Chapter 2: Time value of money Flashcards by Emil Melchor

Is a process that considers risked and return to determine the worth or value of an asset

Time value of money

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A critical consideration for financed and investment decisions

Time value of money

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Are money market instruments issued at value less than their stated face value.

Discount instrument

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Occurs when interest paid on the investment during the first period is added to the principal; then during the second period, interest is on this new sum

Compound interest

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Formula of FV Compound annual

FV = PV (1 + i) n

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The current worth of a future sum of money or stream of cash inflows given a specified rate of return

Present value

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the process of determining the present value of a payment or stream of payments that is to be received in the future.

Discounting

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This is the method used to figure out how much these future payments are worth today.

Discounting

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the income return of an investment. This refers to the interest or dividends received from a security and is usually expressed as a percentage based on the investment’s cost its current market value or its face value

Yield

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describes what an investment has concretely earned

Return

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are money market instruments that are issued at a value less than their stated face value and mature for their face value

Discount instrument

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gain or loss of an investment over a specified period of time expressed as a percentage increase over the initial capital investment cost

Rate of Return

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Formula of Simple Interest for Future Value

I = P x r x t
= Prt
FV = P + I

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I = ?
P = ?
r = ?
t = ?

I = Amount of Interest
P = Principal
r = rate of interest per annum
t = times the period in years

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Formula of Future Value

FV = P + I

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Suppose we deposit Php 10,000 for a year into an account that will earn 5% per annum interest income on the principal only. What will this deposit be worth at the end of the year?

I = Php 10,000 x .05 x 1
= Php 500

FV = P + I
= Php 10,000 + 500
= Php 10,500

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Formula of Present Value of Simple Interest

PV = FV / (1 + it)

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What is the present value of a money market instrument that will pay 10% per annum simple interest and will pay its holder Php 100,000 in 120 days?

PV = Php100,000
/ (1+0.10 x 120/365)
= Php 100,000 (1.03288)
= $96,816.98

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is a loan of the government

Treasury Bill

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Terms of Purchase in Treasury Bills

28 days (4 weeks), 91 days (13 weeks), or 1 Yr

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How to get Purchase Price (proceeds) of a Treasury Bill

= The value of the Treasury bill - the discount

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Example: If you buy a P10,000, 13-week Treasury bill at 8%, how much will you pay?

P10,000 x .08 x (13/52) = P200

Cost = P10,000 – P200 = P9,800

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Example: If you buy a P10,000, 13-week Treasury bill at 8%, what is the effective rate?

ER = P200 / P 9,800 x (13/52)

= 8.16%

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In some loans, interest is computed once during the life of the loan, using the ________________.

simple interest formula

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interest that occurs when interest paid on the investment during the first period is added to the principal; then during the second period, interest is earned on this new sum.

Compound interest

The value (1+i) n used as a multiplier to calculate an amount’s future value

Future-value interest factor (FVIF i,n)

The value used as a multiplier to calculate an amount’s present value

Present-value interest factor (PVIF i,n)

the future value of the investment at the end of n year; the future value of the present sum

FVn

the present value or original amount invested at the beginning of the period; the current value of the future sum/payment; moving future money back to the present; discounted back to the present

PVn

the annual interest or discount rate

the number of years until payment will be received or during which compounding occurs

the number of times compounding occurs during the year

Future/Maturity Value Annual Periods:

FVn = PV (1+i)n

Compound Interest Future Value of Non-annual periods

FVn = PV (1+ i/ m)^nxm

How much is Php 12,000 @ 12% per annum compounded semi- annually for two years?

Non-annual Periods: FVn = PV (1+ i/m)^nm FVn = 12,000 (1+ 0 .12/2)^2(2) = Php 12,000 (1+.06)4 = Php12,000(1.26247696) =15,149.73

How much is Php 12,000 @ 12% per annum for two years?

FVn = P + I = 12, 000 + PRT =12,000 + (12,000 x .12 x 2) = 12,000 + 2,880 = 14,880

How much is P12,000 @12% per annum compounded semi-annually for two years? What is the Effective Rate?

