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Tutorial 2 : Time Value of Money

Conducted by : Mr. Chong Lock Kuah, CFA

1

Key Points

• The first step in time value analysis is to set up a time line, which will help you to visualize what is happening in a particular problem

CF2CF1 CF3

I/YR

0 1 2 3 Year

Cash flows

1. Time 0 is today; Time 1 is the end of period 1; or the beginningof period 2

2. Negative CF (-CF) are cash outflows

2

Key Points

• FV is the amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate

• PV is the value today of a future cash flow or series of cash flows

• Annual Percentage Rate (APR) is the contracted, quoted or statedinterest rate. It is equal to periodic interest rate times the number of period per year

• Effective Annual Rate (EAR or EFF) is the annual interest rate actually being earned (or paid), as opposed to the quoted rate. An EAR is the interest rate expressed as if it were compounded once per year

3

Key Points

� Solving time value of money problems using financial calculator TI II Plus is a preferred method

� Before using a financial calculator, make sure that the calculator is set up as follows:

� P/Y is set equal to 1, hit

� Make sure that the calculator is in “End Mode” (default setting)

� Set the calculator to display 8 decimal places, hit

� Clear all memory hit and hit

� Always make sure that the interest rate and the time period match. For example, when there are more than one periods in a year, use periodic interest rate (APR/m where m is the compounding frequency per year) and input N= number of periods per year x number of yrs

� For example, APR =10% and the compounding frequency is twice per year and you enter in the financial calculator as I/Y =5 and then enter N= 2x number of years

2ND

.

FORMAT

1

2ND 8

2ND FV

CLR TVM

I/Y

P/Y

2ND CE/C

CLR WORK

4

Key Points

• When calculating the PV of an ordinary annuity,

100100 100

10%

0 1 2 3 Year

Ordinary Annuity

Using a financial calculator input: N = 3, I/Y = 10, PMT =100, FV = 0, and then solve for PV = -$248.69

5

Key Points

• When calculating the PV of an annuity due,

100100

10%

0 1 2 3 Year

Annuity Due

Using a financial calculator

Hit and then enter :

N = 3, I/Y = 10, PMT =100, FV = 0, and then solve for PV = -$273.55

100

2ND PMT

BGN

2ND ENTER

SET

6

Key Points

• When calculating the PV of series of uneven cash flows,

We have to enter individual cash flows using key

10%

0 1 2 3 Year

0 100 300

4

300 -50

NPV = 530.09

NPV = -CF0 + PV(CF)

Since CF0 = 0, NPV=PV(CF)

CF

CF 0 ENTER

SET

100 300ENTER

SET

ENTER

SET

ENTER

SET

ENTER

SET

2 50 ENTER

SET

NPV calculator will display I=0.000000, hit

And then hit

10 ENTER

SET

CPT

QUIT

↓

INS

↓

INS

↓

INS

↓

INS

+/-

RESET

↓

INS

↓

INS

7

Key Points

• When calculating the FV of series of cash flows when payments occur annually, but compounding occurs each 6 months,

I/YR = 10%

0 1 2 3

100

4 5

100 100

6 period

Here we can’t use normal annuity valuation techniques. You can use the EAR and treat the cash flows as an ordinary annuity or use the periodic rate and compound the cash flows individually.)

APR=10%, to compute EAR, using financial calculator, hit

Finally, enter N=3, I/YR=10.25, P/Yr =1, PV=0, PMT=-100, to find FV=331.80

2ND 2

ICONV

calculator will display nom = 0.000000,

10 ENTER

SET

C/y 2 ENTER

SET

CPT

QUIT

EAR (or EFF) = 10.25

↓

INS

↓

INS

↓

INS

↓

INS

8

RWJ Chap 5 Q1

First City Bank pays 8 percent simple interest on its savings account balances, whereas Second City Bank pays 8 percent interest compounded annually. If you made a $5,000 deposit in each bank, how much more money would you earn from your Second City Bank account at the end of 10 years?

The simple interest per year is:$5,000 × 0.08 = $400So after 10 years you will have: $400 × 10 = $4,000 in interest.The total balance will be $5,000 + $4,000 = $9,000With compound interest we use the future value formula: FV = PV(1 +r)t

FV = $5,000(1.08)10 = $10,794.62The difference is:$10,794.62 − $9,000 = $1,794.62

RWJ Chap 5 Q15

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, Sotheby's sold the Edgar Degas bronze sculpture Petite Danseuse de Quartorze Ans at auction for a price of $10,311,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,377,500. What was his annual rate of return on this sculpture?

?%

1999 2000 2001 2002

-12,377,500

2003

10,311,500

4 12377500 PV

AMORT

0 PMT

BGN

10311500

CPT

QUIT

I/Y

P/Y

I/Y =-4.46%

N

xP/Y

+/-

RESET

FV

CLR TVM

10

RWJ Chap 5 Q15

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2003, Sotheby's sold the Edgar Degas bronze sculpture Petite Danseuse de Quartorze Ans at auction for a price of $10,311,500. Unfortunately for the previous owner, he had purchased it in 1999 at a price of $12,377,500. What was his annual rate of return on this sculpture?

