Time Value of Money - University of Waterlookvetzal/AFM271/tvom.pdf · Time Value of Money (Text...

Time Value of Money

(Text reference: Chapter 4)

Topics

Background

One period case - single cash flow

Multi-period case - single cash flow

Multi-period case - compounding periods

Multi-period case - multiple cash flows

Perpetuities

Annuities

Mortgages

Amortization schedulesAFM 271 - Time Value of Money Slide 1

Background

the economic value of a cash flow depends on when itoccurs (i.e. the timing of the cash flow)

⇒ $1 today > $1 tomorrow > $1 a year from now > .. .

present value calculations allow us to determine thevalue today of a stream of cash flows to be rec’d/paid inthe future, by taking into account the time value ofmoney

this is an important concept which is widely used bothin corporate and in personal financial decision-making

AFM 271 - Time Value of Money Slide 2

Cont’d

notation:PV = present value = value today of a stream ofcash flowsFV = future value = value at some future time of astream of cash flowsC = cash flowr = interest rate (a.k.a. discount rate)T = number of yearsm = number of periods in a yearn = total number of periods (n = m×T )

we will often apply subscripts to indicate time, e.g. Ct isa cash flow occurring at time t


One Period Case - Single Cash Flow

cash flow assumptions: timing and size are given in allour examples, and for now there is no risk

as long as we measure investment alternatives at thesame point in time, we can rank them consistently

example: you have $100,000. An investment costsC0 = $100,000 today, and returns C1 = $104,000 in ayear. A bank offers a 5% annual interest rate ondeposits. Should you make the investment (option A) orput your money in the bank (option B)?

FV analysis:

PV analysis: PV = C1/(1+ r)


Cont’d

net present value: NPV = (PV of future cash flows) - (cost of

investment today)

NPV < 0 ⇒ don’t invest (since the cost of the investment

today exceeds the value today of all its future cash flows)

calculate NPV for options A and B:

observations:

PV analysis and FV analysis both yield the same

conclusion; the difference is only the time at which the cash

flows are compared

NPV also gives the same conclusion (A and B cost the

same, so we are really just comparing their PVs)AFM 271 - Time Value of Money Slide 5

Cont’d

another example: a piece of land costs $20,000 and willbe worth $21,500 a year from now. The bank pays a 4%annual interest rate. Should you invest?

PV analysis:

FV analysis:

NPV analysis:


Multi-Period Case - Single Cash Flow

two types of interest:simple interest: interest earned only on originalprincipal

FV after 1 yr = C0 +C0 × r

FV after 2 yrs = C0 +C0 × r +C0 × r

...

FV after n yrs = C0 +

n terms︷︸︸︷

C0 × r +C0 × r + · · ·+C0 × r

= C0 × (1+n× r)


Cont’d

compound interest: interest earned on originalprincipal and on previously earned interest

FV after 1 yr = C0 × (1+ r)

FV after 2 yrs = C0 × (1+ r)× (1+ r)

...

FV after n yrs = C0 ×

n terms︷︸︸︷

(1+ r)× (1+ r)×·· ·× (1+ r)

= C0 × (1+ r)n


Cont’d

the power of compounding ($100 deposited at 6% and10%)

Time (years) 1 2 4 10 50 1006% simple 106 112 124 160 400 7006% compound 106 112.36 126.25 179.08 1842.02 33930.2110% simple 110 120 140 200 600 110010% compound 110 121 146.41 259.37 11739.09 1378061.23

other examples:text p. 85: Julius Caesar lent one penny to someone;assuming 6% annual interest, what would be owedon this loan 2,000 years later?the island of Manhattan was purchased in 1626 forthe equivalent of $24; what would this amount beworth in 2005, assuming 5% annual interest?


Cont’d

discounting moves a cash flow that is expected to occurin the future back to today (i.e. finding PV)

compounding moves a cash flow from present to futurevalue amounts (i.e. finding FV)

cash flows at different points in time cannot becompared or aggregated, unless first brought to thesame point in time (via discounting and/orcompounding)

example: C0 = 120, C5 = 130. Adding these up to 250 ismeaningless (like adding £120+$130), but we can findthe value of the combined CFs at any point in time bydiscounting and/or compounding (like using exchangerates for currency conversion).


