Finance 3310

Lecture Notes

Lecture 5

I. Time Line and definitions

II. Rate of return

III. Future Value

IV. Present Value

V. Perpetuities

VI. Annuities:

a) present value

• ordinary

• annuity due

b)future value

• ordinary

• annuity due

VII. Nominal rate, Effective rates and Compounding

Time Line and Definitions

Consider the following time line:

------------cf1------- cf2 -------- cf3 ------- cf4 ....... cfn

|________|________|________|________|________|

0 ----------1---------- 2---------- 3 ----------4 --..... --- n

PV --------------------------------------------------------FV

* The numbers represent the end of a period. eg, 3 = end of period 3; period 3 begins at 2

* It is important to define where t=0 and t=n are



Definition of terms:

PV (P0) = present value of cash flows = value at t=0

FV (FVn) = future value of cash flows = value of cash flows at t=n

n = number of periods

R = interest rate per period (usually stated as annual rate)

• Daily = R/365

• Monthly = R/12

• Semi-annually = R/2, etc.

CF = cash flow per period; also PMT or PYMT

Rates of Return - Intro

Rate of return = growth rate

Growth rate = Value / Beginning value

Ex. Inflation = Price level / Beginning price level = % change in prices

Growth rate of GDP = GDP / Begin. GDP

Rate of return = growth rate of money

 Value = ending value - beginning value = FV - PV

Rate of return =  money / Beginning money

• R = (FV - PV) / PV

• R = FV / PV - 1

• 1 + R = FV / PV (one period)

1 + r = ( n periods)

Would you rather have $100 today, or $100 one year from now? Almost everyone would rather have the $100 today. If asked 'why' a typical response might be that if you had the $100 today you could invest it now and end up with more than $100 in one year. This simple idea is the foundation of the time value of money.

As the title time value of money implies, a dollar today is different (has a different value) than a dollar somewhere else in time. The Cardinal Sin of finance is to add or subtract or compare cash flows occuring at different points in time. No one would add apples to oranges or dollars to yen, nor should they add dollars now to dollars one year from now. We first 'translate' future dollars to current dollars, or current dollars to future dollars, and then we can add or subtract or compare them. Future value involves 'translating' current dollars into future dollars.

Future Value

Suppose you invest $100 today, and r = 5%. How much will you have in 1 year?

0 ------------------ 1

|______________|

PV = $100 ---------------- FV= ?

Most people know that you will have $105, representing the $100 you start with plus $5 of interest. In symbols we can write:

FV = 105 = 100 + 5

FV = PV + R * PV

FV = PV * (1+R) (recall FV/PV = (1+r))

Now, suppose we want to know FV2?

|__________|__________|

PV -------- FV1 --------- FV2

$100 ----- $105 ----------- ?

----------PV(1+r) --------- ?

Same as: |____________| (same as one period problem)

------- $105--------------- ? (= FV₂)

FV2 = 105 * (1+R) = $110.25

In symbols:

• FV2 = FV1 * (1+R)

• FV2 = PV * (1+R) *(1+R)

• FV2 = PV * (1+R)²

Now consider n periods:

FVn = PV * (1+r)ⁿ

The term (1+r)ⁿ is a future value factor. It translates current dollars into equivalent future dollars. This process is called compounding.

Note what happens if we solve for r:

(1+R)ⁿ =

(1 + R) =

Ex 1: What is the future value of $4000 invested at 6% for two periods?

PV -------------------------------- FV2

|____________|______________|

0 --------------- 1------------------- 2

FV2 = PV * (1+R)2

FV2 = $4,000 * (1+6%)2

FV2 = $4,000 * 1.1236 = $4,494.40

Ex 2: What is the future value of $3,000 invested at 7.5% for 25 years?

PV ------------------------------------------------ FV25

|_________|_________|_________|_________|

0 ---------- 1------------ 2 ----------- 3 --.......-- 25

FV25 = PV * (1+R)²⁵

FV25 = $3,000 * (1+7.5%)²⁵

FV25 = $3,000 * 6.09834 = $18,295.02

Ex 3: What is the value at t=5 of $6,000 invested at t=3 for 7%?

