Review
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80 minute exam: 6:30‐7:50pm ET, Wednesday, Oct 20
60 minute class 7:40pm‐9:00pm
One two‐sided sheet of paper allowed with formulas, notes etc.
No discussion with anyone
Calculator
A few conceptual questions: Any topics discussed in class (reading book is enough)
Numerical problems
Introduction – 1, 2
TVM – 3, 4, 5, 6, 9
Capital Budgeting – 7, 8
CAPM ‐ 10,11
Page 1 of 19
Fin 638: Midterm Review
Time Value of Money
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤
𝐶
1
𝑟
𝐶
1
𝑟
Arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making any investment. A
normal market is a competitive market in which there are no arbitrage opportunities.
The Law of One Price states that if equivalent investment opportunities trade simultaneously in different competitive
markets, then they must trade for the same price in both markets.
Page 2 of 19
Fin 638: Midterm Review
𝑭𝒖𝒕𝒖𝒓𝒆 𝑽𝒂𝒍𝒖𝒆
𝑪𝟎
𝟏
𝒓
𝒏
𝑪𝟏
𝟏
𝒓
𝒏 𝟏
⋯
𝑪𝒏
𝟏
𝟏
𝒓
𝑪𝒏
Example: Suppose we plan to save $1000 today and $1000 at the end of each of the next two years. If we earn a fixed
10% interest rate on our savings, how much will we have three years from today?
Year 1
Date
Cash Flow
Year 2
Year 3
0
1
2
1000
1000
1000
3
Future value = $1000 × 1.13 + $1000 × 1.12 + $1000 × 1.1 = $3641
𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆
𝑪𝟎
𝑪𝟏 / 𝟏
𝒓
⋯
𝑪𝒏 𝟏 / 𝟏
𝒓
𝒏 𝟏
𝑪𝒏 / 𝟏
𝒓
𝒏
Page 3 of 19
Fin 638: Midterm Review
Perpetuities and Annuities
A perpetuity is a stream of equal (or constant) cash flows that occur at regular intervals and last forever.
𝑪
𝑪
𝑪
𝑪
⋯
𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂 𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚
𝟏 𝒓 𝟐
𝟏 𝒓 𝟑
𝒓
𝟏 𝒓
Note: The first payment or cash flow of ‘C’ occurs at the end of the first period (or in one period from today.
A growing perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate forever.
𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂 𝑮𝒓𝒐𝒘𝒊𝒏𝒈 𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚
𝑪
𝟏
𝑪
𝒓
𝟏
𝟏
𝒈
𝒓
𝟐
𝑪
𝟏
𝟏
𝒈
𝒓
𝟑
𝟐
⋯
𝑪
𝒓
𝒈
Example: How much should you pay for a stream of dividends from a stock. The first dividend of $5 will be paid two
years from today. Subsequent dividends will be paid annually and will grow at 2% per year. The discount rate is 12%
per year.
5
/1.12 $44.64
𝑃
0.12 0.02
Page 4 of 19
Fin 638: Midterm Review
An annuity is a stream of equal (or constant) cash flows that lasts for a finite period of time. The cash flows occur at
regular intervals.
𝑪
𝑪
𝑪
𝟏
𝑪
⋯
𝟏
𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝑨𝒏𝒏𝒖𝒊𝒕𝒚
𝟏 𝒓 𝟐
𝟏 𝒓 𝑵 𝒓
𝟏 𝒓 𝑵
𝟏 𝒓
𝑭𝒖𝒕𝒖𝒓𝒆 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒂𝒏 𝑨𝒏𝒏𝒖𝒊𝒕𝒚
𝑪
𝟏
𝒓
𝒏 𝟏
𝑪
𝟏
𝒓
𝒏 𝟐
⋯
𝑪
𝟏
𝒓
𝑪
𝑪
𝒓
𝟏
𝒓
𝑵
𝟏
𝑷𝒓𝒊𝒏𝒄𝒊𝒑𝒂𝒍 𝒓
.
