Financial Management Unit 3 (ALL printed) Flashcards
FV FORMULA
FV FORMULA
At the beginning of your freshman year, your favorite aunt and uncle deposit $10,000 into a 4-year bank certificate of deposit (CD) that pays 5% annual interest. You will receive the money in the account (including the accumulated interest) if you graduate with honors in 4 years. How much will there be in the account after 4 years ……………………………………………..?
Using the formula approach, we know that FVN = PV (1 + 1)N. In this case, you know that N = 4,
PV = $10, 000, and I = 0.05. It follows that the future value after 4 years will be
FV4 = $10, 000(1.05) 4 = $12,155.06
FUTURE VALUE / PRESENT VALUE FORMULA
FUTURE VALUE / PRESENT VALUE FORMULA
The future value of an annuity can be found using the step-by-step approach or using a formula, a financial calculator, or a spreadsheet. As an illustration, consider the ordinary annuity diagrammed earlier, where you deposit $100 at the end of each year for 3 years and earn 5% per year ………………………………………………?
ANS-REF
Your grandfather urged you to begin a habit of saving money early in your life. He suggested that you put $5 a day into an envelope. If you follow his advice, at the end of the year you will have $1,825 (365 × $5).
Your grandfather further suggested that you take that money at the end of the year and invest it in an online brokerage mutual fund account that has an annual expected return of 8%.
You are 18 years old. If you start following your grandfather’s advice today, and continue saving in this way the rest of your life, how much do you expect to have in the brokerage account when you are 65 years old ………………………………………………?
ANS-REF ATTACHED
For an ordinary annuity with five annual payments of $100 and a 10% interest rate, how many years will the first payment earn interest? What will this payment’s value be at the end? Answer this same question for the fifth payment ………………………………………………?
REFERENCE
FV = PV * (1 + r)An
where:
FV = future value
PV = present value (the amount of
the payment)
r = interest rate
n = number of years
So, the future value of the first payment will be:
FV = $100 * (1 + 0.10)^4
FV = $100 * 1.4641
FV = $146.41
ANS =
The fifth payment, made at the end of the fifth year, will not earn any interest because it is made at the end of the period. Therefore, its value at the end will be the same as its present value, which is
$100
Assume that you plan to buy a condo 5 years from now, and you estimate that you can save $2,500 per year. You plan to deposit the money in a bank account that pays 4% interest, and you will make the first deposit at the end of the year. How much will you have after 5 years? How much will you have if the interest rate is increased to 6% or lowered to 3% ………………………………………………?
1) FV = P * [(1 + r)^n - 1 / r
or
FV = P * (((1+ r)^n -1 / r
2) In this case, P = $2,500 (the
amount you save each year), n = 5
years (the number of years you will be investing).
1. If r = 0.04 (the annual return of
the bank account), the future value of your investment after 5 years would be:
FV = $2,500 * (((1 + 0.04) ^5 - 1) / 0.04
FV = $13,540.81
2) If the interest rate is increased to 6%, the future value will be:
FV = $2,500 * (((1 + 0.06)^ 5 - 1) / 0.06 = $14.092.73
If the interest rate is lowered to 3%, the future value will be:
FV = $2,500 * (((1 + 0.03)^5 - 1] / 0.03 = $13,272.84
So, after 5 years, you will have $13,540.81 if the interest rate is 4%, $14.092.73if the interest rate is 6%, and $13,272.84 if the interest rate is 3%.
ANS-REF ATTACHED
ANS-REF
You just won the Florida lottery. To receive your winnings, you must select ONE of the two following choices:
You can receive $1,000,000 a year at the end of each of the next 30 years.
You can receive a one-time payment of $15,000,000 today.
Assume that the current interest rate is 6%. Which option is most valuable ……………………………………………………….?
The most valuable option is the one with the largest present value. You know that the second option has a present value of $15,000,000, so we need to determine whether the present value of the $1,000,000, 30-year ordinary annuity exceeds $15,000,000.
