Chapter 2: Time value of money Flashcards
Is a process that considers risked and return to determine the worth or value of an asset
Time value of money
A critical consideration for financed and investment decisions
Time value of money
Are money market instruments issued at value less than their stated face value.
Discount instrument
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
Formula of FV Compound annual
FV = PV (1 + i) n
The current worth of a future sum of money or stream of cash inflows given a specified rate of return
Present value
the process of determining the present value of a payment or stream of payments that is to be received in the future.
Discounting
This is the method used to figure out how much these future payments are worth today.
Discounting
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
describes what an investment has concretely earned
Return
are money market instruments that are issued at a value less than their stated face value and mature for their face value
Discount instrument
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
Formula of Simple Interest for Future Value
I = P x r x t
= Prt
FV = P + I
I = ?
P = ?
r = ?
t = ?
I = Amount of Interest
P = Principal
r = rate of interest per annum
t = times the period in years
Formula of Future Value
FV = P + I
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
Formula of Present Value of Simple Interest
PV = FV / (1 + it)
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
is a loan of the government
Treasury Bill
Terms of Purchase in Treasury Bills
28 days (4 weeks), 91 days (13 weeks), or 1 Yr
How to get Purchase Price (proceeds) of a Treasury Bill
= The value of the Treasury bill - the discount
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
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%
In some loans, interest is computed once during the life of the loan, using the ________________.
simple interest formula
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
i
the number of years until payment will be received or during which compounding occurs
n
the number of times compounding occurs during the year
m
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
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 B
Given:
> Interest = 3.15%
> Paid quarterly
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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