PV------------------------- FV

|_________|_________|_________|_________|_________|

0----------- 1 ----------- 2 ----------- 3 ----------- 4 ----------- 5

FV = PV * (1+7%)2

FV = $6,000 * 1.1449 = $6,869.40

CALCULATOR: Your calculator has a Y^x function. In our context, Y = (1+r) and x = n, the number of periods. You can enter the following keystrokes to solve the above problem:

You Input Calculator displays

1.07, Y^x , 2 , = 1.1449

x , 6000 , = 6869.40

Or, if you want to kill an ant with a sledgehammer, you can use your time value of money functions as follows:

2, n -----N = 2.00

7, I/Y -----I/Y = 7.00

-6000, PV ----- PV = 6,000

CPT, FV----- FV = 6869.4

Notice that PV is entered with a minus sign. Your calculator assumes initial cash flows are outflows and later cash flows are inflows.

Present Value

Suppose you have $105 one year from now and you want to know what that would be worth today if r = 5%.

0 ------------------ 1

|______________|

PV--------------$105

FV = $105

R = 5%

PV = ?

Using the formula for FV:

FV = PV * (1+r)

$105 = PV * (1+5%) = PV * 1.05

PV = $105/1.05 = $100

In other words, take our formula for FV and simply solve it for PV.

PV = FV / (1+r)

PV = FV *

or, PV = FV * (1+r)-1

Two periods:

PV ------------ FV1 ------------ $110.25 = FV2

|____________|______________|

0 --------------- 1------------------- 2

Since we only know how to calculate FV for one period we will break the problem into two one-period problems.

FV2 = FV1 * (1+R)

but, FV1 = PV * (1+R)

so, FV2 = PV * (1+R)*(1+R)

FV2 = PV * (1+R)² ; now, solve for PV:

PV = FV2 / (1+R)² = FV2 * 1/(1+R)²

PV = $110.25 / (1+R)² = $110.25/1.1025

PV = $100

For n periods:

PV = FV * or, PV = FV * (1+R)^-n

The expression is the present value factor, or discount factor. It translates future dollars into equivalent present dollars. This process is called discounting.

ex 1: What is the present value of $4,494.40 received two years from now if the discount rate (r) is 6%?

PV = FV * (1+R)^-2 = FV * 1/(1+R)²

PV = FV * (1+ .06)^-2

PV = $4,494.40 * .889996 = $4,000

Ex 2: What is the value today of $100,000 to be received 20 years from now if r = 10%?

PV = FV * (1+R)-20

PV = 100,000 x (1.1)-20

PV = 100,000 x .14864 = $14,864

Would you rather have $100,000 in 20 years, or $14,864 today?

Perpetuities

A perpetuity is a stream of consecutive, equal cash flows received for a perpetual (infinite) number of periods.

Math fact: suppose |x| < 1; then: =

Recall, = x⁰ + x¹ + x² + x³ + x⁴ + ...

Present values (& future values) are additive ; in other words, we can add apples to apples, yen to yen, etc.

Present value of perpetuity:

---------- cf ------- cf ------- cf ------- cf -------- cf-------- cf

|_______|_______|_______|_______|_______|_______|

0 ------- 1 -------- 2 -------- 3 -------- 4 --------- 5 ..... ---n

(Although the following is just algebra, the student may skip down to 'Present Value of Perpetuity' formula.)

We can find the present value by adding the present value of each cash flow, one at a time.

PV = SUMMATION (CF* (1/(1+R)t

where t = 1 to infinity

But, 1/(1+R) < 1, so let x=1/(1+R)

PV = SUMMATION (CF* x^t )

where t = 1 to infinity

Note that this looks very similar to our math fact, except that our math fact starts counting with t=0.

So add CF*x⁰ = CF to both sides

PV + CF = SUMMATION (CF* x^t) ; Now, using our math fact,

PV + CF = CF * 1/(1-x)

PV = CF [( 1/(1-x)) - 1]

PV = CF [x/(1-x)]

Now, just substitute 1/(1+R) for x;

PV

PV

Students may skip to here without loss of continuity.

