310
Tax rate = 40% = 0.4
Interest amount = 600,000 * 0.8 = 48,000
Annual sales = 3,000,000
Net Profit = 3% = 3,000,000 * 3% = 90,000
Net profit + tax = profit before tax = 90,000 / (1 – Tax) = 90,000 / (1 – 0.4) = 90,000 / 0.6 = 150,000
Profit before tax + interest = Earning before interest and tax = 150,000 + 48,000 = 198,000
TIE ratio = EBIT/Interest = 198000/48000 = 4.125
èThe bank will refuse to renew
311
Partial Income 
Statement Information 
Sales 
$600,000 
Cost of goods sold (COGS) 
$450,000 
Balance Sheet 
Balance Sheet Information 

Cash 
$28,000 
Accounts Payable 
$110,000 
Accounts Receivable (AR) 
$60,000 
Longterm Debt 
$50,000 
Inventories 
$160,000 
Common Stocks 
$140,000 
Fixed assets (FA) 
$152,000 
Retained Earnings 
$100,000 
Total Assets 
$400,000 
Total liabilities and Equity 
$400,00 
312
Current Assets/Current Liabilities = 3.0 x 810,000 /current liabilities = 3.0 x Current liabilities = 270,000
Current assets – Inventories / Current Liabilities = 1.4 x 810,000 – Inventories / 270,000
= 1.4 x Inventories = 810,000 – (1.4)*(270,000)
= 432,000
Current assets = Cash + Marketable Securities + Accounts Receivable + inventories
810,000 = 120,000 + accounts receivable + 432,000
Accounts Receivable = 810,000 – 120,000 – 432,000 = 2580,000
DSO = Accounts Receivable / (Sales / 365)
36.5 = 258,000 / (Sales / 265)
è Sales = (258,000 x 365) / 36.5 = 2,580,000
313
Current Assets/Current liabilities = 2,925,000/1,453,000 = 2.01
Days sales outstanding (DSO) = Accounts Receivable / (Sales/365)
= 1,575,000 / (7,500,000 / 365) = 1570000 / 20547.94 = 76.74 è 77 days
Inv TO = COGS/Inventories = 6,375,000 / 1,125,000 = 5.67
FA Turnover = Sales/Fixed assets = 7,500,000 / 1,350,000 = 5.56
TA Turnover = Sales/Total assets = 7,500,000 / 4,275,000 = 1.75
PM = Net income/Sales = 113,021 / 7,500,000 = 0.015 = 1.5%
ROA = Net income/Total assets = 113,021 / 4,275,000 = 0.0264 = 2.64%
ROE = Net income/Common equity = 113,021 / 1,752,750 = 0.0645 = 6.45%
Debt Ratio = Total current liabilities / Total assets = 1,453,500 / 4,275,000 = 0.34 = 34%
Total liabilities /Total Assets = (Total current liabilities + Longterm Debt) / Total Assets
= (1,453,500 + 1,068,750) / 4,275,000 = 2,522,250 / 4,275,000 = 0.59 = 59%
Lozano Chip Company:
ROE = PM x TATO x EM = PM x TATO x (TA / Common Equity)
6.45% = 1.5% x 1.75% x (4,275,000 / 1,752, 750) = 1.5% * 1.75% * 2.44
Industry:
ROE = PM x TATO x EM = PM x TATO x (TA / Common Equity)
9% = 1.2% x 3% x EM è EM = 2.5
Lozano has better PM but greater DSO, fixed asset turnover, TATO than the industry
Ratio 
Lozano 
Industry Average 
Current assets/Current liabilities 
2.03 
2.0 
Days sales outstanding (365day year) 
76.65 
35.0 days 
COGS/Inventory 
5.67 
6.7 
Sales/Fixed assets 
5.55 
12.1 
Sales/Total assets 
1.75 
3.0 
Net income/Sales 
1.5% 
1.2% 
Net income/Total assets 
2.64% 
3.6% 
Net income/Common equity 
6.44% 
9.0% 
Total debt/Total assets 
34% 
30.0% 
Total liabilities/Total assets 
59% 
60.0% 
314
Quick ratio = (cash and equivalents + marketable securities + accounts receivable) / Current liabilities
= ($439000 + $72000)/ $602000 = $511000/ $602000 = 0.8
Firm has less liquid assets per dollar of current liabilities.
Current ratio (CA/CL) = Current Assets/ current liabilities = $1,405,000 /$602000 = 2.3
The Current ratio of the industry is 2.7 and the current ratio of the firm is 2.3. The firm has adequate resources to cover its current liabilities. However in comparison to the industry it is less efficient and capable of servicing its short term liabilities.
Inventory turnover = Cost Of goods sold/Average Inventory. = 3580000/894000 = 4.0
