Chapter Four - The Time Value of Money

What is a financial timeline?

a linear representaton of when cash flows occur. it helps visualize the timing of payments and receipts

what does date 0 represent on a timelime?

today, or present time

what are cash inflows and outflows?

inflows: positive cash flows (+) - money received

outflows: negative cash flows (-) - money paid or spent

why use a timeline? why is drawing a timeline useful?

it makes it easier to identify when each cash flow occurs, which is essential for calculating present and future values

what are the three rules?

1. compare or combine cash flows only at the same point in time

2. move cash flows forward (compound them)

3. move cash flows backward (disocunt them)

Rule #1; what is the first rule of time travel?

you can only compare or combine values at the same point in time

example: $1,000 today cannot be directly compared with $1,210 recieved later without accounting for the time value of money

Rule #2; how do you move cash flow forward in time?

compound it

what is Future Value (FV)?

the value of a cash flow at a future point in time after accounting for interest earned

formula: FV = PV(1 + r)^t

where;

PV = present value

r = interest or discount rate

t = number of periods

what is compound interest?

interest is earned on both: the original principal and the interest that has already accumulated

what is the difference between simple and compound interest?

simple interest: interest is earned only on the orginal principal

compound interest: interest is earned on the principal and accumulated interest

FUTURE VALUE EXAMPLE: what is the future value of $5000 invested for 5 years at 10%?

FV = 5000(1 + 0.10)^5

FV = $8,053

so, $5000 today becomes approximately $8053 in 5 years

rule #3; how do you move a cash flow backwards in time?

discount it

what is present value (PV)?

the value today of cash flow that will be received in the future

PV = FV / [(1 + r)^t]

PRESENT VALUE EXAMPLE: what is the present value of $10,000 received in 5 years if the discounted rate is 10%?

PV = 10000 / [(1 + 0.10)^5]

PV = $6,209

so, the $10,000 recived in five years is worth about $6,209 today at a 10% rate

how can multiple cash flows be combined?

move each cash flow to the same point in time, then add them together

how do you find the PV of multiple future cash flows?

1. discount each cash flow back to today

2. add their present values

PVtotal = PV1 + PV2 + ... + PVn

how do you find the FV of multiple cash flows?

1. compound each cash flow forward to the same future date

2. add the future values

FVtotal = FV1 + FV2 + ... + FVn

how do you find the present value of a stream of cash flows?

find the PV of each individual cash flow and add them together

PV= t=1∑n Ct / (1+r)^t

where Ct is the cash flow receivedd at time t

How do you find the future value of a cash flow stream?

compound each cash flow forward to the desired future date and add the resulting values

EXAMPLE OF A FUTURE VALUE OF A STREAM: Three $ 2,000 payments occur at the ends of years 1, 2, and 3. If r = 5%, what is their value at the end of year 3?

year 1 payment: 2000(1 + 0.05)^2 = 2,205

year 2 payment: 2000(1 + 0.05) = 2,100

year 3 = 2000

total: FV = 2205 + 2100 + 2000 = 6305

what does Net Present Value (NPV) measure?

NPV compares the present value of cash inflows or benefits with the present value of cash outflows or costs

what is the formula for NPV?

NPV = PV(benefits) - PV(costs)

NPV = t=0 ∑ n Ct/(1+r)^t

why dont we discount a cash flow occuring at t = 0?

becasye it is already in present value terms

(1 + r)^0 = 1

so, PV = C0

NPV EXAMPLE: you pay $5000 today and receive $3000 in year 1, $2000 in year 2 and $1000 in year 3. if r = 7% what is NPV?

PV = 3000 / (1 + 0.07) + 2000 / [(1 + 0.07)^2] + 1000 / [(1 + 0.07)^3]

PV = 5,366.91

NPV = 5366.91 - 5000

NPV = 366.91

what does a positive, zero or negative NPV mean?