Cont’d

some examples:

how much must be invested today in order to receive $1,000

in 5 years if interest is compounded at 7% per annum?

what is the annual compound interest rate equivalent to a

simple interest rate of 6% for a five year investment?

suppose a bank’s annual compound interest rate is 4%.

What is the PV of $100 invested for 5 years at 4% simple

interest?


Multi-Period Case - Compounding Periods

stated annual rate: (SAR)

not the whole story unless the compound frequency is given

effective annual rate: (EAR)

if compounding occurs m times per year, then

EAR = (1+SAR/m)m−1

example: SAR = 10%, C0 = $1

annual compounding

FV1 = $1× (1+0.10) = $1.10 EAR = 10%semi-annual compounding

FV1 = $1× (1+0.10/2)2 = $1.1025 EAR = 10.25%quarterly compounding

FV1 = $1× (1+0.10/4)4 = $1.1038 EAR = 10.38%AFM 271 - Time Value of Money Slide 12

Cont’d

we can also have monthly, weekly, daily compounding, etc.

example: Mastercard statement says that the annual interest

rate is 18.4%, the daily interest rate is .05041%, and interest is

compounded daily. What is EAR?

in the limit: limm→∞

(

1+ SARm

)m×T= eSAR×T

this is called continuous compounding:

FV1 = $1× e0.10 = $1.1052 EAR = 10.52%

why does EAR increase as compounding period decreases

(i.e. as compounding frequency increases)?


Cont’d

converting between compounding frequencies: e.g. 10%compounded semi-annually is equivalent to what rate

compounded weekly?

compounding over several years at non-annual compounding

frequencies: FVT = C0 × (1+SAR/m)m×T

FV of $100 invested for 6 years at 5% compounded

monthly:

continuous compounding over many years: FVT = C0 × eSAR×T

C7 = $1,000, what is PV today if interest rate is 5.5%compounded continuously?


Multi-Period Case - Multiple Cash Flows

time line:

t = 0

C0

t = 1

C1

t = 2

C2

t = 3

C3

. . .

. . .

t = k

Ck

. . .

. . .

t = n

Cn

t = k implies end of year k, beginning of year k +1

PV = C0 + C11+r + C2

(1+r)2 + C3(1+r)3 + · · ·+ Cn

(1+r)n

e.g. if r = 6%, calculate PV of the following cash flows:

t = 0

-$100

t = 1

$150

t = 2

-$80

t = 3

$300


Cont’d

Summarizing to here:

simple interest: FV = C0 × (1+n× r)

discrete compounding/discounting:

FVT = C0 ×

(

1+SAR

m

)mT

PV0 =CT

(1+ SAR

m

)mT

= C0 × (1+ r)n =CT

(1+ r)n

continuous compounding/discounting: FVT = C0 × erT ,

PV0 = CT × e−rT

with multiple CFs, treat each one separately and add (as in the

example on slide 15). There are no convenient formulas if cash

flows vary in general, but there are for “nice” cash flows.AFM 271 - Time Value of Money Slide 16

Perpetuities

a perpetuity is a stream of equal cash flows that occurevery period forever, 1st payment one period from now:

t = 0

C0 = 0

t = 1

C1 = C

t = 2

C2 = C

t = 3

C3 = C

t = 4

C4 = C

t = 5

C5 = C

PV of a perpetuity is

PV =C

(1+ r)+

C(1+ r)2 +

C(1+ r)3 + · · ·

=Cr

e.g. if r = 3%, find PV of a perpetuity paying $200/year:


Cont’d

a growing perpetuity is a stream of cash flows thatoccurs every period forever, has 1st payment oneperiod from now, and grows at a rate of g per period:

0

0

1

C

2

C(1+g)

3

C(1+g)2

4

C(1+g)3

PV of a growing perpetuity is

PV =C

(1+ r)+

C(1+g)

(1+ r)2 +C(1+g)2

(1+ r)3 +C(1+g)3

(1+ r)4 + · · ·

=C

(r−g)


Cont’d

the formula above is valid only if g < r

e.g. you wish to purchase a preferred share of ABC Co.The share is expected to pay a dividend of $2.50 nextyear, thereafter growing at a rate of 3%. Assume aninterest rate of 8%. How much should you be preparedto pay for the share? (Assume that you will neverreceive your initial capital back.)

note that we could have g < 0 (a decreasing perpetuity)- reconsider the example above but assume dividendswill decline at a rate of 2% per year forever:


