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Chapter 9 Mathematics of Finance

9-1 Time Value of Money Problems

- Whenever we take out loans, make investments, or

deal with money over time, common questions

arise. - How much will this be worth in 10 years?
- How will inflation eat into my retirement savings

over the next 30 years? - How much do I need to save each month to have

1,000,000 at age 60? - How much do I need to pay each month on a 30 year

mortgage of 400,000 if the interest rate is

5.25 - These are called Time Value of Money (TVM)

problems.

Common Notations

- P Present Value the current value of an

investment or loan or sum of money. - F Future Value The value of an investment,

loan or sum of money in the future. - r Annual Percentage Rate
- m Number of periods per year. Example If you

make monthly loan payments, them m 12

Common Notationscontd

- i Interest Rate per Period Example The annual

interest rate is 6 and there are 12 payments per

year. Then i .06/12 .005, or 0.5 per month. - i r/m
- t Time, measured in years
- n Total number of periods. Example You have a

10 year loan that is paid monthly. Then you have

n 1012120 total periods. n mt - R The amount of a Rent. This is the regular

payment made on a loan or into an investment.

Example

- Suppose you deposit 1000 into an account that

compounds interest quarterly. The annual rate of

interest is 2.3 and you are going to keep it in

the account for 4.5 years. At the end of this

time, the account will be worth 1108.72 - P 1000 F 1108.72 r .023
- m 4 i .023/4 .00575 t 4.5
- n 18

9-2 Percent Increase/Decrease

- If a quantity increases by some percent, we can

create a multiplier that helps us convert a

beginning value to an ending value. - To create the appropriate multiplier
- Percent Increase 1 i
- Percent Decrease 1 - i

Example

- This year, the SCCC student population is 11,350.
- The administration estimates that will increase

by 2 next year. How many students can we expect

next year? - The multiplier 10.02 1.02
- New student population 11,350(1.02) 11577

Example

- The current balance of my retirement account is

244,350. - If the value of the account drops by 5.2 over

the next year, what will be the new value? - Multiplier 1 - .052 .948
- New Value 244,350(.948) 231,643

9-3 Compound Interest

- When we invest money, interest may be applied to

the account on a regular basis. For example, if

we invest in an account that pays interest

monthly, we say the interest compounds monthly.

Anything in the account at the time of

compounding gets interest added to it. In the

monthly case, we have 12 compoundings per year,

with each compounding representing 1/12 of the

total annual interest rate.

Example

- We invest 1000 in an account that compounds

monthly. The annual interest rate is 3.6. If we

keep it in the account for 5 years, adding or

removing nothing, how much will be in the account

at the end of 5 years? - First, we need to note that the interest per

period is i .036/12 .003. - Lets begin by building a table

Example

Period Previous Balance New Balance

0 0 1000

1 1000 1000(1.003)

2 1000(1.003) 1000(1.003)(1.003)

3 1000(1.003)(1.003) 1000(1.003) (1.003) (1.003)

Example

Period New Balance

0 1000 1000(1.003)0

1 1000(1.003) 1000(1.003)1

2 1000(1.003)(1.003) 1000(1.003)2

3 1000(1.003) (1.003) (1.003) 1000(1.003)3

Example

Period New Balance

3 1000(1.003) (1.003) (1.003) 1000(1.003)3

4 1000(1.003)4

5 1000(1.003)5

After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have

60 1000(1.003)60

After n total periods, we have After n total periods, we have After n total periods, we have

n 1000(1.003)n

Compound Interest Formula

- If P dollars earn an annual interest rate of i

per period for n periods, with no additional

principal added or removed, then the future

value (F) is given by - F P(1i)n

Example

- A bank account is opened with 4,000 in the

account. It earns 6 annual interest. If it earns

interest quarterly (four times per year), then

what is in the account after 10 years?

Example

- Suppose you invest 500 today at an annual rate

of 1.5, compounded daily. How long before the

balance doubles?