PV = = Present value of Perpetuity

Example 1: What is the present value of a $150 periodic payment, received forever if the discount rate is 7%?

PV = $150/.07 = $ 2,142.86


Example 2: Suppose, because of your sincere love of UALR and this class, you wanted to establish an endowment that would generate a $300 scholarship per year for lucky future students. What would the endowment have to be if R=6%?

PV = $300/.06 = $5,000

ANNUITY

An annuity is a series of constant payments made at fixed intervals for a specified number of periods.

------------- cf ---------- cf --------- cf -------- cf ......... cf

|_________|_________|________|________|________|

0 ...........1 ............2 ..........3 ..........4 ......... n

PV ..........................................................FV

Ordinary: Payments made at the end of the period (the above depicts an ordinary annuity)

Annuity due: Payments made at beginning of period (move each cf one to the left above)

Conceptually, we find PV of each CF, and then add.

EX: Suppose R = 7%; What is PV of the following?

.............1,000 ........1,000 ........1,000

|__________|__________|__________|

0 ............1 ..............2 .............3

PV = 1000/(1.07) + 1000/(1.07)² + 1000/(1.07)³ =

934.58 + 873.44 + 816.30 = 2,624.32

Consider the following annuity:

............cf........... cf........... cf ..........cf

|________|_________|_________|_________|

0 ..........1 ...........2 ............3 ........... n

Note that we can write an annuity as the difference between two perpetuities as follows:

1 ---|-----|------|-----|-----|-----|-----|------------>
0.....................n

2.................... |-----|-----|-----|------------>

So, we just subtract the second perpetuity, 2, from the first, 1. But PV2 occurs at point 'n' in time, while PV1 is now.

The present value of each perpetuity is CF/R

CF/R ......................................CF/R

|________|________|________|________|

0......... 1 ...........2 ..........3 ......... n

so, PV(annuity) = CF - PV(CF/R)

PV(annuity) =

We can rewrite this as:

PV(annuity) =

Previous example: What is PV of 1,000 received at end of year for 3 years?

PV = 1,000 x {1 - 1/(1.07)³} / .07

= 1,000 x {1 - 1/1.22504} / .07

= 1,000 x {1 - .8163} / .07

= 1,000 x .18137 / .07 = 1,000 x 2.62432

= $2,624.32

Calculator:

Pmt = 1,000

R = 7

n = 3

PV = ?

Example 1: What is the present value of $4,000 received at the end of every period for 10 periods, if R = 7%?

PV = $4,000 * [1 - 1/(1.07)¹⁰] /.07

PV = $4,000 * 7.024 = $28,096

Calculator: pymt = 4000, n=10, R=7, PV=? (28,094.33)

How do we interpret the number 7.024?

Example 2: What is the present value of a series of $5,000 payments, received starting at the end of year 6 and continuing for 10 years, if r=8%?

0 ... ......5......... 6 ..........7 ..........8 ............9 .... ......15

|________|_______|________|________|_________|_________|

PV=? ............5,000 ......5,000 ....5,000 .......5,000 ..... 5,000

We want value at t=0 of annuity from 6 to 15

1) First find PV at t=5 (or of annuity due at t=6)

PV(5,000, 10 periods,8%)

= $5,000 * [1- 1/(1+.08)10] /.08

= $5,000 * 6.71 = $33,550.41

2) Then, discount back to t=0.

PV(t=0) = $33,550.41 * 1/(1+8%)⁵

= $33,550.41 * .6806 = $22,833.85

How do we interpret the 6.71 above?

How do we interpret the .6806 above?

Annuity due: Payments occur at the beginning of the period. The same number of cash flows are received, but each is received one period earlier. Visually, everything is shifted to the left one period

cf ........ cf........... cf........... cf........... cf

|________|_________|_________|_________|_________|

0......... 1............ 2............ 3........... n-1.......... n

Thus, we could invest each cash flow for one extra period at R ===>

PV (annuity due) = PV(ordinary) x (1+R)

PV(annuity due) = CF * [1 - 1/(1+R)ⁿ] / R * (1+R)

Or,

PV(annuity due) = PV(ordinary w/ n-1 cf's) + CF

Your calculator does not know where you are on the number line. You simply tell it how many consecutive cash flows. It returns an answer that is at the beginning of the period relative to the first cash flow.