ANS-REF
1) 1-1 / (1.06)^30 = .8258898691
2) .8258898691 / 0.06 = 13.76483115
3) 13.76483115 * 1,000,000 = 13,764,831.15
Finding Annuity Payments, PMT ……………………………………………………….?
Suppose we need to accumulate $10,000 and have it available 5 years from now. Suppose further that w
can earn a return of 6% on our savings, which are currently zero. Thus, we know that FV = 10, 000,
PV = 0, N = 5, and I/YR = 6. We can enter these values in a financial calculator and press the PMT key find how large our deposits must be
Thus, you must save $1,773.96 per year if you make deposits at the end of each year, but only $1,673.55 if the deposits begin immediately. Note that the required annual deposit for the annuity due can also be
calculated as the ordinary annuity payment divided by (1 + I): $1, 773.96/1.06 = $1, 673.55.
PV of a perpetuity ANS-REF
ANS-REF
What’s the present value of a perpetuity that pays $1,000 per year beginning 1 year from now, if the appropriate interest rate is 5%? What would the value be if payments on the annuity began immediately ……………………………………………………….?
PV = C / r
where:
C = cash flow per period (in this
case, $1,000 per year)
r = interest rate (in this case, 5%
or 0.05)
So, if the payments begin 1 year from now, the present value of the perpetuity is:
PV = $1,000 / 0.05 = $20,000
If the payments on the annuity began immediately, it would be considered a perpetuity due. The present value of a perpetuity due is calculated as:
PV = C/ r * (1 + r)
So, the present value of the perpetuity due is:
PV = $1,000 / 0.05 * (1 + 0.05) =
$20,000 * 1.05 = $21,000
So, the present value of the perpetuity is $20,000 if payments begin 1 year from now, and $21,000 if payments begin immediately.
What’s the present value of a 5-year ordinary annuity of $100 plus an additional $500 at the end of Year 5 if the interest rate is 6%? What is the PV if the $100 payments occur in Years 1 through 10 and the $500 comes at the end of Year 10 ……………………………………………………….?
Given information for annuity and future cash flows.
Annuity payment= $100 for 5 years
Future cash flows= $500 at the end of 5 years
Discount rate= 6%
The present value of annuites and future cashflows is:
1) 100 (1 - (1+0.06)^-5 = 25.27418271 / 0.06
= 421.2363786
2) 421.2363786 + 500 / (1+ 0.06)^5
= 794.87
Here, PMT= Annual cash flows, r=discount rate, FV= future value, and n= period.
1) 100 (1 - (1 + 0.06) ^-10 = 44.16052231
= 44.16052231 / 0.06 = 736.0087051
2) 736.0087051 + 500 / (1+ 0.06)^10
= $1015.21
Here, PMT= Annual cash flows, r=discount rate, FV= future value, and n= period.
What’s the present value of the following uneven cash flow stream: $0 at Time 0, $100 in Year 1 (or at Time 1), $200 in Year 2, $0 in Year 3, and $400 in Year 4 if the interest rate is 8% ……………………………………………………….?
Cash flows over time:
* CF. = $0
* CF1 = $100
* CF2 = $200
. CF3 = $0
- CF4 = $400
Interest rate (r) = 8%
Where:
- CFt is the cash flow in period t,
- r is the discount rate or interest rate,
* t is the time period.
PV= 0 + 100 / 1.08 + 200 / 1.08^2 + 0 / 1.08^3 + 400 / 1.08^4
= 558.07
What is the future value of this cash flow stream: $100 at the end of 1 year, $150 due after 2 years, and $300 due after 3 years, if the appropriate interest rate is 15% ……………………………………………………………..?
FUTURE VALUE OF THE CASHFLOWS =
100(1.15^2) + 150(1.15^1) + 300(1.15^0)
= 132.25 + 172.5 + 300
= $604.75
An investment costs $465 and is expected to produce cash flows of $100 at the end of each of the next 4 years, then an extra lump sum payment of $200 at the end of the fourth year. What is the expected rate of return on this investment ……………………………………………………………..?
