What is the better loan option between 8% compounded annually and 7.85% compounded quarterly?
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We cannot compare these nominal interest rates directly because they have different compounding periods. We need to calculate the Annual Percentage Yield (APY) for a proper comparison.
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What is the better loan option between 8% compounded annually and 7.85% compounded quarterly?
We cannot compare these nominal interest rates directly because they have different compounding periods. We need to calculate the Annual Percentage Yield (APY) for a proper comparison.
What is an annuity?
A sequence of equal cash flows, occurring at the end of each period.
What is the Present Value (PV) of $1,000 at the beginning of each of the next 3 years with an opportunity cost of 8%?
$2,783.26
What is the formula for calculating the Present Value of a perpetuity?
The formula is PV = PMT / i, where PMT is the payment per period and i is the interest rate.
What is the present value of receiving $10,000 annually forever at an 8% required return?
$125,000
What is the formula for calculating the Present Value of an annuity?
PV = PMT (PVIFA i, n)
What type of payments do you receive if you buy a bond?
Equal semi-annual coupon interest payments over the life of the bond.
If you invest $1,000 each year at 8%, how much would you have after 3 years?
After 3 years, you would have approximately $3,246.64.
What are the four variables in single sum present value and future value problems?
FV, PV, i, and n.
What is compound interest?
Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
Why is receiving 1 in the future?
It is due to opportunity costs, specifically the interest that could have been earned if the $1 was received sooner.
If you invest $1,000 each year at 8%, how much would you have after 3 years?
$3,246.40
What is the Present Value (PV) of receiving $1,000 at the end of each of the next 3 years with an opportunity cost of 8%?
$2,577.10
What type of payments do you make if you borrow money to buy a house or a car?
A stream of equal payments.
What can we do if we can measure opportunity cost?
We can translate 1 in the future into its equivalent today (discounting).
What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
The present value (PV) can be calculated using the formula for the present value of an annuity. PV = C × [(1 - (1 + r)^-n) / r], where C is the cash flow per period (2,577.10.
How is future value calculated in compound interest?
Future value is calculated using the formula FV = P(1 + r/n)^(nt), where P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
How many variables are typically provided in single sum problems?
Three variables are provided, and you solve for the fourth.
What is the benefit of understanding the four variables in time value problems?
It makes solving time value problems much easier.
What is the Future Value of the ordinary annuity at year 3 with an interest rate of 8%?
$3,246.40
What is a perpetuity?
A perpetuity is a fixed payment received every period (month, year, etc.) forever.
If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?
$3,506.11
If you sold land for 5,000, what is your annual rate of return?
To calculate the annual rate of return, use the formula: Rate of Return = (FV - PV) / PV / n, where FV is the future value (5,000), and n is the number of years (5).
What is the formula for calculating Present Value (PV)?
PV = FV (PVIF i, n) or PV = FV / (1 + i)^n
What is the formula to calculate the Annual Percentage Yield (APY) for a loan compounded quarterly?
APY = (1 + quoted rate/m)^m - 1
What is the Present Value of the ordinary annuity at year 0 with an interest rate of 8%?
$2,577.10
What happens to PVIFA as n gets very large?
It becomes zero.
If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
$106
What is the formula for calculating Present Value?
The formula for Present Value (PV) is PV = FV / (1 + r)^n, where FV is the future value, r is the interest rate, and n is the number of periods.
What is the FV of $1,000 earning 8% with continuous compounding after 100 years?
$2,980,957.99
How can a perpetuity be compared to an annuity?
A perpetuity can be thought of as an annuity that goes on forever.
Is this an annuity?
No, this is not an annuity because the cash flows are different.
What is the formula for Present Value Interest Factor of Annuity (PVIFA)?
(PVIFA i, n ) = 1 - (1 + i)^(-n) / i
What is the formula for calculating Present Value (PV) of an annuity?
PV = PMT (PVIFA i, n)
If you receive $100 one year from now with an opportunity cost of 6%, what is the Present Value?