The Inventory turnover Ratio for the industry is 7 whereas for the firm it is 4.0. Low Inventory turnover implies poor sales or excess inventory. Excess inventory is also bad as the capital is tied up in inventory which has zero rate of return.
Days sales Outstanding (DSO) = 365/Credit Sales/Average Accounts Receivable
= 365/ (4290000/439000) = 37 days
The industry on an average takes around 32 days to convert sales into cash whereas the firm takes 37 days to convert sales into cash. This means that as compared to the industry in general the firm is less efficient to bring cash into the business.
Fixed asset Turnover (FATO) = Net sales/ Net property plant and equipment
= 4,290,000 / 431,000 = 9.9
TATO = Sales/Total assets = 4,290,000 / 1,836,000 = 2.33
ROA = Net income/Total assets = 108,408 / 1,836,000 = 0.059 = 5.9%
ROE = Net income/ (Common Stock + Retained Earnings)
= 108,408 / (575,000 + 254,710) = 0.13 1 = 13.1%
Profit Margin on Sales (PM) = Net income/Sales = 108,408 / 4,290,000 = 0.025 = 2.5%
Debt Ratio = (Notes Payable + Longterm Bonds) / Total Assets
= (100,000 + 404,290) / 1,836,000= 0.2746 = 27.46%
Total liabilities /Total Assets = (Total current liabilities + Longterm Debt) / Total Assets
= (602,000 + 404,290) / 1,836,000 = 0.548 = 54.8%
P/E Ratio = 5% (from book)
Price/Cash flow ratio = Price per share / Cash flow per share =
Market/book ratio = Market cap / Total common equity =
Minicase
Ratio Analysis 
2015 
2016 
2017E 
Industry Average 
Current 
2.3 
1.5 
2.58 
2.7 
Quick 
0.8 
0.5 
0.93 
1.0 
Inventory Turnover 
4.8 
4.5 
4.10 
6.1 
Days Sales Outstanding 
37.4 
39.5 
45.5 
32.0 
Fixed Assets Turnover 
10.0 
6.2 
8.41 
7.0 
Total Assets Turnover 
2.3 
2.0 
2.00 
2.5 
Debt Ratio 
35.6% 
59.6% 
22.75% 
32.0% 
Liabilitiestoassets ratio 
54.8% 
80.7% 
43.8% 
50.0% 
TIE 
3.3 
0.1 
6.3 
6.2 
EBITDA Coverage 
2.6 
0.8 
5.5 
8.0 
Profit Margin 
2.6% 
1.6% 
3.6% 
3.6% 
Basic Earning Power 
14.2% 
0.6% 
14.3% 
17.8% 
ROA 
6.0% 
3.3% 
7.2% 
9.0% 
ROE 
13.3% 
17.1% 
12.8% 
17.9% 
Price/Earnings (P/E) 
9.7 
6.3 
12.0 
16.2 
Price/Cash Flow 
8.0 
27.5 
8.1 
7.6 
Market/Book 
1.3 
1.1 
1.5 
2.9 
a. Why are ratios useful? What are the five major categories of ratios?
Answer: Ratios are used by managers to help improve the firm’s performance, by lenders to help evaluate the firm’s likelihood of repaying debts, and by stockholders to help forecast future earnings and dividends. The five major categories of ratios are: liquidity, asset management, debt management, profitability, and market value.
b. Calculate the 2017 current and quick ratios based on the projected balance sheet and income statement data. What can you say about the company’s liquidity position in 2015, 2016, and as projected for 2006? We often think of ratios as being useful (1) to managers to help run the business, (2) to bankers for credit analysis, and (3) to stockholders for stock valuation. Would these different types of analysts have an equal interest in the liquidity ratios?
Answer: Current Ratio_{06 }= Current Assets/Current Liabilities
= $2,680,112/$1,039,800 = 2.58´.
Quick Ratio_{06} = (Current Assets – Inventory)/Current Liabilities
= ($2,680,112  $1,716,480)/$1,039,800 = 0.93´.
The company’s current and quick ratios are higher relative to its 2015 current and quick ratios; they have improved from their 2016 levels. Both ratios are below the industry average, however.
c. Calculate the 2017 inventory turnover, days sales outstanding (DSO), fixed assets turnover, and total assets turnover. How does Computron’s utilization of assets stack up against other firms in its industry?
Answer: Inventory Turnover_{06} = Sales/Inventory
= $7,035,600/$1,716,480 = 4.10´.
DSO_{06} = Receivables / (Sales/365)
= $878,000 / ($7,035,600/365) = 45.5 Days.
Fixed Assets Turnover_{06} = Sales/Net Fixed Assets