𝟏
𝟏
𝟏 𝒓 𝑵
Example: The interest rate is 5% for the next one year, 4% for the year after that, and 3% for the year after that. Consider
a cash flow of $1000 after two years. What is its value three years from now?
𝑪𝒂𝒔𝒉 𝑭𝒍𝒐𝒘 𝒐𝒇 𝒂𝒏 𝑨𝒏𝒏𝒖𝒊𝒕𝒚
$1000 × 1.03 = $1030
In this problem, if we had to find present value of an annuity that pays $100 in each of the next 3 years, we won’t be
able to use the annuity formulas because of changing interest rate.
𝑃𝑉
1
100
1.05
1
100
1.04
100
1.03
$275.72
Page 5 of 19
Fin 638: Midterm Review
Interest Rates
Example: Convert
a) 5% every six months to a rate with annual compounding
b) 5% every six months to a rate with quarterly compounding
c) 5% every six months to a rate with monthly compounding
1
1.05
1.05
.
1.05
/
1
1
0.1025
0.024695
0.008165
10.25% per annum
2.4695% per quarter
0.8165% per month
Nominal interest rate: the rate at which the money will grow. Real interest rate: growth rate of purchasing power.
Inflation rate: rate at which prices increase over time.
1 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑎𝑡𝑒
1 𝑅𝑒𝑎𝑙 𝑅𝑎𝑡𝑒 1 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒
Page 6 of 19
Fin 638: Midterm Review
Bonds
-
Bond Certificate states the terms of the bond (including the dates and amounts of all payments from the bond)
Maturity Date is the final repayment date
Term is the time remaining until the final repayment date
Coupons are the promised interest payments (typically paid semi‐annually)
Face Value is the principal amount paid at maturity (and determines the coupon payments)
Coupon Rate (set by the issuer and expressed as an APR) determines the amount of each coupon payment. The
coupon payment (CPN) per period equals:
CPN 
Coupon Rate  Face Value
Number of Coupon Payments per Year
As interest rates and bond yields rise, bond prices fall. As interest rates and bond yields fall, bond prices rise
When the bond price is
We say the bond trades
This occurs when
greater than the face value
“above par” or “at a premium”
Coupon Rate > Yield to Maturity
equal to the face value
“at par”
Coupon Rate = Yield to Maturity
less than the face value
“below par” or “at a discount”
Coupon Rate < Yield to Maturity
Example: ABC Chemicals Inc. issued 8% bonds on December 31, 1993. The bonds had a maturity of 20 years and were
issued at par. The 20th coupon payment was due on December 31, 2003. To everyone’s surprise, the company defaulted
on this payment. The bonds were trading at $1150 each just before this news was made public. The company disclosed
Page 7 of 19
Fin 638: Midterm Review
that it is under short‐term financial distress. After negotiations with the bondholders, it has agreed to resume coupon
payments starting June 30, 2004 at a higher rate to compensate for the missed coupon payment.
Note: Express all interest rates in percentage terms to two decimal places.
1. What was the stated yield to maturity offered by the bonds at the time of issue?
2. Jessica had purchased 10 bonds at the time of issue. She sold the bonds on December 31, 2003 at $1150 each.
What is the semiannual rate of return that she earned on her investment?
3. At what stated yield to maturity were the bonds trading on December 31, 2003 before the news of default became
public? (Hint: The price of $1150 is inclusive of the coupon due immediately.)
4. The bondholders want to make sure that they do not suffer in the restructuring of the bonds. They want their
bonds to have the same price of $1150 with the new terms. What is the new coupon rate that will achieve this?
Assume that the required rate on bonds is unaffected by the news of default.
1. Bonds were issued at par. This means the yield to maturity equals the coupon rate. The stated YTM is 8% per
annum compounded semiannually (corporate bonds have semiannual coupons).