To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)t

Solving for r, we get:r = (FV / PV)1/t – 1r = ($10,311,500 / $12,377,500)1/4 – 1 = – 4.46%Notice that the interest rate is negative. This occurs when the FV is less than the PV.

RWJ Chap 6 Q24

You are to make monthly deposits of $300 into a retirement account that pays 10 percent interest compounded monthly. If your first deposit will be made one month from now, how large will your retirement account be in 30 years?

360 10/12 PV

AMORT

0 PMT

BGN

300

FV

CLR TVM

CPT

QUIT

FV= 678,146.38

N

xP/Y

I/Y

P/Y

+/-

RESET

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RWJ Chap 6 Q24

You are to make monthly deposits of $300 into a retirement account that pays 10 percent interest compounded monthly. If your first deposit will be made one month from now, how large will your retirement account be in 30 years?

This problem requires us to find the FVA. The equation to find the FVA is:FVA = C{[(1 + r)t – 1]/r}FVA = $300[{[1 + (0.10/12)]360 – 1}/(0.10/12)] = $678,146.38

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RWJ Chap 6 Q36

You've just joined the investment banking firm of Dewey, Cheatum, and Howe. They've offered you two different salary arrangements. You can have $95,000 per year for the next two years, or you can have $70,000 per year for the next two years, along with a $45,000 signing bonus today. The bonus is paid immediately, and the salary is paid at the end of each year. If the interest rate is 10 percent compounded monthly, which do you prefer?

Explanation:

Since we have an APR compounded monthly and an annual payment, we must first convert the interest rate to an EAR so that the compounding period is the same as the cash flows.

2ND 2

ICONV

10 ENTER

SET

C/y 12 ENTER

SET

CPT

QUIT

EAR (or EFF) = 10.4713

↓

INS

↓

INS

↓

INS

↓

INS

The present value of the first arrangement = 163,839.10

2 10.4713 0PMT

BGN

95000 FV

CLR TVM

CPT

QUIT

PV= -163,839.10PV

AMORT

2 10.4713 0PMT

BGN

70000 FV

CLR TVM

CPT

QUIT

PV= -120,723.549PV

AMORT

The present value of the second arrangement =$45,000 + 120,723.549 = 165,723.549

N

xP/Y

N

xP/Y

I/Y

P/Y

I/Y

P/Y

15

RWJ Chap 6 Q36

You've just joined the investment banking firm of Dewey, Cheatum, and Howe. They've offered you two different salary arrangements. You can have $95,000 per year for the next two years, or you can have $70,000 per year for the next two years, along with a $45,000 signing bonus today. The bonus is paid immediately, and the salary is paid at the end of each year. If the interest rate is 10 percent compounded monthly, which do you prefer?

Explanation:

Since we have an APR compounded monthly and an annual payment, we must first convert the interest rate to an EAR so that the compounding period is the same as the cash flows.EAR = [1 + (0.10 / 12)]12 − 1 = .104713 or 10.4713%

PVA1 = $95,000 {[1 − (1 / 1.104713)2] / 0.104713} = $163,839.09

PVA2 = $45,000 + $70,000{[1 − (1/1.104713)2] / 0.104713} = $165,723.54

RWJ Chap 6 Q40

You're prepared to make monthly payments of $340, beginning at the end of this month, into an account that pays 6 percent interest compounded monthly. How many payments will you have made when your account balance reaches $20,000?

Explanation:Here we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the number of payments.

6/12 PMT

BGN

340 FV

CLR TVM

CPT

QUIT

N= 51.69

N

xP/Y

I/Y

P/Y

0 PV

AMORT

20000+/-

RESET

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RWJ Chap 6 Q40

You're prepared to make monthly payments of $340, beginning at the end of this month, into an account that pays 6 percent interest compounded monthly. How many payments will you have made when your account balance reaches $20,000?

Explanation:Here we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the number of payments.

Using the FVA equation:

FVA = $20,000 = $340[{[1 + (.06/12)]t − 1 } / (.06/12)]Solving for t, we get:1.005t = 1 + [($20,000)/($340)](.06/12)

t = ln 1.294118 / ln 1.005 = 51.69 payments

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RWJ Chap 6 Q42

You need a 30-year, fixed-rate mortgage to buy a new home for $240,000. Your mortgage bank will lend you the money at a 6.35 percent APR for this 360-month loan. However, you can afford monthly payments of only $1,150, so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment. How large will this balloon payment have to be for you to keep your monthly payments at $1,150?

360 6.35/12 1150 PMT

BGN

240000FV

CLR TVM

CPT

QUIT

FV= -368,936.54

PV

AMORT

N

xP/Y

I/Y

P/Y

+/-

RESET

19

RWJ Chap 6 Q42

You need a 30-year, fixed-rate mortgage to buy a new home for $240,000. Your mortgage bank will lend you the money at a 6.35 percent APR for this 360-month loan. However, you can afford monthly payments of only $1,150, so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment. How large will this balloon payment have to be for you to keep your monthly payments at $1,150?