Using financial calculator
N=4
PV = -465
PMT = 100
FV = 200
I/YR= 9.05
Note = Calculator setting setup
1) TVM (2) OTHER (3) Change P/YR to 1(annually)
ASSIGNMENT
An analyst evaluating securities has obtained the following information. The real rate of interest is 2.9% and is expected to remain constant for the next 5 years. Inflation is expected to be 2.7% next year, 3.7% the following year, 4.7% the third year, and 5.7% every year thereafter. The maturity risk premium is estimated to be 0.1 × (t – 1)%, where t = number of years to maturity. The liquidity premium on relevant 5-year securities is 0.5% and the default risk premium on relevant 5-year securities is 1%
……………………………………………………………..?
a. What is the yield on a 1-year T-bill? Round your answer to one decimal place.
b. What is the yield on a 5-year T-bond? Round your answer to one decimal place.
c. What is the yield on a 5-year corporate bond? Round your answer to one decimal place.
A) The yield on a 1-year T-bill is the sum of the real rate of interest and the expected inflation for the next year.
The real rate of interest is given as 2.9%.
The expected inflation for the next year is given as 2.7%.
Adding these two components, the yield on a 1-year T-bill is 2.9% + 2.7% = 5.6%
So, the yield on a 1-year T-bill is 5.6%.
B) The real rate of interest is given as 2.9%.
The average expected inflation over the next 5 years can be calculated as (2.7% + 3.7% + 4.7%
+ 5.7% + 5.7%) / 5 (next 5 years) = 4.5%
The maturity risk premium for a 5-year security is calculated as 0.1 x (5 - 1)% = 0.4%. Adding these three components the yield on a 5-year T-bond is 2.9% + 4.5% + 0.4% = 7.8%
C) Using the maturity risk premium for a 5-year security is calculated as 0.1 x (5 - 1)% = 0.4%.
The liquidity premium is given as 0.5%.
The default risk premium is given as 1%.
Adding these five components, the yield on a 5-year corporate bond is 2.9% + 4.5% + 0.4% + 0.5% + 1%
= 9.3%
ASSIGNMENT
Today, interest rates on 1-year T-bonds yield 1.4%, interest rates on 2-year T-bonds yield 2.4%, and interest rates on 3-year T-bonds yield 3.5% ……………………………………………………………..?
A) a. If the pure expectations theory is correct, what is the yield on 1-year T-bonds one year from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
C) If the pure expectations theory is correct, what is the yield on 1-year T-bonds two years from now? Be sure to use a geometric average in your calculations. Do not round intermediate calculations. Round your answer to four decimal places.
A) Solving this equation for Y gives Y
= (1 + 2.4/100)^2 / (1 + 1.4/100) -
1] * 100 = 3.4099%
C) According to the pure expectations theory, the yield on the 3-year T-bond (3.5%) is the geometric average of the yield on the 1-year T-bond today (1.4%), the yield on a 1-year T-bond one year from now (which we calculated in the previous question as 3.4099%), and Y.
So, we have (1 + 3.5/100) = [(1 +1.4/100) * (1 + 3.4099/100) * (1 + Y/100)]^(1/3).
Solving this equation for Y gives Y
= [(1 + 3.5/100)^3 / (1 + 1.4/100) /
(1 + 3.4099/100) - 1] * 100 =
5.7355%
ASSIGNMENT
Today, interest rates on 1-year T-bonds yield 1.4%, interest rates on 2-year T-bonds yield 2.4%, and interest rates on 3-year T-bonds yield 3.5% ……………………………………………………………..?
B) Let’s denote the yield on a 2-year T-bond one year from now as Y.