9-4 Rule of 72

- Given some investment that grows at an annual

interest rate, r, (not expressed as a decimal),

then the amount of time in years it takes for the

investment to double is approximately

Example

- How long will it take for an investment to double

if it earns 5 annual interest? - Note that estimating the doubling time does not

require that we know how much is originally

invested!

Example

- If you want your investment to double in 30

months, what annual interest rate do you need to

secure?

9-5 Yield

- Because each compounding acts on the original

balance and any interest that has been previously

earned, the net interest earned will not be the

same as the annual interest rate at the end of

the investment. The true interest rate earned

is called the Yield.

Example

- Invest 100 for 1 years, compounded monthly at an

annual rate of 12. - F 100(1.01)12 112.68
- This represents a yield of 12.68, which is

higher than the original 12 stated above. The

yield is often called the Annual Percentage Yield

(APY). Always ask what this is when you take out

a loantime, compounding and bank fees can

substantially increase your rate of interest and

therefore your total payments due! - APR 12
- APY 12.68

Yield Formula

- The formula for yield in the t 1 year case is

Where did the formula come from?

Why is it important to know the APY?

- The APY, or yield, is helpful since it simplifies

calculations. If we know the APY, then it does

not matter how many times we compound per year

because the APR will give us actual percentage

increase at the end of a year. - APR Annual Percentage Rate or nominal rate or

rate - APY Annual Percentage Yield or effective rate

or yield

Example

- If 375 is invested with an APY of 5.22 for 8

years and 3 months, what is the Future value of

the investment?

9-6 Annuities

- When additional payments or deposits (rents) are

made at regular intervals into an investment,

then we call these annuities - Ordinary Annuity Payment is due at the END of

each period. - Annuity Due Payment is due at the START of each

period. - This will complicate our Future Value

calculations.

Example

- We invest in an account that compounds annually.

The annual interest rate is 3. If we add 800 at

the end of each year (an ordinary annuity), how

much will be in the account at the end of 5 years

total? - Lets look at a picture of what is going on here.

Example

Year 1

Year 2

Year 3

Year 4

Year 5

Start

Investment Period

800

800

800

800

800

Each of these 800 investments earns interest for

a different period of time. Hence, the value of

each of these deposits is different at the end of

the 5-year period.

Example

The End

End Year 1

End Year 2

End Year 3

End Year 4

End Year 5

Start

Investment Period

800

800

800

800

800

This one is worth 800(1.03)4 at the End 900.41

This one is worth 800(1.03)3 at the End 874.18

This one is worth 800 at the End 800

This one is worth 800(1.03)1 at the End 824

This one is worth 800(1.03)2 at the End 848.72

Example

- We can add all of these up
- 800(1.03)4 800(1.03)3 800(1.03)2

800(1.03)1 800 - 900.41 874.18 848.72 824 800
- 4247.31

A General Formula

- Now imagine if the monthly payments were

deposited and monthly interest credited. We would

then have 512 60 different deposits to find

the values for so we can add them up. - To avoid this inefficiency, we instead use the

following formula, which is equivalent to going

through that process.

A General Formula

- If R dollars are paid at the end of each period,

with an interest rate of I per period, then the

Future Value of the Annuity is

Where did the formula come from?

- We can generalize the example before and think

about adding R R(1i) R(1i)2 R(1i)3

R(1i)n-1. Let us call this sum S. - Hence, (1i)S - S R(1i)n - R
- S (1i - 1) R(1i)n - R
- Hence, the sum S is
- (R(1i)n - R)/i

Check

- We invest in an account that compounds annually.

The annual interest rate is 3. If we add 800 at

the end of each year (an ordinary annuity), how

much will be in the account at the end of 5 years

total?

Example

- What is the future value if you invest 95 per

month for 7 years at an annual rate of 3.75? - R 95
- i .0375/12
- n 127 84

Note that I try to keep as many decimal places as

possible until the end

FV for Annuities Due

- When payments or deposits are made at the

beginning of a period (rather than at the end as

in the previous examples), an adjustment is

needed. - We can view each payment as if it were made at

the end of the preceding period. This would

require one more payment (n1 total) than usual

and would require that we subtract the last

payment so we dont overpay.