= { 1 + (0.12/2) } ^2 - 1 = { 1 + 0.06} ^2 - 1 = 1.1236 – 1 = 0.1236 or 12.36%

Formula of effective rate

ie = [1 + i/m]^m - 1

The annualized interest rate that uses simple interest ratios to annualize an interest rate quoted on a fraction of a year

Annual Percentage Rate (APR)

The quoted rates

Nominal rate

The actual rate of interest that includes the adjustment to the nominal rate for the frequency of compounding

Effective rate (ie)

This is normally the advertised or quoted rate.

Annual Percentage Rate (APR)

By law lenders have to show this rate to customers.

Annual Percentage Rate (APR)

It is used so that customers can easily compare financial products.

Annual Percentage Rate (APR)

shows the cost of borrowing if interest is charged on an annual basis.

Annual Percentage Rate (APR)

is higher than the quoted rate (APR)

Effective Annual Rate (EAR)

Often, interest is not charged once a year but on a quarterly or monthly basis

Effective Annual Rate (EAR)

Takes the APR (or quoted rate) and adjusts it to take into account the frequency of interest charges.

Effective Annual Rate (EAR)

Formula of Present Value in Annual Periods

PVn = FVn [ 1 / (1+i) ]^n or PVn = FVn / ( 1 + i )^n

Formula of Present Value in Non-annual Periods

PVn = FVn [ 1 / (1+ i/m) ^ nm ] or PVn = FVn / (1 + i/m)^nm

Payment is made later

Ordinary Annuity

Payment is made immediately

Annuity Due

A regular stream of payments over a fixed time

Annuity

Let us assume that an interest rate of 5% per annum remains constant over the period, the future value of an ordinary annuity of $1 payments at the end of the end of next 4 periods is

FVA = 1 [ (1 + i)^4 - 1 / 0.05] = 4.31

A company has an annuity that will pay you Php 5000 per year for the next ten years. Assuming that the interest rates will average 9% per annum, how much would you expect to pay for the annuity?

PVA = 5,000 [1 - (1+0.09)^-10 / 0.09] PVA = 32,088.29

You may decide to set yourself a goal to save Php 9000 in four years (i.e. an FV) and will do so by saving regular amounts at an interest rate of 8% per annum compounded annually. If equal payments are made for each year, how much should you invest at the end of each year to reach your target?

PMT = [9,000 x 0.08 / (1 + 0.08)^4 -1 ] = 1,997.29

Suppose you are beginning your four-year university degree with Php 25, 000 in the bank. If you can invest your funds at 9% per annum, how much money can be withdrawn each year to provide for living expenses without exhausting your funds before you finish your studies?

PMT = [ 25000 x 0.09 / 1- (1 +0.09)-4] PMT = 7,716.72

Using these cash flows below, what will the account balance be at the end of year 5 if you can earn 11% per annum over the period 1= 1,000 2 = 3,000 3 = 2,000

1000 x (1.11)4 = $1,518.07 3000 x (1.11)3 = $4,102.89 2000 x (1.11)2 = $2,464.20 Future value of stream = 8,085.16

Find the present value of the following cash flow stream where k equals 8%. Assume year-end cash flows: 1= 500 2 = 700 3 = 11,000

500 x 1/1.08 = 462.96 700 x 1/(1.08)2= 600.14 11,000 x 1/(1.08)3 =8,732.13 Present Value of stream = 9,795.23

A security that promises regular cash flows forever

Perpetuity

Infinite stream of equal cash flows; Example : consol

Perpetuity

Formula of Perpetuity

PV = C/k

The headline rate of interest quoted by deposit takers before tax is deducted

Gross interest

The interest amount paid to customers once tax has been deducted from the gross interest

Net interest

For most deposits, tax on the interest is deducted __________ i.e. the bank or building society remove the tax amount before interest is paid to the depositor.

‘at source’

True or False Generally, interest received by an individual is subject to income tax.