_{ } = $7,035,600/$836,840 = 8.41´.
Total Assets Turnover_{06} = Sales/Total Assets
_{ } = $7,035,600/$3,516,952 = 2.0´.
The firm’s inventory turnover ratio has been steadily declining, while its days sales outstanding has been steadily increasing. While the firm’s fixed assets turnover ratio is below its 2015 level, it is above the 2016 level. The firm’s total assets turnover ratio is below its 2015 level and equal to its 2016 level.
The firm’s inventory turnover and total assets turnover are below the industry average. The firm’s days sales outstanding is above the industry average (which is bad); however, the firm’s fixed assets turnover is above the industry average. (This might be due to the fact that Computron is an older firm than most other firms in the industry, in which case, its fixed assets are older and thus have been depreciated more, or that Computron’s cost of fixed assets was lower than most firms in the industry.)
d. Calculate the 2017 debt, timesinterestearned, and EBITDA coverage ratios. How does Computron compare with the industry with respect to financial leverage? What can you conclude from these ratios?
Answer: Debt Ratio = (Notes Payable + Longterm Bonds) / Total Assets
= ($300,000 + $500,00) / $3,516,952
= 0.2275 = 22.75%
Liabilitiestoassets = Total Liabilities/Total Assets
= (Total Liabilities & Equity – Total Equity) / Total Assets
= ($3,516,952  $1,977,152) / $3,516,952
= $1,539,800 / $3,516,952 = 0.4378 = 43.8%
Tie_{06} = EBIT/Interest = $502,640/$80,000 = 6.3´.
EBITDA Coverage_{06} =
= ($502,640 + $120,000 + $40,000)/($80,000 + $40,000) = 5.5´.
The firm’s debt ratio is much improved from 2016, and is still lower than its 2003 level and the industry average. The firm’s TIE and EBITDA coverage ratios are much improved from their 2015 and 2016 levels. The firm’s TIE is better than the industry average, but the EBITDA coverage is lower, reflecting the firm’s higher lease obligations.
e. Calculate the 2017 profit margin, basic earning power (BEP), return on assets (ROA), and return on equity (ROE). What can you say about these ratios?
Answer: Profit Margin_{06} = Net Income/Sales = $253,584/$7,035,600 = 3.6%.
Basic Earning Power_{06} = EBIT/Total Assets = $502,640/$3,516,952
= 14.3%.
ROA_{06} = Net Income/Total Assets = $253,584/$3,516,952 = 7.2%.
ROE_{06} = Net Income/Common Equity = $253,584/$1,977,152 = 12.8%.
The firm’s profit margin is above 2015 and 2016 levels and is at the industry average. The basic earning power, ROA, and ROE ratios are above both 2015 and 2016 levels, but below the industry average due to poor asset utilization.
f. Calculate the 2017 price/earnings ratio, price/cash flow ratios, and market/book ratio. Do these ratios indicate that investors are expected to have a high or low opinion of the company?
Answer: EPS = Net Income/Shares Outstanding = $253,584/250,000 = $1.0143.
Price/Earnings06 = Price Per Share/Earnings Per Share
= $12.17/$1.0143 = 12.0.
Check: Price = EPS P/E = $1.0143(12) = $12.17.
Cash Flow/Share06 = (NI + DEP)/Shares
= ($253,584 + $120,000)/250,000 = $1.49.
Price/Cash Flow = $12.17/$1.49 = 8.2.
BVPS = Common Equity/Shares Outstanding
= $1,977,152/250,000 = $7.91.
Market/Book = Market Price Per Share/Book Value Per Share
= $12.17/$7.91 = 1.54x.
Both the P/E ratio and BVPS are above the 2015 and 2016 levels but below the industry average.
g. Perform a common size analysis and percent change analysis. What do these analyses tell you about Computron?
Answer: For the common size balance sheets, divide all items in a year by the total assets for that year. For the common size income statements, divide all items in a year by the sales in that year.
Common Size Balance Sheets 