According to the pure expectations theory, the yield on the 3-year T-bond (3.5%) is the geometric average of the yield on the 1-year T-bond today (1.4%), the yield on a 2-year T-bond today (2.4%), and Y.
So, we have (1 + 3.5/100)^(1/3) =
[(1 + 1.4/100) * (1 + 2.4/100) * (1 +
Y/100)]^(1/3).
Solving this equation for Y gives Y
= [(1 + 3.5/100)^(3) / (1 +
1.4/100) / (1 + 2.4/100) - 1] * 100
= 6.7783
ASSIGNMENT What is the present value of a security that will pay $12,000 in 20 years if securities of equal risk pay 8% annually? Do not round intermediate calculations. Round your answer to the nearest cent ……………………………………………………………..?
1) PLUG INTO CALCULATOR
N= 20
FV = 12,000
I/YR= 8%
Then press PV
ANSWER= 2,574.58
OR
PV = FV / (1 + r)^n
PV = 12,000. (1+0.08)^20
= 2,574.58
Your parents will retire in 19 years. They currently have $300,000 saved, and they think they will need $1,900,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don’t save any additional funds? Round your answer to two decimal places ………………………………………………………………..?
Present Savings = $300,000
future savings = $1,900,000
time = 19 years
r = (FV / PV)^(1/n) - 1
Substituting the given values into the formula gives:
r= ($1,900,000 / $300,000)^(1/19) - 1
= 10.20
As his adviser, which contract would you recommend that he accept?
Select the correct answer ………………………………………………………………..?
Which ever answer gives the player the HIGHEST PRESENT VALUE is the contract to choose
Contract 1:
Explanation:
Contract 1:
PV1 = $3,500,000/(1+0.09) +
$3,500,000/(1+0.09)^2 +
$3,500,000/(1+0.09)^3 +
$3,500,000/(1+0.09)^4
PV1 = $3,211,009.17+ $2,945,879.976 + $2,702,642.18 + $2,479,488.239 =
11,339,019.57
Contract 2:
PV2 = $2,500,000/(1+0.09) +
$3,000,000/(1+0.09)^2 +
$4,000,000/(1+0.09)13 +
$5,000,000/(1+0.09)14
PV2 = $2,293,577.98 +
$2,523,424.35 + $3,024,565.47 +
$3,478,341.02
PV2 = $11,319,908.82
Contract 3:
PV3 = $6,000,000/(1+0.09) +
$1,500,000/(1+0.09)^2 +
$1,500,000/(1+0.09)13 +
$1,500,000/(1+0.09)14
PV3 = $5,504,587.16 +
$1,261,712.18 + $1,156,213.86 +
$1,043,170.26
PV3 = $8,965,683.46
You have saved $3,000 for a down payment on a new car. The largest monthly payment you can afford is $350. The loan will have a 12% APR based on end-of-month payments. What is the most expensive car you can afford if you finance it for 48 months? For 60 months? Do not round intermediate calculations. Round your answers to the nearest cent ………………………………………………………………..?
The formula to calculate the loan amount is:
Loan Amount =
Monthly Payment * [1 - (1 + r)^-n) / r]
where:
-r is the monthly interest rate (annual interest rate / 12 / 100)
- n is the number of payments (loan term in months)
For a 48-month loan:
- r = 12% / 12 / 100 = 0.01
- n = 48
Loan Amount = $350 * [ (1 - (1 + 0.01)1-48 ) / 0.01 ] = $13,290.88582 / 13,290.9
So, the most expensive car you can afford with a 48-month loan is $13,290.9 (loan amount) + $3,000 (down payment) =
$16,290.9
Loan Amount = $350 * [ (1 - (1 + 0.01)^-60 ) / 0.01 ] = $15,734.3
So, the most expensive car you can afford with a 60-month loan is $15,734.3 (loan amount) + $3,000 (down payment) =
$18,734.26
Interest rates on 4-year Treasury securities are currently 5.5%, while 6-year Treasury securities yield 7.95%. If the pure expectations theory is correct, what does the market believe that 2-year securities will be yielding 4 years from now? Calculate the yield using a geometric average. Do not round intermediate calculations. Round your answer to two decimal places ………………………………………………………………..?