$94.34
What is the process for handling uneven cash flows?
We have to discount each cash flow back separately.
What is the future value of an annuity due if you invest $1,000 at the beginning of each of the next 3 years at an interest rate of 8%?
$3,506.11
What is the APY for a loan with a quoted rate of 7.85% compounded quarterly?
APY = (1 + 0.0785/4)^4 - 1 = 0.0808, or 8.08%
What is the formula for PVIFA?
PVIFA = 1 i (1 - 1/(1 + i)^n) i
What is the formula to calculate Future Value (FV) for a single sum?
FV = PV (1 + i)^n
Why is Present Value important in finance?
Present Value is important because it allows investors to determine the current worth of future cash flows, helping in investment decisions and financial planning.
What is an annuity due?
An annuity due is a type of annuity where cash flows occur at the beginning of each period.
What is the present value if you place $100 in an account that pays 9.6% interest, compounded monthly?
$100
How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate.)
To find the PV, discount each cash flow back to the present value using the formula PV = CF / (1 + r)^n, where CF is the cash flow, r is the discount rate, and n is the time period.
What formula is used to calculate the future value with continuous compounding?
FV = PV (e^(in))
What is the process for handling uneven cash flows?
We have to discount each cash flow back separately.
What happens to the PVIFA formula when n approaches infinity?
As n gets very large, the formula approaches 1/i, indicating the present value of a perpetuity.
What is the Present Value (PV) of receiving $1,000 at the end of each of the next 3 years with an opportunity cost of 8%?
$2,577.10
How do you calculate the Present Value using the formula PV = FV / (1 + i)^n?
PV = 100 / (1.06)^1 = $94.34
What are the cash flows in the given example?
The cash flows are -10,000 at year 0, 2,000 at year 1, 4,000 at year 2, 6,000 at year 3, and 7,000 at year 4.
How do you calculate the future value of an annuity due?
FV = PMT (FVIFA i, n) (1 + i)
What is the difference between Begin Mode and End Mode in annuities?
Begin Mode means payments are made at the beginning of each period, while End Mode means payments are made at the end of each period.
How does the APY of a quarterly loan compare to an 8% loan with annual compounding?
The quarterly loan is more expensive than the 8% loan with annual compounding.
Using the FVIF table, what is the Future Value Interest Factor (FVIF) for 6% over 1 year?
1.06
How does the interest rate affect Present Value?
As the interest rate increases, the Present Value decreases, meaning future cash flows are worth less today.
What is the formula to calculate the Future Value of an annuity?
FV = PMT (FVIFA i, n)
In an annuity due with a 3-year timeline and cash flows of $1000, when do the cash flows occur?
The cash flows occur at the beginning of each year.
What is the value of 'PV' in the calculation?
$1,000
What is the future value if your account grows to $500?
$500
What is the formula to calculate Present Value (PV) from Future Value (FV)?
PV = FV / (1 + i)^n
What is the difference between Begin Mode and End Mode in financial calculations?
Begin Mode calculates cash flows at the beginning of each period, while End Mode calculates them at the end of each period.
What is the Time Value of Money?
The concept that money available today is worth more than the same amount in the future due to its potential earning capacity.
What are the cash flows in the given example?
The cash flows are -10,000 at year 0, 2,000 at year 1, 4,000 at year 2, 6,000 at year 3, and 7,000 at year 4.
How do you calculate PV using the formula PV = PMT / i (1 - (1 + i)^-n)?
PV = 1000 / 0.08 (1 - (1.08)^-3)
In an ordinary annuity, when are the payments made?
Payments in an ordinary annuity are made at the end of each period.
What does a higher number of periods do to Present Value?
A higher number of periods decreases the Present Value, as future cash flows are discounted more heavily.
How long will it take for 500 at 9.6% interest compounded monthly?
It will take 202 months.
If you invest $1,000 each year at an interest rate of 8% for 3 years, what will be the Future Value?
$3,246.40
What is the formula for future value of an ordinary annuity?