True

Additional Rate (45%)

Over Php 150,000

Basic Rate (20%)

Php 15,000 - 50,000

High rate (40%)

Php 50,001 - 120,000

Find the accumulated value of P30,000 for 3 years at 14% compounded a. annually

FV= PV (1+i)^n =30,000 (1+0.14)^3 = 30,000 (1.481544) = 44,446.32

Find the accumulated value of P30,000 for 3 years at 14% compounded b. semiannually

} FVn = PV (1+i/m)nm = 30,000 (1+.14)^3(2) 2 = 30,000 (1+.07)^6 = 30,000 (1.500730352) = 45,021.91

Find the accumulated value of P30,000 for 3 years at 14% compounded c. quarterly

}FVn = PV (1+ i/m)nm } = 30,000 (1+.14)^3(4) 4 } = 30,000 (1+.035)^12 = 30,000 } (1.511068657) = 45,332.06

Find the accumulated value of P30,000 for 3 years at 14% compounded e. Ordinary daily

} FV = PV (1 + i/m)^nxm } FV = 30,000 (1+0.14/360)^(3)(360) } FV = 30,000 (1.000388889)^1080 } FV = 30,000 (1.521837299) } FV = 45,655.12

True or False The more the number of years, the bigger would be the terminal value

True

Summer invested Php 2,500 for 2 years with an interest income of 9% per annum How much will she get after two years?

}FV=P +IorP+(PRT) } =Php2,500 +(2500x0.09x2) } = Php 2,500 + 450.00 } =Ph2,950.00

Jack decides that he is going to buy a cheaper flat and place the remaining Php 40,000 in a deposit account. Jack also decides that he is unlikely to withdraw any of the money within the next year unless there was an emergency. Bank A Given: > Interest = 3.25% > Bonus = 1.8% > Paid yearly

40,000 x 0.0325 = 1,300 interest for the year 41,30 0 x 0.018 = 743.40 bonus interest Jack’s cash deposit after one year = 42,043.40

Interest } 40,000 x (0.007875) = 315 (for 3 months interest) } 40,315 x (0.007875) = 317.48 (after 6 months interest) } 40,632.48 x (0.007875) = 319.98 (after 9 months interest) } 40,952.46 x (0.007875) = 322.50 (after 12 months interest) Jack’s cash deposit after one year = 41,274.96

Jack decides that he is going to buy a cheaper flat and place the remaining Php 40,000 in a deposit account. Jack also decides that he is unlikely to withdraw any of the money within the next year unless there was an emergency. Bank C Given: > Interest = 2.80% > Bonus = 2.3% > Paid anniversary

Interest } 40,000 x 0.0280 =1,120 Bonus } 41,120 x 0.023 = 945.76 Jack’s cash deposit after one year = 42,065.76

Jack decides that he is going to buy a cheaper flat and place the remaining Php 40,000 in a deposit account. Jack also decides that he is unlikely to withdraw any of the money within the next year unless there was an emergency. Bank D Given: > Interest = 2.75% > Bonus = 2.22% > Paid monthly

Interest } 40,000 x (1+0.002291666667)^12 } 40,000 x (1.002291666667)^12 } 41,113.97 = Interest } 41,113.97 x 0.0222 = 912.73 Bonus } Jack’s cash deposit after one year is 42,026.70

Jack decides that he is going to buy a cheaper flat and place the remaining Php 40,000 in a deposit account. Jack also decides that he is unlikely to withdraw any of the money within the next year unless there was an emergency. Bank E Given: > Interest = 2.75% > Bonus = 1.5% > Paid yearly

40,000 x 0.0275 = 1,100 interest for the year 41,100 x 0.015 = 616.50 bonus interest Jack’s cash deposit after one year = 41,716.50

Generally, interest received by an individual is subject to _________

Income Tax

Formula of Annuities (Future Value)

FVA = PMT x [ (1 + i)^n - 1 / i]

Formula of Annuities (Present Value)

PVA = PMT x [ 1 - (1 + i)^-n / i ]

Formula of PMT (Future Value)

PMT = [ FVA x i / (1+i)^n-1 ]

Formula of PMT (Present Value)

PMT = [ PVA x i / 1 - (1 + i)^-n]

Stream of Cash Flows (Future Value)

FV = PV (1 + i)^n

Stream of Cash Flows (Present Value)

PV = FV x [1 / (1 + i)^n]

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