NPV > 0; PV of benefits exceeds PV of costs

NPV = 0; PV of benefits equals PV of costs

NPV < 0; PV of benefits is less than PV of costs (an NPV below zero leads to the conclusion not to invest)

what is a perpetuity?

a constant cash flow that occurs at regular intervals forever

what is the formula for the PV of a perpetuity?

PV = C/r

C = contsant periodic cash flow

r = discount or interest rate

PERPETUITY EXAMPLE: an endowment must provide $100,000 every year forever. if r = 4%, how much must be donated?

PV = 100,000 / 0.04

PV = $2,500,000

what is an annuity?

a constant cash flow that occurs at regular intervals for a finite number of periods

what is the key difference between perpetity and annuity?

PERPETUITY:

- payments continue forever

- constant payments

- PV = C/r

ANNUITY:

- payments continue for n periods

- constant payments

- uses annuity formula

what is the present value of an n-period annuity?

PV = C [ (1 - 1 / [(1+r)^n]) / r ]

where:

C = payment each period

r = discount rate per period

n = number of payments

** the standard annuity starts with its first first payment one period from today

why does C = rP for a perpetuity?

if you have a principal amount P invested at rate r, the interest earned each period is: C = rP

Rearranging: P = C/r

what is the future value of an n=period annuity at the time of the last payment?

FV = C [ ([(1+r)^n] - 1)/ r]

this is useful for determing how a series of regular deposits grows over time

when is the FV annuity formula useful?

when you make equal periodic deposits and want to know how much the account will be worth at the time of the final payment

what happens when an annuity's first payment does not occur immediately?

first calculate the annuitys value at the point immediately before the first payment, then discount that amount back to today

TRUST FUND EXAMPLE: a trust fund pays $25,000/year for 40 years, beginning in 5 years at r = 4%. why cant you directly use the ordinary annuity PV formula to get todays value?

The ordinary annuity formula gives the value one period before the first payment, not necessarily today's value. The annuity formula gives a value at date 4, which must then be discounted four additional periods to date zero.

what is a growing perpetuity?

a perpetuity in which the paymeny increases at a constant growth rate g every period

what is the formula of a PV of a growing perpetuity?

PV = C1 / r - g

where,

C1 = first payment, occuring one period from now

r = discount rate

g = constant growth rate

*** requires r > g

GROWING PERPETUITY EXAMPLE: a mine generates $15,000 in year 1. its cash flow declines by 5% annually forever, and r = 6%. what is its PV?

g = -5% = -0.05

PV = 15000 / (0.06 - (-0.05))

PV = 15000/ 0.11

PV = $136,364

if an endowment needs to increase its annual payment by 2% due to inflation, how does the formula change?

treat the payments as a growing perpetuity:

PV = C1 / r - g

for $100,000, r = 4% and g = 2%

PV = 100,000 / (0.04 - 0.02)

PV = $5,000,000

what is a growing annuity?

a series of cash flows that: occurs at regular intervals, lasts for a fininte number of periods, and grows at a constant rate g each period

what is the PV formula for a growing annuity?

PV = C1 / r - g [ 1 - (1 + g/ 1 + r)^n]

where,

C1 = first payment

r = discount rate

g = growth rate

n = number of payments

what is a growing annuty's FV used for?

to determine the value at the time of the final payment when periodic contributions grow at a constant rate.

RETIREMENT EXAMPLE: You want to begin saving for your retirement. You plan to contribute $12,000 to the account at

the end of this year. You anticipate you will be able to increase your

annual contributions by 3% each year for the next 45 years. If your expected annual return is 8%, how much do you expect to have in your retirement account when you retire in 45 years?

C1 = $12,000

Number of years: n = 45

Growth in contributions: g = 3%

annual return: r = 8%

PV = (12000 / 0.08 - 0.03) (1 - [(1 +[0.03 / 1 + 0.08)^45])

PV = $211, 567

do time value of money concepts still work when payments arent annual?

yes, the same concepts apply, buth both the interest rate and number of periods must be adjusted to match the new time interval

If interest is compounded m times per year, how do you adjust the annual rate?

rperiod = rannual / m

and nperiods = (years)(m)