FV = PMT (1 + i)^n - 1 / i
What is the formula for calculating Present Value (PV)?
PV = FV (PVIF i, n) or PV = FV / (1 + i)^n
What does PV stand for in the context of Future Value calculations?
Present Value
What is the present value (PV) of the cash flow stream at period 0?
-10,000.00
How does an annuity due differ from a regular annuity?
An annuity due has cash flows at the beginning of each period, while a regular annuity has cash flows at the end of each period.
What is the formula for calculating Future Value (FV) using Present Value (PV)?
FV = PV (FVIF i, n)
What is the interest rate used in the calculation?
8%
What is the formula for calculating Present Value (PV)?
PV = FV (PVIF i, n) or PV = FV / (1 + i)^n
What is the interest rate for the account in the scenario?
9.6%
In Begin Mode, how is the present value (PV) of an annuity due calculated?
The present value of an annuity due in Begin Mode is calculated by taking the present value of an ordinary annuity and multiplying it by (1 + r), where r is the interest rate.
If you sold land for 5,000 five years ago, what is the annual rate of return?
The annual rate of return is 19%.
What is the difference between Begin Mode and End Mode in annuities?
Begin Mode assumes payments are made at the beginning of each period, while End Mode assumes payments are made at the end of each period.
What does compounding refer to in finance?
The process of earning interest on both the initial principal and the accumulated interest from previous periods.
If an annuity has payments of 1000 for 5 years, what is the total amount received?
The total amount received would be 1000 multiplied by 5, which equals 5000.
What formula is used to calculate the present value in this scenario?
PV = FV / (1 + i)^n
What does FVIFA stand for in the Future Value formula?
Future Value Interest Factor of Annuity
What is the present value (PV) of the cash flow at period 1?
1,818.18
What is the Present Value (PV) of $1,000 to be received 15 years from now at an opportunity cost of 7%?
$362.45
If you deposit $100 in an account earning 6% with quarterly compounding, how much will you have after 5 years?
$134.68
How often is the interest compounded in this scenario?
Monthly
What does PVIF stand for in the context of Present Value calculations?
PVIF stands for Present Value Interest Factor.
If you receive $100 five years from now with an opportunity cost of 6%, what is the Present Value?
$74.73
How does the future value (FV) of an annuity due differ in Begin Mode compared to End Mode?
In Begin Mode, the future value of an annuity due is calculated by taking the future value of an ordinary annuity and multiplying it by (1 + r), where r is the interest rate.
In an ordinary annuity, when are the payments made?
In an ordinary annuity, payments are made at the end of each period.
What is the exponent in the continuous compounding formula for this scenario?
8
What is the Time Value of Money?
The Time Value of Money is the concept that money available today is worth more than the same amount in the future due to its potential earning capacity.
What is a cash flow stream?
A series of cash inflows and outflows over a period of time.
How does the number of years affect the total payments in an annuity?
The total payments in an annuity increase with the number of years, as each year adds another payment.
What is the difference between Begin Mode and End Mode in financial calculations?
Begin Mode calculates cash flows at the beginning of each period, while End Mode calculates them at the end of each period.
What is the formula for calculating the Present Value of an annuity due?
PV = PMT (PVIFA i, n) (1 + i)
What is the value of 'i' in the given problem?
i = 0.008 (which is 9.6% annual interest compounded monthly).
Using the formula FV = PMT (1 + i)^n - 1 / i, what is the Future Value if PMT is $1,000, i is 0.08, and n is 3?
$3,246.40
How do you calculate PV using the formula PV = FV / (1 + i)^n?
PV = 1000 / (1.07)^15 = $362.45
What is the present value (PV) of the cash flow at period 2?
3,305.79
What is the Future Value (FV) of $100 after 20 years at an interest rate of 1.5%?
$134.68
What is the goal amount for the account to grow to?
$500
How do you derive the interest rate (i) from the Present Value formula?
By rearranging the formula: i = (FV/PV)^(1/n) - 1.
What does PVIF stand for in the context of Present Value calculations?
