Financial Ratios,
Formulas, and
Calculations
During this course, we will review key financial ratios, formulas, and
calculations, including but not limited to: time value of money concepts,
mortgage payments, amortization schedules, college planning calculations,
retirement planning calculations, Monte Carlo Simulation, bond valuation,
return analysis, insurance/annuity calculations, net present value (NPV), and
internal rate of return (IRR). With a mastery of this material, you can help
your clients enhance project or portfolio returns, maximize cash flow,
minimize interest payments, and avoid excessive fees.
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Financial Ratios, Formulas, and Calculations – 2017
Table of Contents
I.
Future Value ............................................................................................................................... 4
1)
2)
3)
4)
5)
6)
7)
8)
9)
II.
FV Concepts 1 ...................................................................................................................................... 4
FV Concepts 2 ...................................................................................................................................... 4
FV Concepts 3 ...................................................................................................................................... 4
Future Value – Compounded Annually 1..................................................................................... 4
Future Value – Compounded Annually 2..................................................................................... 5
Future Value – Compounded Annually 3..................................................................................... 5
Future Value – Compounded Monthly 1 ...................................................................................... 5
Future Value – Compounded Monthly 2 ...................................................................................... 6
Future Value – Compounded Monthly 3 ...................................................................................... 6
Present Value ............................................................................................................................ 7
10)
11)
12)
13)
14)
15)
16)
17)
III.
Payment .................................................................................................................................. 10
18)
19)
IV.
Present Value – Compounded Annually 1 ................................................................................ 7
Present Value – Compounded Annually 2 ................................................................................ 7
Present Value – Compounded Annually 3 ................................................................................ 7
Present Value – Compounded Semiannually 1 ....................................................................... 8
Present Value – Compounded Semiannually 2 ....................................................................... 8
Present Value – Compounded Semiannually 3 ....................................................................... 8
Present Value – Compounded Monthly 1.................................................................................. 9
Present Value – Compounded Monthly 2 .................................................................................. 9
Payment Example 1 ...................................................................................................................... 10
Payment Example 2 ...................................................................................................................... 10
Term Calculations ................................................................................................................ 10
20)
21)
22)
23)
24)
Term Calculation – Compounded Annually 1 ........................................................................ 10
Term Calculation – Compounded Annually 2 ........................................................................ 11
Term Calculation – Compounded Annually 3 ........................................................................ 11
Term Calculation – Compounded Monthly 1 ......................................................................... 11
Term Calculation – Compounded Monthly 2 ......................................................................... 12
V. Solving for Interest Rates .................................................................................................... 12
25)
26)
VI.
Annuities ................................................................................................................................. 13
27)
28)
29)
30)
31)
32)
2
Interest Rates Example 1 ............................................................................................................ 12
Interest Rates Example 2 ............................................................................................................ 12
Future Value of an Ordinary Annuity – Compounded Annually ....................................... 13
Future Value of an Annuity Due – Compounded Annually................................................. 13
Present Value of an Ordinary Annuity – Compounded Annually ..................................... 13
Present Value of an Ordinary Annuity – Compounded Semiannually ............................ 14
Present Value of an Annuity Due – Compounded Annually............................................... 14
Present Value of an Annuity Due – Compounded Monthly ................................................ 14
Financial Ratios, Formulas, and Calculations – 2017
33)
34)
35)
VII.
Bond Valuation .................................................................................................................... 16
36)
37)
38)
39)
VIII.
Bond Concept 1 .............................................................................................................................. 16
Bond Concept 2 .............................................................................................................................. 16
Bond Valuation 1 ........................................................................................................................... 16
Bond Valuation 2 ........................................................................................................................... 16
Equity Valuation ................................................................................................................ 17
40)
41)
IX.
Annuity Due with Lump Sum Deposit – Compounded Monthly ....................................... 15
PV of Inflation-Adjusted Retirement Needs of an Annuity Due........................................ 15
PV of Inflation-Adjusted Annuity Due – Compounded Annually ...................................... 15
Equity Valuation 1......................................................................................................................... 17
Equity Valuation 2......................................................................................................................... 17
Return Calculations.............................................................................................................. 18
42)
43)
44)
45)
46)
Return Calculation 1..................................................................................................................... 18
Return Calculation 2..................................................................................................................... 18
Return Calculation 3..................................................................................................................... 18
Return Calculation 4..................................................................................................................... 18
Return Calculation 5..................................................................................................................... 19
X. Mortgage Payments ............................................................................................................... 19
47)
48)
49)
50)
51)
XI.