..........cf.......... cf........... cf........... cf.......... cf

|_______|________|_________|_________|_________|_________|

-1....... 0.......... 1............ 2............ 3...... .... n-1.......... n

If you told your calculator to find the present value of n consecutive cf's, it would return an answer that is valid for t = -1 on the above number line. To move from -1 to 0, compound for one year ==> multiply by (1+R).

Future Value of Annuity

Consider the same annuity; cf every period for n periods

............cf........... cf........... cf ..........cf

|________|_________|_________|_________|

0 ..........1 ...........2 ............3 ........... n
PV ..................................................FV

Conceptually, we can find FV of each cf and then add.

But we know how to find the present value, PV(annuity), thus, all we need to do is translate the PV to FV

We know how to find FV; -------> FV = PV(1+R)ⁿ

Thus: FV(annuity) = PV(annuity) * (1+R)ⁿ

FV(annuity) = cf*[1/R - 1/R(1+R)ⁿ ] * (1+R)ⁿ

FV(annuity) = cf*[(1+R)ⁿ - 1] / R

Example 1: Thrifty Kim deposits $1,200 in savings at the end of every year. If she expects to earn 10%, how much will she have in 15 years.

This is FV of annuity for 15 years at R=10%.

FV(1,200, 15,10%) = $1,200 * [(1+.10)15 - 1]/.10

= $1,200 * [4.1772 - 1]/.10

= $1,200 * 3.1772/.10

= $1,200 * 31.7725 = $38,126.98

Or: Calculator:

Pmt = 1,200

R = 10

n = 15

FV = ?

Ex 1: suppose your intend to work for 20 more years and then retire. You will deposit a sum at the end of every year for 20 years which will earn 10%. At the beginning of the 21st year, (end of 20th year), you will take out $15,000 per year for 10 years. During retirement the money will only earn 7%. How much do you need to deposit for the first 20 years?

|_____|_____|_____|_____|_____|_____|_____|_____|_____|_____|

0 .....1 .....2 .......3 .....4. --. 20 .....21 ....22 ....23 ....24 .--. 29

In effect, two annuities:

15 ....15 ....15.....15 ....15.... 15

|_____|_____|_____|_____|_____|

20 ...21 ....22 ....23 ....24 .--. 29

pv(annuity due)

PV(annuity due) = $15,000 * [1/.07 - 1/.07(1+.07)10] * 1.07

= $15,000 * [14.286 - 7.262]*1.07

= $15,000 * 7.5155 = $112,733.07

====> $112,733.07 = value at t=20

................................................$112,733.07

|_______|_______|_______|_______|_______|

0 ........1 .........2 ........3 .........4 ...--... 20

?........ ? .........? .........? .........?..........?

What is payment amount? R=10%, n=20, FV = $112,733

payment = $1,968.28 (check)

Other time value of money problems:

a)Solving for n or R:

Calculator: Just plug in what you know and solve for what you don't know.

b) Uneven series of cash flows

Find PV (or FV) of each cf separately, then add.

Example: Suppose you discover a (strange) investment opportunity that promises the following cash flows. What would you be willing to pay for it if R=9%? (what is its pv?)

3,000 per year, for 10 yrs, starting at end of 4

6,000 at the end of yr 14 (t=14)

10,000 at end of yr 15 (t=15)

2,000 per yr for 5 yrs, starting at end of year 16

50,000 at end of year 21

We have two annuities and 3 simple present values. We need to add the pv of each group of cash flows.