Annuities

ordinary annuity (a.k.a. annuity in arrears): a stream ofequal cash flows that occur every period, for a specifiednumber of periods, at the end of each period. Examplesinclude car leases, mortgages, pensions, etc. Our PVformula below (slide 22) assumes that the 1st cash flowis one period from now.

e.g. find PV of an annuity paying $1,000 per year for 4years, r = 7%:

0

0

1

1,000

2

1,000

3

1,000

4

1,000

5

0


Cont’d

e.g. find FV at the end of year 4 of an annuity paying$1,000 per year for 4 years, r = 7%:

0

0

1

1,000

2

1,000

3

1,000

4

1,000

5

0


Cont’d

a simplifying formula for the PV of an annuity - find aformula for the PV of an ordinary annuity paying C for atotal of n periods, assuming an interest rate of r perperiod:

PV of annuity = C×1− (1+ r)−n

rAFM 271 - Time Value of Money Slide 22

Cont’d

a simplifying formula for the FV of an annuity - find aformula for the FV (at the end of n periods) of anordinary annuity paying C for a total of n periods,assuming an interest rate of r per period:

FV of annuity after n periods = C×(1+ r)n

−1r

exercise: use the formulas on slides 22 and 23 to verifythe calculations on slides 20 and 21


Cont’d

note that our PV formulas for annuities and perpetuitiesall assume that the 1st payment is one period from nowand that the rate of interest per period is r

ordinary annuity examples:you have an outstanding balance on your Visaaccount of $2,400. If you can only afford to repay atthe rate of $150 per month, how long will it take youto repay the entire amount if the interest rate on theoutstanding balance is 24% compounded monthly?


Cont’d

redo the previous example, but assume that theinterest rate is 24% compounded semi-annually:

you are 25 years old and wish to have savings of$200,000 by the time you retire on your 55thbirthday. You are willing to put money aside for next20 years. Assuming an interest rate of 6% until yourretirement, how much must you set each year?


Cont’d

annuity due (a.k.a. annuity in advance): payments areat the start of each period

PV annuity due = (1+ r)× PV ordinary annuityPV(FV) of annuity due > PV(FV) of ordinary annuitye.g. Sue has won a lottery, which pays $25,000 peryear for 8 years, starting today. Calculate PV todayand FV (in 8 years) of the lottery payout (assumer = 5%).


Cont’d

delayed annuity: payments start after a delay by acertain number of periods

this involves a two step calculation; e.g. Bill willgraduate in 4 years, and will thereafter earn anannual income of $75,000 for 35 years (assume allincome is received at end of a year). Calculate thePV of Bill’s lifetime earnings (assume r = 4%).


Cont’d

infrequent annuity: payments occur less often thanonce per year

this also involves a two step calculation; e.g. youplan to purchase a new $35,000 car every 5 yearsfor next 30 years starting in 5 years. Calculate thePV of your car purchases (assume r = 6.5% perannum).


Cont’d

growing annuity: like an ordinary annuity, but payments grow at

a rate of g per period

0

0

1

C

2

C(1+g)

3

C(1+g)2

n

C(1+g)n−1

n+1

0

PV = C×

1−

[(1+g)(1+r)

]n

r−g

e.g. XYZ Fund will pay out distributions over next 10 years.

The first distribution, $80 per unit, will be paid out a year

from today. Subsequent distributions will grow at a rate of

8%. No further cash flow is expected once the ten

distributions have been paid out. Find the PV of one fund

unit (assume r = 11%).AFM 271 - Time Value of Money Slide 29

Mortgages

many varieties, but these are basically annuities with monthly

payments, semi-annual compounding, a 25 year maturity

(usually), and a term (typically 5 years) less than the maturity

e.g. find the monthly payment on a $225,000 mortgage,

assuming the interest rate on the initial 5 year term is 6%

find the monthly payment after the initial 5 year term, if the

interest rate changes to 8%


Amortization Schedules

applicable to loans being paid off over a number of periods,

with constant monthly payments (e.g. mortgages, leases)

the loan is said to be “amortized” over the loan period

does not apply to debt where only interest is paid

periodically, with principal repaid as a lump sum at maturity

(e.g. corporate bonds)

see spreadsheet handout (try to recreate it yourself)

note that as time passes:

principal balance decreases (eventually to zero when the

last payment is made)

interest portion of each payment becomes smaller

principal portion of each payment becomes larger


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