Mortgages – Monthly Payment .................................................................................................. 19
Mortgages – Interest..................................................................................................................... 19
Mortgages – Principal .................................................................................................................. 20
Mortgages – Interest Deduction ................................................................................................ 20
Mortgage – Total Interest Payment.......................................................................................... 20
Cost of Capital ........................................................................................................................ 21
52) Cost of Capital – Cost of Equity ........................................................................................................ 21
53) Cost of Capital – Weighted Average Cost of Capital.............................................................. 21
XII.
54)
55)
56)
57)
XIII.
58)
59)
60)
XIV.
Net Present Value ................................................................................................................ 21
NPV Concept 1 ................................................................................................................................ 21
NPV Concept 2 ................................................................................................................................ 22
NPV Example 1 ............................................................................................................................... 22
NPV Example 2 ............................................................................................................................... 22
Internal Rate of Return .................................................................................................... 23
IRR Concept 1 ................................................................................................................................. 23
IRR Concept 2 ................................................................................................................................. 23
IRR Example 1 ................................................................................................................................ 23
Statistics ............................................................................................................................... 24
61) Statistics – Normal Distribution ................................................................................................ 24
62)
Statistics - Sharpe Ratio....................................................................................................................................24
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Financial Ratios, Formulas, and Calculations – 2017
I. Future Value
1)
FV Concepts 1
The process of discounting or finding the present value of a cash flow to be received in the
future is really the reverse of compounding. True or False?
2)
FV Concepts 2
One of the potential benefits of investing early for retirement is that an investor can receive
greater benefits from the compounding of interest. True or False?
3)
FV Concepts 3
Disregarding risk, if money has time value, the future value of some amount of money always
will be more than the amount originally invested, and the present value of some amount to be
received in the future is always less than that future amount to be received. True or False?
4)
Future Value – Compounded Annually 1
John purchased a rare gold coin for $100,000. He expects it to increase in value at a rate of 8.5%
compounded annually for the next 7 years. How much will the coin be worth at the end of the
seventh year if his expectations are correct?
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Financial Ratios, Formulas, and Calculations – 2017
5)
Future Value – Compounded Annually 2
What is the future value of $11,000 invested for 30 years assuming an annual interest rate of
12.5%?
6)
Future Value – Compounded Annually 3
What is the future value of $13,500 invested for 15 years assuming an annual interest rate of
6.5%?
7)
Future Value – Compounded Monthly 1
Fred invested $25,000 in an interest-bearing promissory note earning a 9% annual rate of
interest compounded monthly. How much will the note be worth at the end of 8 years assuming
all interest is reinvested at the 9% rate?
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Financial Ratios, Formulas, and Calculations – 2017
8)
Future Value – Compounded Monthly 2
What is the future value of $15,000 invested for 18 years assuming an annual interest rate of
12%, compounded monthly?
9)
Future Value – Compounded Monthly 3
Calculate the future value of $1,000 invested for 100 years assuming an annual interest rate of
8%, compounded monthly.
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Financial Ratios, Formulas, and Calculations – 2017
II.
Present Value
10)
Present Value – Compounded Annually 1
Sandy wants to give her daughter $50,000 to start her own business in 10 years. How much
should she invest today at an annual interest rate of 9% (compounded annually) to have
$50,000 in 10 years?
11) Present Value – Compounded Annually 2
Calculate the present value of $10 million to be received in 50 years assuming an annual interest
rate of 10%.
12)
Present Value – Compounded Annually 3
What is the present value of $75,000 to be received in 20 years if the estimated annual interest
rate is going to be 6%?
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Financial Ratios, Formulas, and Calculations – 2017
13)
Present Value – Compounded Semiannually 1
John expects to receive $105,000 in 7 years. What is the current value of this sum if it is
discounted at 9% compounded semiannually?
14)
Present Value – Compounded Semiannually 2
Penelope expects to receive $10,000,000 from a rich uncle in 7 years. What is the current value
of this sum if it is discounted at 6% compounded semiannually?
15)
Present Value – Compounded Semiannually 3
Robert expects to receive $400,000 from a trust fund in 12 years. What is the current value of
this fund if it is discounted at 8% compounded semiannually?
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Financial Ratios, Formulas, and Calculations – 2017
16)
Present Value – Compounded Monthly 1
Billy Mack expects to receive $105,000 in 9 years. His opportunity cost is 10% compounded
monthly. What is this sum worth to Billy today?