Assets 


2015 
2016 
2017E 
Ind. 

Cash 
0.6% 
0.3% 
0.4% 
0.3% 

Short Term Investments 
3.3% 
0.7% 
2.0% 
0.3% 

Accounts Receivable 
23.9% 
21.9% 
25.0% 
22.4% 

Inventories 
48.7% 
44.6% 
48.8% 
41.2% 

Total Current Assets 
76.5% 
67.4% 
76.2% 
64.1% 

Gross Fixed Assets 
33.4% 
41.7% 
34.7% 
53.9% 

Less Accumulated Depreciation 
10.0% 
9.1% 
10.9% 
18.0% 

Net Fixed Assets 
23.5% 
32.6% 
23.8% 
35.9% 

Total Assets 
100.0% 
100.0% 
100.0% 
100.0% 

Liabilities And Equity 
2015 
2016 
2017E 
Ind. 

Accounts Payable 
9.9% 
11.2% 
10.2% 
11.9% 

Notes Payable 
13.6% 
24.9% 
8.5% 
2.4% 

Accruals 
9.3% 
9.9% 
10.8% 
9.5% 

Total Current Liabilities 
32.8% 
46.0% 
29.6% 
23.7% 

LongTerm Debt 
22.0% 
34.6% 
14.2% 
26.3% 

Common Stock (100,000 Shares) 
31.3% 
15.9% 
47.8% 
20.0% 

Retained Earnings 
13.9% 
3.4% 
8.4% 
30.0% 

Total Equity 
45.2% 
19.3% 
56.2% 
50.0% 

Total Liabilities And Equity 
100.0% 
100.0% 
100.0% 
100.0% 
Common Size Income Statement 
2015 
2016 
2017E 
Ind. 

Sales 
100.0% 
100.0% 
100.0% 
100.0% 

Cost Of Goods Sold 
83.4% 
85.4% 
82.4% 
84.5% 

Other Expenses 
9.9% 
12.3% 
8.7% 
4.4% 

Depreciation 
0.6% 
2.0% 
1.7% 
4.0% 

Total Operating Costs 
93.9% 
99.7% 
92.9% 
92.9% 

EBIT 
6.1% 
0.3% 
7.1% 
7.1% 

Interest Expense 
1.8% 
3.0% 
1.1% 
1.1% 

EBT 
4.3% 
2.7% 
6.0% 
5.9% 

Taxes (40%) 
1.7% 
1.1% 
2.4% 
2.4% 

Net Income 
2.6% 
1.6% 
3.6% 
3.6% 
Computron has higher proportion of inventory and current assets than industry. Computron has slightly more equity (which means less debt) than industry. Computron has more shortterm debt than industry, but less longterm debt than industry. Computron has lower COGS than industry, but higher other expenses. Result is that Computron has similar EBIT as industry.
For the percent change analysis, divide all items in a row by the value in the first year of the analysis.
Percent Change Balance Sheets 

Assets 


2015 
2016 
2017E 

Cash 
0.0% 
19.1% 
55.6% 

Short Term Investments 
0.0% 
58.8% 
47.4% 

Accounts Receivable 
0.0% 
80.0% 
150.0% 

Inventories 
0.0% 
80.0% 
140.0% 

Total Current Assets 
0.0% 
73.2% 
138.4% 

Gross Fixed Assets 
0.0% 
145.0% 
148.5% 

Less Accumulated Depreciation 
0.0% 
80.0% 
162.1% 

Net Fixed Assets 
0.0% 
172.6% 
142.7% 

Total Assets 
0.0% 
96.5% 
139.4% 

Liabilities And Equity 
2015 
2016 
2017E 

Accounts Payable 
0.0% 
122.5% 
147.1% 

Notes Payable 
0.0% 
260.0% 
50.0% 

Accruals 
0.0% 
109.5% 
179.4% 

Total Current Liabilities 
0.0% 
175.9% 
115.9% 

LongTerm Debt 
0.0% 
209.2% 
54.6% 

Common Stock (100,000 Shares) 
0.0% 
0.0% 
265.4% 

Retained Earnings 
0.0% 
52.1% 
45.4% 

Total Equity 
0.0% 
16.0% 
197.9% 

Total Liabilities And Equity 
0.0% 
96.5% 
139.4% 
Percent Change Income Statement 
2015 
2016 
2017E 