Present Value Interest Factor
How does the Present Value (PV) differ in End Mode compared to Begin Mode?
In End Mode, the Present Value is calculated based on payments made at the end of each period, which typically results in a lower PV compared to Begin Mode.
What does compounding refer to in finance?
Compounding refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods.
How does discounting affect future cash flows?
Discounting reduces the value of future cash flows to reflect their present value, accounting for the time value of money.
What is the formula for calculating Future Value (FV) using Present Value (PV)?
FV = PV (FVIF i, n)
In an annuity due, when are the payments made?
Payments in an annuity due are made at the beginning of each period.
What does 'N' represent in the mathematical solution?
N represents the number of months it takes for the investment to grow.
How do you calculate the Present Value of an annuity due using the given example?
PV = 1,000 (PVIFA .08, 3) (1.08) = $2,783.26
How do you calculate Future Value (FV) with compounding interest?
FV = PV (1 + i/m)^(m x n)
What is the present value (PV) of the cash flow at period 3?
4,507.89
How do you calculate the Present Value of $100 received in 5 years at a 6% interest rate?
PV = 100 / (1.06)^5 = $74.73
What is the Future Value (FV) in End Mode for an ordinary annuity?
The Future Value in End Mode is calculated based on the total amount accumulated at the end of the last period, considering payments made at the end of each period.
What is discounting in the context of finance?
Discounting is the process of determining the present value of a future sum of money or stream of cash flows given a specified rate of return.
If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?
$133.82
What is the main difference between an Ordinary Annuity and an Annuity Due?
An Ordinary Annuity pays at the end of each period, while an Annuity Due pays at the beginning of each period.
If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
$134.89
If an annuity has payments of 1000 for 4 years, what is the total cash flow over the period?
The total cash flow over 4 years would be 4000.
What is the natural logarithm of 5 used for in the solution?
It is used to solve for N in the equation ln 5 = N ln (1.008).
What does PVIFA stand for in the context of Present Value calculations?
PVIFA stands for Present Value Interest Factor of Annuity.
What is the present value (PV) of the cash flow at period 4?
4,781.09
How does the Time Value of Money affect investment decisions?
The Time Value of Money influences investment decisions by highlighting the importance of earning returns on investments over time, making future cash flows less valuable than immediate cash flows.
What is the formula to calculate Future Value (FV) for a single sum?
FV = PV (1 + i)^n
If you receive $1000 annually for 4 years, is it an Ordinary Annuity or an Annuity Due?
It depends on when the payments are made; if at the end of each year, it's an Ordinary Annuity; if at the beginning, it's an Annuity Due.
What is the Future Value (FV) when PV is $100, interest rate is 0.5% per month, and the period is 60 months?
$134.89
How does the timing of cash flows affect the present value of an annuity?
Cash flows received at the beginning of each period (Begin Mode) have a higher present value compared to those received at the end (End Mode).
What is the value of PMT in the provided example for calculating Present Value?
PMT is 1,000.
What is the total present value of the cash flow stream?
$4,412.95
What is the formula for calculating future value of a single sum?
The formula for calculating future value (FV) is FV = PV * (1 + r)^n, where PV is the present value, r is the interest rate, and n is the number of periods.
What does PV stand for in the Future Value formula?
Present Value
How would the value of an Annuity Due compare to an Ordinary Annuity with the same cash flows?
The value of an Annuity Due is higher than that of an Ordinary Annuity because payments are received earlier.
What interest rate is used in the Present Value calculation example?
The interest rate used is 8% (0.08).
What is the formula for calculating present value of a single sum?
The formula for calculating present value (PV) is PV = FV / (1 + r)^n, where FV is the future value, r is the interest rate, and n is the number of periods.
What does 'i' represent in the Future Value formula?
The interest rate
What is the cash flow amount for the given annuities?
$1000 for each year.
In the calculation FV = 100 (1.06)^5, what does '5' represent?
The number of years the money is invested
For how many years are the annuities paid in the example?
4 years.