17)
Present Value – Compounded Monthly 2
Calculate the present value of $30,000 to be received in 5 years assuming an annual interest
rate of 12%, compounded monthly.
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Financial Ratios, Formulas, and Calculations – 2017
III.
Payment
18) Payment Example 1
If a 5-year regular annuity has a present value of $1,000, and if the interest rate is 10 percent,
what is the amount of each annuity payment?
19)
Payment Example 2
Suppose the present value of a 2-year ordinary annuity is $100. If the discount rate is 10
percent, what must be the annual cash flow?
IV.
Term Calculations
20) Term Calculation – Compounded Annually 1
Terry purchased a painting for $28,000. Today, he sold the painting for $50,000. Terry estimated
his average annual opportunity cost on the painting was 11%. Approximately how many periods
did Terry own the painting?
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Financial Ratios, Formulas, and Calculations – 2017
21)
Term Calculation – Compounded Annually 2
Paul purchased an antique for $15,000. Today, he sold the antique for $71,000. Paul estimated
the average annual compound rate of return on the antique was 9%. Approximately how many
periods did Kerry own the antique?
22)
Term Calculation – Compounded Annually 3
Barry purchased several gold coins for $3,000. Today, he sold the coins for $5,500. Barry
estimated the average annual rate of return (compounded monthly) on the coins was 15%.
Approximately how many periods did Barry own the coins?
23)
Term Calculation – Compounded Monthly 1
Today, Stewart put all of his cash into an account earning an annual interest rate of 10%
compounded monthly. Assuming he makes no withdrawals from or additions to this account.
Approximately how many periods must Stewart wait to double his money? (Suggestion: Use $1
as the amount to be invested.)
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Financial Ratios, Formulas, and Calculations – 2017
24)
Term Calculation – Compounded Monthly 2
Today, Robert put all of his cash into an account earning an annual interest rate of 15%
compounded monthly. Assuming he makes no withdrawals from or additions to this account.
Approximately how many periods must Robert wait to double his money? (Suggestion: Use $1
as the amount to be invested.)
V.
Solving for Interest Rates
25) Interest Rates Example 1
In the 60’s, the average tuition for one year at a state university was $1,800. Thirty years later,
in 1988, the average cost was $13,700. What was the growth rate in tuition over the 30-year
period?
26)
Interest Rates Example 2
Assuming you need $50,000 five years from now to buy a new Lexus and you currently have
$25,000, what annual rate of return would you need to achieve your goal?
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Financial Ratios, Formulas, and Calculations – 2017
VI.
Annuities
27)
Future Value of an Ordinary Annuity – Compounded Annually
Rudy has been investing $1,000 at the end of each year for the past 20 years in an equity mutual
fund. How much is the fund worth now assuming he has earned 12.5% compounded annually on
his investment?
28)
Future Value of an Annuity Due – Compounded Annually
Hector has been investing $2,000 at the beginning of each year for the past 27 years. How much
has he accumulated assuming he has earned 8% compounded annually on his investment?
29)
Present Value of an Ordinary Annuity – Compounded Annually
Phillip expects to receive $8,000 at the end of each of the next 5 years from a trust fund. His
opportunity cost is 11.5% compounded annually. What are these payments worth today?
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Financial Ratios, Formulas, and Calculations – 2017
30)
Present Value of an Ordinary Annuity – Compounded Semiannually
Jim, who was injured in an automobile accident, won a judgment that provides him $3,500 at
the end of each 6-month period for the next 6 years. If the escrow account that holds Jim’s
settlement award earns an average annual rate of 11 % compounded semiannually, how much
was the defendant initially required to deposit so that Jim would be compensated for his
injuries?
31)
Present Value of an Annuity Due – Compounded Annually
Julie wants to withdraw $10,000 at the beginning of each year for the next 9 years. She wants to have
$50,000 left at the end of the 9 years. She expects to earn 11.5% compounded annually on her
investment. What lump sum should Julie deposit today?
32)
Present Value of an Annuity Due – Compounded Monthly
Carla wants to withdraw $5,000 at the beginning of each month for the next 10 years. She
expects to earn 10% compounded monthly on her investments. What lump sum should she
deposit today?
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Financial Ratios, Formulas, and Calculations – 2017
33)
Annuity Due with Lump Sum Deposit – Compounded Monthly
Alex received an inheritance of $500,000. He wants to withdraw equal periodic payments at the
beginning of each month for the next 15 years. He expects to earn 12% compounded monthly
on his investments. How much can he receive each month?