Sales 
0.0% 
70.0% 
105.0% 

Cost Of Goods Sold 
0.0% 
73.9% 
102.5% 

Other Expenses 
0.0% 
111.8% 
80.3% 

Depreciation 
0.0% 
518.8% 
534.9% 

Total Operating Costs 
0.0% 
80.5% 
102.7% 

EBIT 
0.0% 
91.7% 
140.4% 

Interest Expense 
0.0% 
181.6% 
28.0% 

EBT 
0.0% 
208.2% 
188.3% 

Taxes (40%) 
0.0% 
208.2% 
188.3% 

Net Income 
0.0% 
208.2% 
188.3% 
We see that 2017 sales grew 105% from 2003, and that NI grew 188% from 2015. So Computron has become more profitable. We see that total assets grew at a rate of 139%, while sales grew at a rate of only 105%. So asset utilization remains a problem.
h. Use the extended Du Pont equation to provide a summary and overview of Computron’s financial condition as projected for 2017. What are the firm’s major strengths and weaknesses?
Answer: Du Pont Equation =
= 3.6% ´ 2.0 ´ ($3,516,952/$1,977,152)
= 3.6% ´ 2.0 ´ 1.8 = 13.0%.
Strengths: The firm’s fixed assets turnover was above the industry average. However, if the firm’s assets were older than other firms in its industry this could possibly account for the higher ratio. (Computron’s fixed assets would have a lower historical cost and would have been depreciated for longer periods of time.) The firm’s profit margin is slightly above the industry average, despite its higher debt ratio. This would indicate that the firm has kept costs down, but, again, this could be related to lower depreciation costs.
Weaknesses: The firm’s liquidity ratios are low; most of its asset management ratios are poor (except fixed assets turnover); its debt management ratios are poor, most of its profitability ratios are low (except profit margin); and its market value ratios are low.
i. What are some potential problems and limitations of financial ratio analysis?
Answer: Some potential problems are listed below:
1. Comparison with industry averages is difficult if the firm operates many different divisions.
2. Different operating and accounting practices distort comparisons.
3. Sometimes hard to tell if a ratio is “good” or “bad.”
4. Difficult to tell whether company is, on balance, in a strong or weak position.
5. “Average” performance is not necessarily good.
6. Seasonal factors can distort ratios.
7. “Window dressing” techniques can make statements and ratios look better.
j. What are some qualitative factors analysts should consider when evaluating a company’s likely future financial performance?
Answer: Top analysts recognize that certain qualitative factors must be considered when evaluating a company. These factors, as summarized by the American Association Of Individual Investors (AAII), are as follows:
1. Are the company’s revenues tied to one key customer?
2. To what extent are the company’s revenues tied to one key product?
3. To what extent does the company rely on a single supplier?
4. What percentage of the company’s business is generated overseas?
5. Competition
6. Future prospects
7. Legal and regulatory environment
Chapter 4 – Time Value of Money
Problems
41
Deposit = $10,000
Bank pays = 10%
Period = 5
FV_{n}= PV(1 + i)_{}^{n}
FV_{5} = $10,000 (1 + 0.1)^{5} = $10,000 (1.1)^{5} = $16,1015.10
42
Future Value = $5,000
Years = 20
Discount % = 7%
Use the following equation to calculate present value:
Present Value = Future Value x (Present Value Interest Factor (PVIF))
= $5,000 x (1/1.07)^20 years
= $5,000 x 0.258
= $1,290
43
N = 18
PV = $250K
FV = 1M
1M * (I/YR + i ) ^ 18 = 250K
(I/YR + i ) ^ 18 = 4
I/YR + i = 1.08
I/YR = 8.01%
44
45
I/YR = 12
PV = $42,180.53
PMT =  5000
FV = 250K
250K * (12) ^ N = $42,180.53
N = 11
46
Annuity: $1725.22
Annuity Due: $ 1845.99
Future Value of an ordinary annuity = http://www.getobjects.com/Components/Finance/TVM/fva.html
You may use table  http://www.principlesofaccounting.com/ART/fv.pv.tables/fvofordinaryannuity.htm
$300*5.75074 = $1725.22
S = R(((1 + i)n  1) / i) *(1 + i)
S = $1725.22 * 1.07
S = $1845.98
47
Here's the Present Value Calculation:
Year 1 $100 / (1.0 + .08) = $ 92.59
Year 2 $100 / (1.0 + .08)^2 = $ 85.73
Year 3 $100 / (1.0 + .08)^3 = $ 79.38