2. The fact that Jessica bought 10 bonds is irrelevant. Let us calculate her return on each bond. For each bond, she
paid $1000, got 19 coupons of $40 each, and sold the bond for $1150 on the date of the 20th coupon payment
without actually getting the coupon. We want to solve the following equation:
40
40
40
1150
1000
⋯
1 𝑟
1 𝑟
1 𝑟
1 𝑟
We need to use a financial calculator or a spreadsheet to solve this. It is easier if we can convert the cash flows to
a standard annuity without the missing last coupon. Use the trick of dividing last payment of $1150 into $40 that
makes up for the last coupon plus an extra $1110. Then, use the following inputs in financial calculator or
spreadsheet: PV = ‐1000, PMT = 40, n = 20, FV = 1110 and solve for rate. The solution is 4.356% per six months.
3. The price of $1150 is the present value of immediate and future coupons and face value at maturity:
Page 8 of 19
Fin 638: Midterm Review
1150
40
40
1
𝑟
40
1 𝑟
⋯
40
1 𝑟
1040
1 𝑟
That means 1110 is the present value of all future cash flows. The above equation can be rewritten as
40
40
1040
⋯
1 𝑟
1 𝑟
1 𝑟
1 𝑟
Use the following inputs in financial calculator or spreadsheet: PV = ‐1110, PMT = 40, n = 20, FV = 1000 and solve
for rate. The solution is 3.244% per six months or a stated YTM of 6.488% per annum compounded semiannually.
4. The new coupon C must be set so that the present value of all future cash flows remains 1150. That is,
𝐶
𝐶
𝐶 1000
𝐶
⋯
1150
1.03244
1.03244
1.03244
1.03244
You can solve this equation on a regular calculator. Or use the following inputs in financial calculator or
spreadsheet: PV = ‐1150, n = 20, FV = 1000, rate = 3.244%, and solve for PMT. The answer is $42.75. The annual
coupon rate is 8.55%.
1110
40
Page 9 of 19
Fin 638: Midterm Review
Stocks
If P0 is the price today, Div1 is the dividend paid at the end of the year, and P1 is the price at the end of the year
𝑃
𝐷𝑖𝑣
1 𝑟
or
𝑃
𝐷𝑖𝑣
𝑃
1 𝑟
𝐷𝑖𝑣
1 𝑟
⋯
𝐷𝑖𝑣
1 𝑟
𝐷𝑖𝑣
1 𝑟
𝐷𝑖𝑣
1 𝑟
𝑃
1
𝑟
⋯
Example: The next dividend (in 1 year) will be $2.00 (a share). Dividends are expected to grow at 20% in the second
year, at 20% in the third year, and at 8% per year (forever) thereafter. If the equity cost of capital is 15%, what is the
value (per share) of the stock today?
We calculate the present value of the growing perpetuity that starts with the third dividend of $2.88. That present value
is in year 2 dollars. Add that to the dividend of $2.4 in year 2 and discount both by 2 years. Separately discount the initial
dividend of $2 by one year.
2
1
2 1.2
2 1.2
$34.66
𝑃
1.15 1.15
0.15 0.08
Page 10 of 19
Fin 638: Midterm Review
Capital Budgeting
Depreciation is not in itself a cash flow. So we add back depreciation to income to get cash flow. Further, depreciation
lowers taxes.
𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑇𝑎𝑥 𝑆ℎ𝑖𝑒𝑙𝑑
𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛
𝑇𝑎𝑥 𝑅𝑎𝑡𝑒
Capital Expenditure is purchase of fixed assets. Cash flow consequences of capital expenditure are:
 Cash outflow from purchase
 Depreciation Tax Shield
 Sale or Salvage
𝑁𝑒𝑡 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠 – 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
𝐶𝑎𝑠ℎ
𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝑅𝑒𝑐𝑒𝑖𝑣𝑎𝑏𝑙𝑒𝑠
𝑃𝑎𝑦𝑎𝑏𝑙𝑒𝑠
Example: A project will result in revenue of $50,000 in year 1 and $75,000 in year 2. Cost of goods sold is 60% of
revenue. The project will use a machine that could otherwise be sold today for $5000. The machine will be depreciated
by $5000 in year 1 after which it will be worthless. A feasibility study was conducted for the project and it cost $2000.
The project requires net working capital equal to 10% of revenue in each year. The tax rate is 20% and the cost of
capital is 10%.