(1 + 0.0795)^6 = (1 + 0.055)^4 * (1+ x)^2
Where x is the yield on the 2-year security 4 years from now.
Solving for × gives:
x = [(1 + 0.0795)^6 / (1 + 0.055)^4]^(1/2) - 1
Calculating this gives:
x= 1.582471437 / 1.238824651
= 1.277397439^1/2
= 1.130220084 - 1
= .1302200845 –> 13.02%
ANS-REF COMPLETE
Due to a recession, expected inflation this year is only 3.75%. However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above 3.75%. Assume that the expectations theory holds and the real risk-free rate (r*) is 3.5%. If the yield on 3-year Treasury bonds equals the 1-year yield plus 0.5%, what inflation rate is expected after Year 1? Round your answer to two decimal places ………………………………………………………………..?
Yield_1yr = r* + Inflation_1yr
Yield_1yr = 3.5% + 3.75% = 7.25%
The yield on a 3-year Treasury bond is the 1-year yield plus 0.5%:
Yield_3yr = Yield_1yr + 0.5%
Yield_3yr = 7.25% + 0.5% = 7.75%
Yield_3yr = (Yield_1yr + (r* + x) + (r* + ×) /3
Solving for x gives:
7.75% = (7.25% + (3.5% + x) +
(3.5% + ×)) / 3
[ 7.75 * 3 = 23.25]
23.25% = 7.25% + 2* (3.5% + x)
23.25% - 7.25% = 7% + 2x
16% = 7% + 2x
9% = 2x
4.5% = x
A company’s 5-year bonds are yielding 9% per year. Treasury bonds with the same maturity are yielding 3.9% per year, and the real risk-free rate (r*) is 2.25%. The average inflation premium is 1.25%, and the maturity risk premium is estimated to be 0.1 × (t - 1)%, where t = number of years to maturity. If the liquidity premium is 1.2%, what is the default risk premium on the corporate bonds? Round your answer to two decimal places ………………………………………………………………..?
Yield_Treasury = r* + Inflation
Premium + Maturity Risk Premium
3.9% = 2.25% + 1.25% + 0.1 × (5 - 1)
3.9% = 2.25% + 1.25% + 0.4%
3.9% = 3.9%
Yield_Corporate = r* + Inflation
Premium + Maturity Risk Premium + Liquidity Premium + Default Risk Premium
The yield on the corporate bond is given as 9%
Default Risk Premium =
Yield_Corporate - r* - Inflation
Premium - Maturity Risk Premium - Liquidity Premium
Default Risk Premium =
9% - 2.25% - 1.25% - 0.4% - 1.2%
Default Risk Premium = 3.9%
Yield_Corporate = 9% (GIVEN)
real risk-free rate (r) = 2.25%** (GIVEN)
**Inflation Premium = 1.25%** (GIVEN)
**maturity risk premium = 0.4% (0.1 x (5 - 1)
Liquidity Premium= 1.2% (GIVEN)
FORMULA future value of an ordinary annuity, FVAn
FORMULA future value DUE
FORMULA present value of an ordinary annuity, PVAn
FORMULA present value DUE
If you require an annual return of 10%, what is the present value of this cash flow stream? Do not round intermediate calculations. Round your answer to the nearest cent …………………………………………………………………….?
In this case, the cash flows (CF) are $670, $380, $230, and $320 at the end of years 1, 2, 3, and 4 respectively, and the annual return
(r) is 10% = 0.10.
So, the present value of the cash flow stream is:
PV = $670 / (1 + 0.10)^1 + $380 / (1 + 0.10)^2 + $230 / (1 + 0.10)^3 + $320 / (1 + 0.10)^4
——————-
PV = $1,314.507206
= $1,314.5