Year 4 $200 / (1.0 + .08)^4 = $147.01
Year 5 $300 / (1.0 + .08)^5 = $204.17
Year 6 $500 / (1.0 + .08)^6 = $315.08
Total PV = $923.98
Here's the Future Value calculation:
(Note: since the payment is at the end of each year, the payment for year 1 will compound for only 5 years, the payment for year 2 will compound for 4 years, and so on)
Year 1 $100 X (1.0 + .08)^5 = $146.93
Year 2 $100 X (1.0 + .08)^4 = $136.05
Year 3 $100 X (1.0 + .08)^3 = $125.97
Year 4 $200 X (1.0 + .08)^2 = $233.28
Year 5 $300 X (1.0 + .08)^1 = $324.00
Year 6 $500 X (1.0 + .08)^0 = $500.00
Total FV = $1,466.23
48
N= 60 months
FV =0
PV = $20,000
Interest rate (I) = 12% 0.12/12 = 0.01 per month or 1% per month
Function:
PMT(I,N,PV, FV, Type)
PMT = $444.89
T(0.01,60,200 00,0,0) = EAR = EFF%
= (1 + Quoted Rate / m) m – 1
= (1+0.12/12)121
= (1+0.01)121
= 1.1268251
=0.126825
=1 2.68%
49
a. an initial $500 compounded for 1 year at 6 percent
A = P*(1+r)^t
A = 500*1.06^1
A = 500*1.06
A = $530
With Excel: =FV(0.06,1,,500)
b. an initial $500 compounded for 2 years at 6 percent
A = P*(1+r)^t
A = 500*1.06^2
A = 500*1.1236
A = $561.80
With Excel: =FV(0.06,2,,500)
c. the present value of $500 due in 1 year at a discount rate of 6 percent
PV = FV/(1+r)^t
PV = 500/(1+0.06)^1
PV = 500/1.06
PV = $471.70
With Excel: =PV(0.06,1,,500)
d. the present value of $500 due in 2 years at a discount rate of 6 percent
PV = FV/(1+r)^t
PV = 500/(1+0.06)^2
PV = 500/1.1236
PV = $445
With Excel: =PV(0.06,2,,500)
410
Time Line Approach ( I did the time line approach manually (notebook))
FV_{1} = Pv + INT
= Pv + Pv(I)
= Pv(1+I)
Formula Approach
FV_{2} = FV1(1+I)
=PV(1+I)(1+I)
=PV(1+I) 2
Financial Calculator / Excel Spreadsheet
a. An initial $500 compounded for 1year for 10 years at 6%.
$500(1.06) 10 = $895.42
b. An initial $500 compounded for 2year for 10 years at 6%.
$500(1.12) 10 = $1,552.92.
c. The present value of $500 due in 10 years at a discount rate of 6%.
$500(1/1.06) 10 = $279.20
d. The present value of $500 due in 10 years at a discount rate of 12%.
$500(1/1.12) 10 = $160.99
411
a. 7%
400 = 200(1+0,07)N
2 = (1+0,07)N
ln(2) = N[(ln(1,07)]
N = 0,693147 / 0,067659 = 10.24477 years
b. 10%
400 = 200(1+0,10)N
2 = (1+0,10)N
ln(2) = N[(ln(1,10)]
N = 0,693147 / 0,09531 = 7,272541 years
c. 18%
400 = 200(1+0,18)N
2 = (1+0,18)N
ln(2) = N[(ln(1,18)]
N = 0,693147 / 0,165514 = 4,187835 years
d. 100%
400 = 200(1+1,00)N
2 = (1+1,00)N
ln(2) = N[(ln(2,00)]
N = 0,693147 / 0,693147 = 1 year
412
413
a. $400 per year for 10 years at 10%
In Excel =PV(rate, nyears, payment, present value, type)
=PV(.10,10, 400,0,0)
= $2,457.83
b. $200 per year for 5 years at 5%
In Excel =PV(rate, n years, payment, present value, type)
= PV(.05,5,200,0,0)
= $865.90
c. $400 per year for 5 years at 0% $400 per year for 5 years at 0%
In Excel =PV(rate, n years, payment, present value, type)
= PV(.0, 5,400,0,0)
= $2,000.00
d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.
1. $400 per year for 10 years at 10%
In Excel =PV(rate,nyears,payment,present value, type)
=PV(.10,10, 400,0,1)
= $2,703.61
2. $200 per year for 5 years at 5%
In Excel =PV(rate, n years, payment, present value, type)
= PV(.05,5,200,0,1)
= $909.19
3. c. $400 per year for 5 years at 0%
In Excel =PV(rate, n years, payment, present value, type)
= PV(.0, 5,400,0,1)
= $2,000.00
414
Year 
Cash Stream A 
Cash Stream B 
1 
100 
300 
2 
400 
400 
3 
400 
400 
4 
400 
400 
5 
300 
100 
a. Present values using Excel:
PV = NPV(interest,Value1,Value2,Value3,Value4,Value5,Value6)
PV_{A} = NPV_{A} (.08, 0,100,400,400,400,300)
= $1,251.25
PV = NPV_{B} (interest,Value1,Value2,Value3,Value4,Value5,Value6)
PV_{B} =NPV B (.08, 0,300,400,400,400,100)
= $1,300.32
b. What is the value of each cash flow stream at a 0% interest rate?
Using Excel:
PV = NPV(interest,Value1,Value2,Value3,Value4,Value5,Value6)
PV_{A} = NPV_{A} (0.00, 0, 100, 400, 400, 400, 300)
= $1,600.00
PV = NPV(interest,Value1,Value2,Value3,Value4,Value5,Value6)
PV_{B }=NPV_{B} (0.00, 0, 300, 400, 400, 400, 100)
= $1,600.00