Explanation:

The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present value of the $1,150 monthly payments is:

PVA = $1,150[(1 − {1 / [1 + (.0635/12)]360}) / (.0635/12)] = $184,817.42

20

Cont’d

The monthly payments of $1,150 will amount to a principal payment of $184,817.42. The amount of principal you will still owe is:$240,000 − 184,817.42 = $55,182.58

This remaining principal amount will increase at the interest rate on the loan until the end of the loan period. So the balloon payment in 30 years, which is the FV of the remaining principal will be:

Balloon payment = $55,182.58 [1 + (0.0635/12)]360 = $368,936.54

21

RWJ Chap 6 Q43

The present value of the following cash flow stream is $6,550 when discounted at 10 percent annually. What is the value of the missing cash flow?

Year Cash Flow1 $1,700 2 ? 3 $2,100 4 $2,800

We have to enter individual cash flows using key to find NPV

NPV = 5,035.65

CF

CF 0 ENTER

SET

1700 0ENTER

SET

ENTER

SET

ENTER

SET

2100 ENTER

SET

NPV calculator will display I=0.000000, hit

And then hit

10 ENTER

SET

CPT

QUIT

↓

INS

↓

INS

↓

INS

↓

INS

↓

INS

↓

INS

ENTER

SET

↓

INS

↓

INS

2800 ENTER

SET

↓

INS

ENTER

SET

Given that the present value of cash stream is $6,550, the present value of year 2 cash flow = $6,550 – 5035.65=$1,514.35

The value of the missing CF is:$1,514.35(1.10)2 = $1,832.36

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RWJ Chap 6 Q43

The present value of the following cash flow stream is $6,550 when discounted at 10 percent annually. What is the value of the missing cash flow?

Year Cash Flow1 $1,700 2 ? 3 $2,100 4 $2,800

Explanation:We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are:

24

Cont’d

PV of Year 3 CF: $2,100 / 1.103 = $1,577.76

PV of Year 4 CF: $2,800 / 1.104 = $1,912.44

So, the PV of the missing CF is:$6,550 − 1,545.45 − 1,577.76 − 1,912.44 = $1,514.35

The question asks for the value of the cash flow in Year 2, so we must find the future value of this amount.

The value of the missing CF is:$1,514.35(1.10)2 = $1,832.36

25

RWJ Chap 6 Q53

Suppose you are going to receive $10,000 per year for five years. The appropriate interest rate is 11 percent.

a. What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due?

b. Suppose you plan to invest the payments for five years. What isthe future value if the payments are an ordinary annuity? What if the payments are an annuity due?

c. Which has the highest present value, the ordinary annuity or annuity due? Which has the highest future value? Will this always be true?

a(i).

b(ii).

5 11 0PMT

BGN

10000 FV

CLR TVM

CPT

QUIT

PV= -36,958.97

PV

AMORT

N

xP/Y

I/Y

P/Y

2ND PMT

BGN

2ND ENTER

SET

5 11 0PMT

BGN

10000 FV

CLR TVM

CPT

QUIT

PV= -41,024.46

PV

AMORT

N

xP/Y

I/Y

P/Y

b(i).

5 11 0PMT

BGN

10000FV

CLR TVM

CPT

QUIT

FV= 62,278.01

PV

AMORT

N

xP/Y

I/Y

P/Y

+/-

RESET

2ND PMT

BGN

2ND ENTER

SET

5 11 0PMT

BGN

10000FV

CLR TVM

CPT

QUIT

FV= 69,128.60

PV

AMORT

N

xP/Y

I/Y

P/Y

+/-

RESET

b(ii).

c.

Annuity due will have the highest PV and FV.

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RWJ Chap 6 Q53

Suppose you are going to receive $10,000 per year for five years. The appropriate interest rate is 11 percent.

a. What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due?

b. Suppose you plan to invest the payments for five years. What isthe future value if the payments are an ordinary annuity? What if the payments are an annuity due?

c. Which has the highest present value, the ordinary annuity or annuity due? Which has the highest future value? Will this always be true?

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Explanation:a(i) If the payments are in the form of an ordinary annuity, the present

value will be:PVA = C({ 1 − [ 1/(1 + r)t ] } / r ))PVA = $10,000[ { 1 − [1 / (1 + 0.11)5] }/ 0.11 ]PVA = $36,958.97

a(ii) If the payments are an annuity due, the present value will be:PVAdue = (1 + r) PVAdue = (1 + 0.11)$36,958.97PVAdue = $41,024.46

b(i) We can find the future value of the ordinary annuity as:FVA = C{[(1 + r)t − 1] / r}FVA = $10,000{[(1 + 0.11)5 − 1] / 0.11}FVA = $62,278.01

30

Cont’d

b(ii) If the payments are an annuity due, the future value will be:FVAdue = (1 + r) FVAdue = (1 + 0.11)$62,278.01FVAdue = $69,128.60

c.

Annuity due will have the highest PV and FV provided r is positive.