...................3 .......3 ........3 .......6 ......10 .......2 .......2 ......50

|______|______|______|______|______|______|______|______|______|

0 ..... 3 ........4 .......5 ..--.. 13 .....14 ......15 ......16 ..--.. 20 ......21

pv(annuity) ==> 3,000 for 5 yrs at t=3

pv(single cf) ==> 6,000 in 14 yrs

pv(single cf) ==> 10,000 in 15 yrs

pv(annuity) == 2,000 for 5 yrs, at t=15

pv(simple) ==> 50,000 in 21 yrs

Time

i) pv(pmt =3,000,n=10, R=9) = $19,252.97 t=3

ii) pv(FV=19,252.97, n=3,R=9) = $14,866.83 t=0

iii) pv(FV=6,000, n=14,R=9) = $1,795.48 t=0

iv) pv(FV=10,000, n=15,R=9) = $2,745.38 t=0

v) pv(pmt=2,000, n=5, R=9) = $7,779.30 t=15

vi) pv(FV=7,779.30, n=15,r=9) = $2,135.71 t=0

vii) pv(FV=50,000, n=21,r=9) = $8,184.90 t=0

Now, add all values at t=0:

====> PV = $29,728.30

Compounding, Nominal & Effective rates

The nominal rate is the stated rate of interest, sometimes called annual percentage rate.(APR)

The effective rate is the actual rate, or rate that would produce the future value from the pv.

The two will differ whenever interest is paid (compounded) over a different period than the nominal rate pertains to.

example: Suppose the annual nominal rate is 10%, but interest is paid semi-annually; invest 1000.

Remember to match rate with period!!!

FV = $1,000 * (1+5%)² i.e., a period is half a year

FV = 1,000 * 1.1025

FV = 1,102.50

Effective rate is rate that causes pv to grow to fv.

so: (1+R) * PV = FV

(1+R) = FV/PV = 1,102.50/1,000 ===> R = 10.25%

Intuitively, the effective rate is higher, because when interest is paid more frequently, you earn more interest on interest. In the example you get 50 after 6 months, and then earn 5% x 50, or 2.50 more.

If you compound annually at the effective rate, you get same result as compounding semi-annually at nominal rate. (Think about it)

You can always use the following:

Just need: PV & FV

Example: Suppose you buy a 100m certificate of deposit that matures in 3 years and has a stated rate of 10%. Interest is compounded quarterly. What is the effective annual rate?

Now, we are compounding quarterly:

FV = (1+R)¹² * PV = (1+2.5%)¹² * 100,000

FV = 1.34489 * 100,000 = 134,489.00

Now, on your calculator: PV = 100,000; FV = 134,489; n = 3 ; and solve for R.

R = 10.38%

Example: Suppose you borrow 10,000 at 10%, which you will repay in one year. You have to pay a fee of $200. What is the effective rate on you loan?

1 + R = FV/PV = 1,100/9800 = 11.22%

Summary

* we are working with streams of cash flows!!!!!

* the only way to compare streams is to first translate all cash flows to PV or FV

* everything has to be at same point in time

* if a single cash flow amount, use simple pv or fv

* if a series of constant cash flows, then use annuity

* if you are comparing two streams, decide which of PV or FV makes most sense; if you already have one, use it

example - saving for college

|_______|_______|_______|_______|_______|_______|_______|

0 1 4 5 6 7 8 9

14 17 18

Suppose your newborn child will begin college at the beginning of her 18th year. Tuition, room & board currently cost $12,500 (pd at beg. of yr)

Assume:

• your child expects to take 4 years to complete college

• tuition costs are expected to grow at 5%

• expect to earn 7% after tax on investment

• suppose you already have $10,000 in bank

1) How much do you need in bank now, in total?

2) How much at beginning of 18th yr (t=17)?

3) How much per year do you need to save?

4) How much will college cost per year at 17,18,19,20?

t=17: tuition = 12,500 * (1+5%)¹⁷ = $28,650

t=18: tuition = 12,500 * (1+5%)¹⁸ = $30,083

t=19: tuition = 12,500 * (1+5%)¹⁹ = $31,587

t=20: tuition = 12,500 * (1+5%)²⁰ = $33,166

Now discount these to t=0:

pv1 = (1+7%)^-17 *28,650 = 9,070

pv2 = (1+7%)^-18 *30,083 = 8,900

pv2 = (1+7%)^-19 *31,587 = 8,734

pv4 = (1+7%)^-20 *33,166 = 8,571

Total value at t=0: $35,275

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