Introduction to the Time Value of Money Lecture Outline

Introduction to the Time Value of MoneyLecture Outline

I. Why is there the concept of time value?

II. Single cash flows over multiple periods

III. Groups of cash flowsIV. Warnings on doing time value

calculations – the “tricks”

I. Why is there a time value to money (TVM)?

Which would you prefer: a cheque from me to you for $1,000,000

dated today or a cheque from me to you for

$1,000,000 dated one year from now Why?

What does the TVM mean?

Time Value Calculations Basics so far:

Single cash flow over one period

Objective: Extend to multiple,

or “n”, periods

)1(1

0 r

CPV

)1(01 rCFV

II. The Basic Time-Value-of-Money Relationships for a Single Cash Flow

FVt+n = Ct (1 + r)n

where r is the interest rate per period n is the duration of the investment, stated in the

time units of the period for r Ct is the cash flow at period t Ct+n is the cash flow at period t+n FVt+n is the future value at time period t+n PVt is the present value at time period t

nnt

t r)(1

CPV

Intuition for the Future Value formula

Future Value of $1 invested for 2 years at 8% per year compounded yearly

Year 0 1 2

FV2=C0(1+r)2 = $1 (1+.08)2 = $1.1664 Note: simple interest is just 16¢ over the two years but

compound interest is 16.64¢ which includes the simple interest plus interest on interest.

The Power of Compounding

Given an interest rate of 8% per year and an initial $1,000 investment, compare the compound interest in the 2nd year and the 20th year.

What is the total compound interest over the 20 year investment?

Explain the power of compounding

Present value of $1 received in 2 years given a discount rate of 8% per year.

Intuition for the Present Value Formula

Year 0 1 2

PV0=C2 ÷ (1+r)2 = $1 ÷ (1+.08)2 = $0.8573

Practice exercises Find the value in 3 years of $10,000

received in 8 years. Find the value in 10 years of $25,000

invested in 2 years. How many years was the money invested if

$10,000 grew to $100,000?

If you invested $500 for 7 years and now have $1000, how much did your investment return?

Housekeeping functions:1. Set to 8 decimal places:

2. Clear previous TVM data:

3. Set payment at Beginning/End of Period:

3. Set # of times interest is calculated (compounded) per year to 1:

TVM in your HP 10B Calculator

Yellow=

DISP 8

MARBEG/END

Yellow

Yellow

PMTP/YR

Yellow1

CC ALL

Always keep set on “End” – the display should never say “Begin”

Using the HP 10B TVM functions N = number of periods or number of

payments I/YR = the effective interest rate or discount

rate per period PV = present value or a present cash flow PMT = repeating cash flow payments (not

yet discussed) FV = future value or a future cash flow

III. Groups of Cash Flows Consider the following series of cash flows:

What is PV0; what is FV10? We could apply our PV and FV formulae to each

individual cash flow – but that would be too painful!

Year 0 1 2 3 . . . 10

$1,000 $1,050 $1,102.50 . . . $1,551.33

Mathematics of Perpetuities and Annuities

Fortunately, math provides a simplified way . . .

Consider a growing perpetuity:0 1 2 3 . . . t . . . ∞

$1,000C1

$1,050C2

=C1(1+g)1

$1,102.5C3

=C1(1+g)2

. . .Ct

= C1(1+g)t-1

. . .C∞

Goes on forever

Sum of an infinite series

PV0 of the growing perpetuity is mathematically equivalent to the sum of an infinite geometric series. The sum is defined and is finite as long as the PV of each subsequent cash flow is a fraction (less than 1) of the PV of the previous cash flow.

I.e., as long as r > g

PV of a growing perpetuity

gr

CPV 1

0

C1 is the first cash flow

PV0 is the PV one period before

the first cash flow

This formula is only correct when r > g.

If r = g or if r < g, then the PV is infinite.

This sums the PV’s of each individual cash flow in the growing perpetuity.

Growing Perpetuity – Example To service the current national debt, the

government plans to make the following series of payments beginning in one year and continuing in perpetuity: $8 billion initially and then growing by 4% each year. The interest rate on long-term debt is 6%. What is the PV0 of these payments?