34)
PV of Inflation-Adjusted Retirement Needs of an Annuity Due
Billy and Marlene are ready to retire. They want to receive the equivalent of $25,000 in today’s
dollars at the beginning of each year for the next 20 years. They assume inflation will average
4% over the long run, and they can earn 10% compounded annually on investments. What lump
sum do they need to invest today to attain their goal?
35)
PV of Inflation-Adjusted Annuity Due – Compounded Annually
Barbara needs an income stream equivalent to $50,000 in today’s dollars at the beginning of
each year for the next 12 years to maintain her standard of living. Assume inflation will average
4.5% and she can earn 12% compounded annually on investments. What lump sum does she
need to invest today to fund her needs?
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Financial Ratios, Formulas, and Calculations – 2017
VII.
Bond Valuation
36)
Bond Concept 1
As interests rates increase, the value of a bond increases. True or False?
37)
Bond Concept 2
When the coupon rate of a bond exceeds the going market rate of interest for a comparable
bond with similar characteristics, does the bond trade at a discount, a premium, or at par?
38) Bond Valuation 1
You intend to purchase a 10-year, $1,000 face value bond that pays interest of $60 every 6
months. If your single annual required rate of return is 10 percent with semiannual
compounding, how much should you be willing to pay for this bond?
39)
Bond Valuation 2
Assume that you wish to purchase a 20-year bond that has a maturity value of $1,000 and
makes semiannual interest payments of $40. If you require a 10 percent simple yield to maturity
on this investment, what is the maximum price you should be willing to pay for the bond?
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Financial Ratios, Formulas, and Calculations – 2017
VIII.
Equity Valuation
40)
Equity Valuation 1
You believe that for the next five years, the annual dividends are expected to be $5.00, $5.50,
$6.00, $6.50, and $7.25. Furthermore, your fundamental analysis points towards a price of $135
five years from now. If the cost of equity is still 9% percent, what should be your estimate of
intrinsic value today?
41)
Equity Valuation 2
The following table shows the calculations for FCFE.
Estimates for XYZ Corporation
Year
Sales Growth Rate
Sales Per Share
Net Profit Margin
Earnings Per Share
Net FCInv Per Share
WCInv Per Share
Debt Financing Per Share
FCFE Per Share
Growth Rate of FCFE
1
2
3
4
5
50.00
10.0%
5.00
2.50
1.25
1.40
2.65
25.0%
62.50
10.0%
6.25
4.38
1.56
1.68
1.99
-25%
25.0%
78.13
10.0%
7.81
5.47
1.95
2.02
2.41
21%
25.0%
97.66
10.0%
9.77
3.42
2.44
0.73
4.63
92%
10.0%
107.42
10.0%
10.74
3.76
2.69
0.77
5.07
9%
The stock value is the present value of the first five years’ FCFE plus the present value of the
terminal value in the fifth year. The terminal value is calculated as follows
TV5 = FCFE6(r - g) = 5.07/(0.105 – 0.03) = $67.60. (Note, the cost of equity is 10.5%)
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Financial Ratios, Formulas, and Calculations – 2017
IX.
Return Calculations
42)
Return Calculation 1
Stock A has annual returns of 10.7%, 12.9% and 23.2% for the fiscal years ended December 31,
of 2005, 2006, and 2007. What was the arithmetic average return over this three-year period?
43)
Return Calculation 2
Stock A has annual returns of 20.7%, and -20.7% for the fiscal years ended December 31, 2006,
and 2007. What was the arithmetic average return over this two-year period?
44)
Return Calculation 3
Stock B has annual returns of 10.7%, 12.9% and 23.2% for the fiscal years ended December 31,
of 2005, 2006, and 2007. What was the geometric average return over this three-year period?
45)
Return Calculation 4
Stock A has annual returns of 20.7%, and -20.7% for the fiscal years ended December 31, of
2006, and 2007. What was the geometric average return over this two-year period?
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Financial Ratios, Formulas, and Calculations – 2017
46)
Return Calculation 5
You invested $1,000,000 ten years ago. You now have $1,750,000. What was your average
annual geometric rate of return over the 10-year period?
X. Mortgage Payments
47)
Mortgages – Monthly Payment
Ed Johnson recently purchased a house for $250,000. He made a down payment of 20% and
financed the balance over 30 years at 7%. How much will Ed’s monthly mortgage payment be?