415. For this problem, you have to financial calculator.. You could download that app or use the online one
a. You borrow $700 and promise to pay back $749 at the end of 1 year.
N=1
PMT=0
PV=700
FV= 749
I/Y = 7%
b. You lend $700 and receive a promise to be paid $749 at the end of 1 year.
N=1
PMT=0
PV=700
FV= 749
I/Y = 7%
c. You borrow $85,000 and promise to pay back $201,229 at the end of 10 years.
N=10
PMT=0
PV=85,000
FV= 201,299
I/Y = 9%
d. You borrow $9,000 and promise to make payments of $2,684.80 at the end of each of the next 5 years.
N=5
PMT=2684.80
PV=9000
FV= 0
I/Y = 14.999 (15%)
416 You have to use the “financial calculator”
a. N = 5 I = 12 PV = 500 PMT = 0 FV = ? FV = $881.17
b. N = 10 I = 6 PV = 500 PMT = 0 FV = ? FV = $895.42
c. N = 20 I = 3 PV = 500 PMT = 0 FV = ? FV = $903.06
d. N = 60 I = 1 PV = 500 PMT = 0 FV = ? FV = $908.35
417
a. 12% nominal rate, semiannual compounding, discounted back 5 years
N = 10 I = 6% = 0.06 FV = $500 PMT = 0
PV = FV_{n }/ (1 + I)^N
= 500 / (1 + 0.06)^10
= 500 / 1.79
= 279.33
b. 12% nominal rate, quarterly compounding, discounted back 5 years
N = 20 I = 3% = 0.03 FV = $500 PMT = 0
PV = FV_{n} / (1 + I)^N
= 500 / (1 + 0.03)^20
= 500 / 1.806
= 276.85
c. 12% nominal rate, monthly compounding, discounted back 1 years
N = 12 I = 1% = 0.01 FV = $500 PMT = 0
PV = FV_{n} / (1 + I)^N
= 500 / (1 + 0.01)^12
= 443.72
418
a. N = 10 I = 6 PV = 0 PMT = 400 FV = $5272.32
b. N = 20 I = 3 PV = 0 PMT = 400 FV = $5374.07
c. The annuity in part b earns more because the money is on deposit for a longer period of time and so earns more interest. Also, because compounding is more frequent, more interest is earned on interest.
419
a. Assuming $100 Deposit:
Universal Bank: N = 1; I1YR = 7; PV = 100; FV = 107
Effective Rate = ((107100))/100) = .07 or 7%
Regional Bank: N = 4; I1YR = 6/4 = 1.5; PV = 100; FV = 106.136
Effective Rate = 6.136%
You would invest in Universal because the effective rate is higher.
b. If funds must be left on deposit until the end of the compounding period (1 year for Universal and 1 quarter for Regional) and you think there is a high probability that you will make a withdrawal during the year, the Regional account might be preferable. If you withdraw 364 days after depositing money in Universal you would receive no interest, however if you were to have deposited with Regional you would have received 3 quarterly interest payments.
420
421
N = 5 PV = 6 PMT = 0 FV = 12
a. Using a calculator, enter N = 5, PV = 6, PMT = 0, and FV = 12; compute I/Y= 14.87% ≈ 15%.
b. Growth rate originally is 15%. When you divide 100% by 5, the growth rate is 20% instead of the original 15% which would make the calculations not correct
422
Amount invested (PV) = $4 million
Time (N) = 10 years
FV = $8 million
I = ?
FV = PV*(1+i)^10
8,000,000 = 4,000,000 * (1+i)^10
2^(1/10) – 1^(1/10) = I
I = 0.07177 = 7.177%
423
PV=85,000
PMT=8,273.59
N=30
I=?
PV = 85,000 = 8,273.59*(1(1+i)^30)/i
= RATE(30,8273.59,85000,0,0)
= 9%
424
PMT = $10,000
I = 7%
N = 4
a. PV = ?
PV=10,000*(1(1+0.07)^4)/0.07=33,872.11
The deposit is $33,872.11
b. How much will be in the account immediately after you make the first withdrawal?
33,872.11 * 1.07 – 10,000 = $26,243.16
After the last withdrawal? the account is zero.
425
Amount of loan = $12,000
I1YR = 9
PV = 12,000
PMT = 1,500
N = 14.77 years or about 15 years (unless she has a rich uncle).
426
W I/YR = 12
PV = 0
PMT = 1250
FV = 10000
N = 5.94 years
$1,250 will be made 94/100th of the way through Year 5.
FV of $1,250 for 5 years at 12%
N = 5, I/YR = 12, PV = 0, and PMT = 1250.
FV = $7,941.06.
Compound this value for 1 year at 12% value in the account after 6 years and
Before the last payment is made; it is $7,941.06(1.12) = $8,893.99.
Make a payment of $10,000  $8,893.99 = $1,106.01 at Year 6.
It will take 6 years, and $1,106.01 is the amount of the last payment.
427
I = 7% = 0.07
PVper = PMT / I = 100 / 0.07 = 1,428.57
I=14% = 0.14
PVper = PMT / 0.14 = 714.28
428
With I=8% => EFF%= (1+8%/4)^4 – 1 = 8.24%
8.24%
    