Page 11 of 19
Fin 638: Midterm Review
NWC
0
5000
7500
0
Year
Revenue
COGS
Depreciation
Taxable Income
Taxes
Unlevered Net
Income
0
1
50000
30000
5000
15000
3000
12000
2
75000
45000
3
0
Add Back
Depreciation
Less Capital
Expenditure
Less Change in NWC
Free Cash Flow
30000
6000
24000
5000
5000
5000
‐10000
2500
14500
‐7500
31500
14500 31500
$29,214.88
1.1
1.1
Tip: After such problems (actually most problems), check if the answer seems reasonable. Suppose you did a
calculation mistake and got an answer of ‐$15,000 or $292,000, comparing with the revenue or income numbers in the
problem should alert you that there is something wrong.
𝑁𝑃𝑉
10000
Page 12 of 19
Fin 638: Midterm Review
Example: A company is thinking of launching product Y. It expects to sell Y for the next 5 years. The sales are expected
to be $2,000,000 in the first four years and $1,000,000 in the final year. The operating costs in any year will be 50% of
revenues. It is expected that 20% of sales will come from customers who switch from current product X of the firm.
The operating costs of product X are 70% of revenues. The initial investment for launching product Y is $2M and can be
depreciated straight line over 5 years. The corporate tax rate is 40%. Find NPV at 12%.
We can create a table for cash flows but here we will just write all the details in text.
Initial cash flow C0 = ‐2,000,000 (initial investment, note no tax effect of that).
In each of the next four years, we have operating profit from sales of product Y is 50% of $2,000,000 = $1,000,000. The
sales of product X will decline by 20% of $2,000,000 = $400,000. The lost profit from reduction in sales of product X is
30% of $400,000 = $120,000. Thus, incremental operating profit is $880,000.
The cash flow impact of this incremental operating profit is (1 – 40%) × $880,000 = $528,000.
The depreciation of $400,000 each year will result in tax savings of 40% × $400,000 = $160,000 (note we multiplied by
tax rate here but by 1‐tax rate in the previous line).
Total incremental free cash flow in each of years 1 through 4 = $528,000 + $160,000 = $688,000.
The only difference in 5th year is that sales decline. Operating profit on product X = 50% of $1,000,000 = $500,000. Lost
operating profit on product Y = 30% of $200,000 = $60,000. Incremental operating profit = $440,000. After‐tax value =
(1 – 40%) × $440,000 = $264,000.
Incremental free cash flow in year 5 = $264,000 + $160,000 = $424,000.
𝑁𝑃𝑉
2000000
688000
1.12
688000
1.12
688000
1.12
688000
1.12
424000
1.12
$330,285
Page 13 of 19
Fin 638: Midterm Review
Risk and Return
Example: Calculating Stock Returns
The dividend payments and prices of a stock are given below:
Year
1999
2000
2001
2002
2003
2004
Year-end Dividend
Year-end Price
$50
$52
$64
$71
$77
$91
$1
$1
$1
$1.1
$1.4
1. What was the average return of the stock?
2. What was the variance of return on the stock?
3. What was the standard deviation of return on the stock?
1. The returns are
52
50
50
1
6%,
10%,
𝑅
2. Variance
0.06
𝜎
0.147
0.25
64
91
52
52
77
77
6%
0.147
1
25%,
1.4
25%
71
64
64
1
12.5%,
77
71
71
1.1
20%.
12.5%
5
10%
0.125 0.147
4
3. Standard deviation is square root of variance:
𝜎 √0.00592
0.0769
20%
0.1
14.7%
0.147
0.2
0.147
0.00592
7.69%
Page 14 of 19
Fin 638: Midterm Review
Example: Calculating Bond Returns
The interest payments and prices of a bond are given below:
Year
1999
2000
2001
2002
2003
2004
Year-end Interest Payment
Year-end Price
$1000
$980
$1,030
$1,065
$1,050
$1,000
$50
$50
$50
$50
$50
1. What was the average return of the bond?
2. What was the variance of return on the bond?
3. What was the standard deviation of return on the bond?
1. The returns are
980
1000
1000
980
50
1065 1030
50
10.2%,
980
1030
1050 1065
50
1100 1050
50
8.25%,
3.29%,
9.52%.