48)
Mortgages – Interest
Ed Johnson recently purchased a house for $250,000. He made a down payment of 20% and
financed the balance over 30 years at 7%. How much of the first payment will go towards
interest?
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Financial Ratios, Formulas, and Calculations – 2017
49)
Mortgages – Principal
Ed Johnson recently purchased a house for $250,000. He made a down payment of 20% and
financed the balance over 30 years at 7%. How much of the first payment will go towards
principal?
50)
Mortgages – Interest Deduction
Ed Johnson recently purchased a house for $250,000 on June 30th. He made a down payment of
20% and financed the balance over 30 years at 7%. How much interest will Ed be able to deduct
on his tax return this year?
51)
Mortgage – Total Interest Payment
Ed Johnson recently purchased a house for $250,000 on June 30th. He made a down payment of
20% and financed the balance over 30 years at 7%. How much total interest will he pay over the
life of the loan? So, how many times (total interest and principal) will he pay for his house over
the life of the loan?
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Financial Ratios, Formulas, and Calculations – 2017
XI.
Cost of Capital
52)
Cost of Capital – Cost of Equity
Using the following information, estimate the cost of equity for XYZ Corporation. (Hint – Use
the CAPM)
• XYZ’s beta is 1.2, the risk-free rate is 4 percent and the market risk premium is 5.5 percent.
• XYZ’s expected long-term growth rate is 3%.
53)
Cost of Capital – Weighted Average Cost of Capital
Kline Company (KC) has a cost of debt of 7.0 (pre-tax) percent and a cost of equity of 12.0
percent. The firm’s capital structure is 35% debt and 65% equity and expects this structure to
remain in the future. The tax rate is 35 percent. Based on the information presented above,
what is the weighted average cost of capital for KC?
XII.
Net Present Value
54) NPV Concept 1
If the calculated NPV is negative, then which of the following must be true? The discount rate
used is:
A. Equal to the internal rate of return.
B. Too high.
C. Greater than the internal rate of return.
D. Too low.
E. Less than the internal rate of return,
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Financial Ratios, Formulas, and Calculations – 2017
55)
NPV Concept 2
The internal rate of return of a capital investment:
a. Changes when the required rate of return changes.
b. Is equal to the annual net cash flows divided by one half of the project’s cost when the
cash flows are an annuity.
c. Must exceed the required rate of return in order for the firm to accept the investment.
d. Is similar to the yield to maturity on a bond.
e. Answers c and d are both correct.
56)
NPV Example 1
What is the net present value of a project that has a $200 start-up cost, expected cash flows of
$235, -$65, and $300, in year 1, 2 and 3, respectively? (Assume a cost of capital of 11.5%)
57)
NPV Example 2
Given the following cash flows, and assuming a 15% required rate of return, which project
would you select?
Year
0
1
2
3
22
Project X
-200,000
75,000
95,000
105,000
Project Y
-100,000
45,000
44,000
43,000
Financial Ratios, Formulas, and Calculations – 2017
XIII.
58)
Internal Rate of Return
IRR Concept 1
The internal rate of return is the discount rate which equates the present value of the cash
outflows (or costs) with the present value of the cash inflows. True or False?
59)
IRR Concept 2
The internal rate of return (IRR) method equates the present value of cash inflows with the
present value of cash outflows by discounting all of the cash flows at the firm’s weighted
average cost of capital. True or False?
60)
IRR Example 1
An insurance firm agrees to pay you $3,310 at the end of 20 years if you pay premiums of $100
per year at the end of each year of the 20 years. Find the internal rate of return to the nearest
whole percentage point.
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Financial Ratios, Formulas, and Calculations – 2017
XIV.
61)
Statistics
Statistics – Normal Distribution
ABC Company has a mean return of 11% for the past five years and a standard deviation of 8%.
Assuming a normal distribution, what is the chance that ABC Company will have a return less
than 3%?
62)
Statistics - Sharpe Ratio
You are trying to decide which mutual fund to invest in, YeeHaw MF or Castle in the Sky MF.
YeeHaw has an average annual return of 15% for the trailing 10 year period, and a standard
deviation of 7.5. Castle in the Sky has an average annual return of 17% for the trailing 10 year
period, and a standard deviation of 15. The risk free rate, on average, over the same period of
time was 4.65%. Which fund would you select, assuming that you are attempting to maximize
returns per unit of risk?
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Financial Ratios, Formulas, and Calculations – 2017