FV for Annuities Due

Example

- If 300 payments are made at the beginning of the

month for 18 years (a college savings fund), what

is the Future Value if the annual interest rate

is 5.5

9-7 Future Value (FV) on Excel

- The FV command will do these computations for us

automatically. - Command Format
- FV(rate, nper, pmt, pv, type)

This is i, the rate per period

This is n, the total of periods

This is R, the amount of rent

This is P, the Present Value

Blank for ordinary annuity, 1 for annuity due

Excel Examples

- If 300 payments are made at the beginning of the

month for 18 years (a college savings fund), what

is the Future Value if the annual interest rate

is 5.5 - What is the future value if you invest 95 per

month (paid at the end of the month) for 7 years

at an annual rate of 3.75?

9-8 PV of Annuities

- Suppose you have an ordinary 20-year annuity that

you pay 500 into at the end of each quarter. The

annual interest rate is 7. - What is the lump-sum of money which should be

deposited at the start of the annuity that would

produce the exact same amount of money at the end

of the period, without any additional payments? - This is know as the Present Value of the Annuity

Present Value of an Ordinary Annuity

Example

- Suppose you set up an ordinary annuity account

which is to last 10 years and earn 4 annual

interest rate. If your rent payment is 150 per

month, how much do you need to deposit as a lump

sum up front to achieve the same end result

without any regular payments?

9-9 Present Value (PV) on Excel

- The PV command will do these computations for us

automatically. - Command Format
- PV(rate, nper, pmt, fv, type)

This is i, the rate per period

This is n, the total of periods

This is R, the amount of rent

This is the FV you want after the last

pmt Optional

Blank for ordinary annuity, 1 for annuity due

- Can you figure out how this comes from the

formula for the Future Value of an Ordinary

Annuity?

Loan Payment Formula

Start here with the original formula

Divide both sides by to get R alone

Here is the formula for the loan payment.

Loan Payment Formula

- Using basic algebra, we can rewrite this as

Example

- Suppose you want to buy a home and take out a

30-year mortgage for 240,000. The annual

interest rate is 5.75. - What is the monthly payment?
- What total amount of money do you pay over the

life of the loan (assuming all regular payments

are made)? - How much of your total payments is interest?

Example (a)

- We use the formula to get 1400.57 per month .

Example (b) and (c)

- The total amount of money we pay is
- 360x1400.57 504,205.20
- Hence, the amount of interest paid is
- 504,205.20 - 240,000 264,205.20

69

- You borrowed 150,000, which you agree to pay

back with monthly payments at the end of each

month for the next 10 years. At 6.25 interest,

how much is each payment?

9-10 Loan Payments (PMT) on Excel

- The PMT command will do these computations for us

automatically. - Command Format
- PMT(rate, nper, pv, fv, type)

This is i, the rate per period

This is n, the total of periods

This is the present value of the loan

This is the FV you want after the last

pmt Default0

Blank for ordinary annuity, 1 for annuity due

9-12 Adjusting for Inflation

- Inflation can seriously devalue a loan or asset

over time. - For example, if the average inflation rate is

3.5, how much will 50 be worth in 5 years (in

terms of todays dollars)? In other words, in

five years how much can I buy with a 50 bill

compared to what I can buy today?

Example

Important Notice that we substituted 50 for F

since that is what we know we will have in the

future. We solve for P since we want to know what

the Future 50 is worth in Present dollars.

- F P(1i)n
- 50 P(1.035)5
- 50 P(1.187686306)
- 50/(1.187686306) P
- 42.10 P

Terms

- Nominal Dollars are those that have not been

adjusted for inflation. - Real Dollars Present Dollars are those that

have been adjusted for inflation and therefore

reflect the spending power of some future amount

of money in terms of todays dollars.

- Well disregard amortization tables.