1065
1050
50
𝑅
𝜎
2. Variance
0.03 0.0685
0.102
3%,
3%
0.0685
1030
10.2%
8.25%
5
0.0825
0.001197
3. Standard deviation is square root of variance:
𝜎 √0.001197
3.29%
0.0685
4
0.0346
9.52%
0.0329
6.85%
0.0685
0.0952
0.0685
3.46%
Page 15 of 19
Fin 638: Midterm Review
Example: Historical Risk and Return
The returns of stocks A, B, and C in five successive years are given below.
Year
2000
2001
2002
2003
2004
1.
2.
3.
4.
5.
A’s return
11%
7%
18%
5%
9%
B’s return
16%
5%
21%
6%
12%
C’s return
9%
6%
12%
6%
7%
What was the average return of stock A?
What was the variance of return on stock A?
What was the standard deviation of return on stock A?
What was the average return of a portfolio P that invested 40% in A, 30% in B, and 30% in C?
What was the standard deviation of return of portfolio P?
1. Average return of A
11%
𝑅
7%
18%
5
0.18
0.1
2. Variance of A
𝜎
0.11
0.1
0.07
0.1
5%
9%
0.05
10%
0.1
4
3. Standard deviation of A
𝜎
√0.003738
0.0611
0.09
0.1
0.003738
6.11%
4. Portfolio returns are
2000: 0.4 × 11% + 0.3 × 16% + 0.3 × 9% = 11.9%
Page 16 of 19
Fin 638: Midterm Review
2001: 0.4 × 7% + 0.3 × 5% + 0.3 × 6% = 6.1%
2002: 0.4 × 18% + 0.3 × 21% + 0.3 × 12% = 17.1%
2003: 0.4 × 5% + 0.3 × 6% + 0.3 × 6% = 5.6%
2004: 0.4 × 9% + 0.3 × 12% + 0.3 × 7% = 9.3%
𝑅
11.9%
6.1%
17.1%
5
5.6%
9.3%
10%
5. Standard deviation of portfolio
𝜎
0.119
0.1
0.061
0.1
𝜎
0.171 0.1
4
√0.002227
0.056
0.04719
0.1
0.093
0.1
0.002227
4.719%
Page 17 of 19
Fin 638: Midterm Review
Example: Expected Return and Risk
The probability distribution of returns for stocks A, B, and C are given below.
State
Boom
Normal
Bust
1.
2.
3.
4.
5.
Probability
0.3
0.4
0.3
A’s return
15%
10%
5%
B’s return
20%
15%
0%
C’s return
7%
7%
7%
What is the expected return of stock A?
What is the variance of return on stock A?
What is the standard deviation of return on stock A?
What is the average return of a portfolio P that invests $400 in A, $400 in B, and $200 in C?
What is the standard deviation of return of portfolio P?
1.
0.3
𝐸𝑅
15%
0.4
10%
0.3
0.4
0.1
0.1
0.3
5%
10%
2.
𝜎
0.3
0.15
0.1
0.05
0.1
0.0015
3.
√0.0015 0.03873 3.873%
𝜎
4. The portfolio weights of A, B, and C are 400/1000 = 0.4, 400/1000 = 0.4, and 200/1000 = 0.2, respectively.
Boom: RP = 0.4 × 15% + 0.4 × 20% + 0.2 × 7% = 15.4%
Normal: RP = 0.4 × 10% + 0.4 × 15% + 0.2 × 7% = 11.4%
Bust: RP = 0.4 × 5% + 0.4 × 0% + 0.2 × 7% = 3.4%
𝐸𝑅
0.3
15.4%
0.4
11.4%
0.3
3.4%
10.2%
5.
𝜎
0.3
0.154
0.102
0.4
0.114 0.102
0.3
0.034
√0.002256 0.0475 4.75%
𝜎
0.102
0.002256
Page 18 of 19
Fin 638: Midterm Review
Historical Risk‐Return Tradeoff
Page 19 of 19
Fin 638: Midterm Review