Question 9

Equilbrium interest rates/discount rates (alt definition)

Answer 9

The opportunity cost of current consumption.

e.g. if market rate of interest on 1 year securities is 5%, earning an additional 5% is the opportunity forgone when current consumption is chosen than saving (postponing consumption).

Question 10

Real risk free rate of interest

Answer 10

A theoretical rate on a single period loan that has no expectation of inflation in it.

Question 11

Real rate of return

Answer 11

Investors increase in purchasing power after adjusting for inflation.

Since expected inflation in future period is not zero, the rates we observe on US T-Bills (example), are risk free rates not real rates of return.

Question 12

Nominal risk-free rates

Answer 12

e.g. T-Bills are nominal risk free rates because they contain an inflation premium.

nominal risk free rate = real risk free rate + expected inflation rate

Question 13

Risk in securities

Answer 13

May have 1 or more types of risk, and each added risk increases the required rate of return on the security.

These types are:

1. Default risk
2. Liquidity risk
3. Maturity risk

Question 14

Default Risk

Answer 14

The risk that a borrower will not make the promised payments in a timely manner.

Question 15

Liquidity Risk

Answer 15

The risk of receiving less than fair value for an investment if it must be sold for cash quickly.

Question 16

Maturity Risk

Answer 16

The prices of longer term bonds are more volatile than those of shorter-term bonds.

Longer maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium.

Question 17

Required interest rate on security

Answer 17

= nominal risk free rate + default risk premium + liquidty premium + maturity risk premium

Risk factors associated with risk premium adjust for greater default risk, less liquidity, and longer maturity relative to very liquid, short-term, default risk-free rate such as T-Bills.

Question 18

Effective annual rate (EAR)/ Effective annual yield (EAY)

Answer 18

The rate of interest that investors actually realise as a result of compounding.

e.g. bank will quote a savings rate as 8% compounded quarterly rather than 2% per quarter.

FIs quote rates as stated annual interest rates along with compounding frequency, then periodic rates.

Question 19

Effective annual rate (EAR) formula:

Answer 19

The annual rate of return actually being earned after adjustments have been made for different compounding periods.

EAR = ( 1 + periodic rate)^m - 1

periodic rate = stated annual rate/m
m = the number of compounding periods per year

Question 20

Computation of EAR

Answer 20

Necessary when comparing investments that have different compounding periods, it allows for apples-to-apples rate comparison.

• EAR stated for 8% compounded annually is not the same as the EAR for 8% compounded semiannually or quarterly.
• When compound interest is being used, stated rate and actual (effective) rate of interest are equal only when interest is compounded annually.
• If not the case, greater the compounding frequency, the greater the EAR will be in comparison to stated rate.

Question 21

Formula: FV of single cash flow

Answer 21

FV = PV(1 + I/Y)^N

PV = amount of money invested today (the present value) at t=0.
I/Y = rate of return per compounding period
N = total number of compounding periods
( I + I/Y)^N = the compounding rate on an investment and is frequently referred to as the future value factor or future value interest factor for a single cash flow at I/Y over N compounding periods.

FV will determine the value of an investment at the end of N compounding periods, given that it can earn a fully compounded rate of return I/Y over all of the periods.

Question 22

Discounting

Answer 22

The process of finding a PV of a cashflow.

The interest rate used is often referred to as opportunity cost, required rate of return, discount rate and cost of capital.

The interest rate represents the annual compound rate of return that can be earned on an investment.

Question 23

Formula: PV of single sum

Answer 23

PV = FV (1/(1+I/Y)^N) = FV/(1 + I/Y)^N

(1/(1+I/Y)^N) = present (interest) value factor or discount factor for a single cashflow at I/Y over N compounding periods.

For a single future cashflow, PV is always less than FV whenever discount rate is positive.

Question 24

Annuities

Answer 24

A stream of equal cashflows that occurs at equal intervals over a given period.

e.g. receiving $1,000 per year at the end of the next 8 years.

PV or FV is solved from a stream of equal periodic cash flows, where the size of the periodic cash flow is defined by the payment (PMT) variable on the calculator.

2 types:

1. ordinary annuities
2. annuities due

Question 25

Ordinary annuity

Answer 25

Most common type.

It is characterised by cash flows that occur at the end of each compounding period, a typical cashflow pattern of many investment and business finance applications.

Question 26

Annunity Due

Answer 26

Where payments or receipts occur at the beginning of each period (i.e. the first payment is today at t=0)

Question 27

PV of an ordinary annuity

Answer 27

Use the future cash flow stream, PMT but discount the cash flows back to the present time (t=0), rather than compounding them forward to the terminal date of the annuity.

Question 28

FV of an annuity due (FVAd)

Answer 28

An annuity where the annuity payments (or deposits) occur at the beginning of each compounding period.

Calculator instructions:

1. Must be in beginning of period mode (BGN).
2. To switch between BGN and END mode on TI, press [2nd] [BGN][2nd][SET], when complete BGN will appear on the right corner.
3. If display indicates the desired mode, press [2nd] [QUIT], to return to END mode.

While annuity due payments are made or received at the beginning of each period, the FV of an annuity due is calculated as of the end of the last period.

Question 29

Formula: FV of annuity due (alt.)

Answer 29

FVAd = FVAo x (1 + I/Y)

i.e. calculate the FV of ordinary annuity, simply multiply the resulting FV by [1 + periodic compounding rate (I/Y)].

Question 29

PV of an annuity due (PVAd)

Answer 29

With an annuity due, there is one less discounting period since the first cash flow occurs at t=0, thus already its PV.

This implies that all else equal, the PV of an annuity due will be greater than PV of an ordinary annuity.

Question 30

Computing: PV of an annuity due (PVAd)

Answer 30

How to calculate:

1. Place the calculator in BGN mode, then input all relevant variables (PMT, I/Y, N).
2. Alt. to treat the cashflow stream as an ordinary annuity over N compounding periods, and simply multiply the resulting PV by [1 + periodic compounding rate (I/Y)].

i.e. PVAd = PVAo x (1 + I/Y)

2nd method: your calculator is in END mode, and won’t run the risk of forgetting to reset it.

Regardless of the procedure used, the computed PV is given as of beginning of first period, t=0.

Question 31

Perpetuity

Answer 31

A financial instrument that pays a fixed amount of money at set intervals over an infinite period of time.

A perpetual annuity.

Most preferred stocks, as promise fixed interest or dividend payments forever.

Question 32

PV of a perpetuity

Answer 32

The discount factor for perpetuity is just one divided by the appropriate rate of return (i.e. 1/r), given this:

PVperpetuity = PMT/I/Y

This is the fixed periodic cash flow divided by the appropriate periodic rate of return.

Question 33

Uneven Cash Flows

Answer 33

• not an annuity as cashflows are different every year, wherein essence this is nothing more than a stream of annual single sum cashflows.
• To find the PV or FV of cashflow stream, all is needed is the sum of PV/FV’s of individual cash flows.

Question 34

The effect of compounding frequency on FV and PV:

Answer 34

• The conceptual foundations of TVM calculations are not affected by compounding period, more frequent compounding does have an impact on PV and FV computations.
• Since, an increase in frequency of compounding increases the effective rate of interest, it also increases the FV of given cash flows and decreases the PV of given cash flow.

Question 35

Cash Flow Additivity Principle

Answer 35

• Refers to the present value of any stream of cash flows equals the sum of the present values of cash flows.
• If we have 2 series of cash flows, the sum of the present values of 2 series is the same as the present values of the 2 series taken together, adding cashflows that will be paid at the same point in time.