Note: the PV0 of all future debt payments is equal to the principal amount currently outstanding.

PV of a Growing Annuity An annuity is a finite series of cash flows – i.e., a series

that has an end – assume the end as at time “n”. We can determine the PV of the growing annuity by

subtracting off the latter part from a growing perpetuity.

Cn

=C1(1+g)n-1

n

C2

=C1(1+g)1

2

. . .

. . .

C1

10

Cn+1

= C1(1+g)n

n+1

. . .

. . .

C∞

∞

PV of a Growing Annuity

… so the PV of the growing annuity is just the PV of the whole growing perpetuity minus the PV of the latter part of the growing perpetuity.

The latter part of the growing perpetuity is just another growing perpetuity that starts at a later time with a different initial cash flow.

PV of a Growing Annuity

n

n

rgr

gC

gr

CPV

)1(

1)1(110

nn

rgr

C

gr

CPV

)1(

1110

n

n

r

g

gr

CPV

)1(

)1(11

0

PV of the whole

growing perpetuity

Subtract off the PV of the latter part of

the growing perpetuity

PV0 is the PV one period before

the first cash flow

Growing Annuity – Example

In 20 years you plan on retiring and you would like income each year that grows at the expected inflation rate of 3%. You desire your year 20 income to be $105,000 and you expect to need 30 years of retirement income. If you are confident your savings will earn 8% per year, how much do you need saved by year 20?

FV of a Growing Annuity If PV0 discounts all the cash flows to time zero and sums

up the discounted amounts . . . then FVn, the future value of all the cash flows taken to

time n, . . . is just PV0(1+r)n

nn

n

n rr

g

gr

CFV )1(

)1(

)1(11

nnn gr

gr

CFV )1()1(1

In effect, this takes all cash flows of the growing annuity, including the last cash flow, forward to the

time period of the last cash flow

FV of Growing Annuity – Example

Given your retirement plans of the previous example, how much do you need to save each year beginning in one year and ending with year 19? Assume your savings will earn 8% and you increase your contributions by 6% each year.

Simple (non-growing) series of cash flows

For constant annuities and constant perpetuities, the time value formulas are simplified by setting g = 0.

r

CPV0 regular perpetuity

n0 r)(1

11

r

CPV

1r)(1r

CFV n

n

regular annuity

We can use the PMT button on the financial calculator for

the annuity cash flows, C

Regular Annuities & Perpetuities – Examples

Your father “loaned” you $20,000 as your down payment on your new house. If you repay him in equal amounts of $2,600 each of the next 10 years, what rate of interest are you, in effect, paying him?

You have won the lottery and are offered cash payments of $1 million per year for the next 20 years (first payment is one year from today). If you could invest at a rate of 10%, how much as a single lump sum would you be willing to receive today in exchange for the 20 yearly cash flows?

Regular Annuities & Perpetuities – Examples

You have just donated to the University of Manitoba and your donation stipulates that the University must spend the income earned from your donation each year. If your donation is $10 million and it earns a 6% rate of return, how much can be spent each year and for how long can this continue?

Regular Annuities & Perpetuities – Examples

IV. Some final warnings Even though the time value calculations

look easy there are many potential pitfalls you may experience

Be careful of the following: PV0 of annuities or perpetuities that do not begin

in period 1; remember the PV formulas given always discount to exactly one period before the first cash flow. If the cash flows begin at period t, then you

must divide the PV from our formula by (1+r)t-1 to get PV0.

Note: this works even if t is a fraction.

Be careful of annuity payments Count the number of payments in an annuity. If

the first payment is in period 1 and the last is in period 2, there are obviously 2 payments. How many payments are there if the 1st payment is in period 12 and the last payment is in period 21 (answer is 10 – use your fingers). How about if the 1st payment is now (period 0) and the last payment is in period 15 (answer is 16 payments).

If the first cash flow is at period t and the last cash flow is at period T, then there are T-t+1 cash flows in the annuity.

Be careful of wording A cash flow occurs at the

end of the third period. A cash flow occurs at

time period three. A cash flow occurs at the

beginning of the fourth period.

Each of the above statements refers to the same point in time!

0 1 2 3 4

C

If in doubt, draw a time line.