0 1 2 3 4
50 50 50 1050
PV_{1,2,3 }= 50* (1(1.0824)^3)/.0824 = 128.29
PV_{4} =1050/(1+.0824)^4=764.87
è PV=128.29+764.87=$893.16
429
This can be done with a calculator by specifying an interest rate of 5 percent per period for 20 periods.
N = 10 x 2 = 20.
I = 10/2 = 5.
PV = 10000.
FV = 0.
PMT = $802.43.
Set up an amortization table:
Beginning Payment of Ending
Period Balance Payment Interest Principal Balance
1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57
2 9,697.57 802.43 484.88
$984.88
You can really just work the problem with a financial calculator using the amortization function. Find the interest in each 6month period, sum them, and you have the answer. Even simpler, with some calculators such as the HP17B, just input 2 for periods and press INT to get the interest during the first year, $984.88.
430
Input N = 5, I = 15, PV = 1000000, and FV = 0 to solve for PMT = $298,315.55.
Beginning Payment of Ending
Year Balance Payment Interest Principal Balance
1 $1,000,000.00 $298,315.55 $150,000.00 $148,315.55 $851,684.45
2 851,684.45 298,315.55 127,752.67 170,562.88 681,121.57
The fraction that is principal is $170,562.88 / $298,315.55 = 57.18%.
431
Here, Number of deposits (n) = 5 deposits;
Semiannual (every 6 months), payment = $ 100;
Nominal interest rate (i) = 12%,
Present value of annuity (PVA) = ?
We have,
FVA = PMT × (1 + i) = $ 100 (1 + 0.06)
= $ 100 × 5.6371 × 1.06 = $ 597.5326
Now remaining period is 15 periods (20 periods  5 periods), so we calculate the future value of this $ 597.5326 for remaining periods.
We have,
FV = PV (1 + i)^{n }= $ 597.5326 (1 + 0.06)^{15 } = $ 1,432.02
Here,
Future value at the end of 10 years = $ 1,432.02;
n = 35 periods because quarterly compounding (in 10 years there are 40 quarters);
Quarterly interest rate = 3%,
PMT = ?, PV = ?
We have,
PV = = 1,432.02/(1+0.03)^{35} = $ 508.91
Now we calculate the payment (PMP)
Here, n = 5 periods, i = 3%, PV = ?; FV = $ 508.91, FVA = $ 508.91
PMT = ?
We have,
FVA = PMT (1 + i)
or, $ 508.91 = PMT (1 + 0.03)
or, $ 508.91 = PMT × 5.3091 × 1.03
\ PMT = $ 508.91/ 5.4684 = $ 93.06
432
First we calculate EAR of bank:
EFF% = (1+(15%/12))^12 = 16.08%
Next we calculate R_{nom} compounding quarterly:
16.08% = ((1+(R/4)^4) – 1
=> 116.08% = (1 + R/4)^4
=> (116.08%)^(1/4) = 1+ R/4
=> R_{nom} = 15.18%
Anne Lockwood should quote her customers with r=15.18%.
433
Save 10 years and receive money in 25 years, calculate the PMT within 10 years=?
Now he has 100,000 in his account with 8% within 10 years, so FV=100,000*(1+.08)^10 = 215,892.5
PMT that he wants to receive with adjusted inflation (5%):
The amount that he wants to receive every year when he retire PMT=FV=40,000*(1.05)^10=65,155.79
PVdue of the PMT:
PVdue=[65,155.79*[1(1.08)^25]/.08)] * (1+.08) = 751,165.3
Total amount he has to save: 751,165.3215,892.5 = 535,272.8, the amount is FV of 10 years with 8%
From now to the time he is 60 years, he has to save every year amount as below
PMT = FV/{[(1+.08)^10 – 1]/0.08}=535,272.8/14.49=36,949.61
434
COULD NOT FOUND!!!
Mini Case
Assume that you are nearing graduation and that you have applied for a job with a local bank. As part of the bank's evaluation process, you have been asked to take an examination which covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.
a. Draw time lines for (a) a $100 lump sum cash flow at the end of year 2, (b) an ordinary annuity of $100 per year for 3 years, and (c) an uneven cash flow stream of $50, $100, $75, and $50 at the end of years 0 through 3.
Answer: (Begin by discussing basic discounted cash flow concepts, terminology, and solution methods.) A time line is a graphical representation which is used to show the timing of cash flows. The tick marks represent end of periods (often years), so time 0 is today; time 1 is the end of the first year, or 1 year from today; and so on.

0 1 2 year
   lump sum 100 cash flow

0 1 2 3
    annuity
100 100 100

0 1 2 3
    uneven cash flow stream
50 100 75 50
A lump sum is a single flow; for example, a $100 inflow in year 2, as shown in the top time line. An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the middle time line. An uneven cash flow stream is an irregular series of cash flows which do not constitute an annuity, as in the lower time line. 50 represents a cash outflow rather than a receipt or inflow.
b. 1. What is the future value of an initial $100 after 3 years if it is invested in an account paying 10 percent annual interest?
Answer: Show dollars corresponding to question mark, calculated as follows:

0 1 2 3
   
100 FV = ?
After 1 year:
FV_{1} = PV + i_{1} = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00.
Similarly:
FV_{2} = FV_{1} + i_{2} = FV_{1} + FV _{1}(i) = FV_{1}(1 + i)
= $110(1.10) = $121.00 = PV(1 + i)(1 + i) = PV(1 + i)^{2}.
FV_{3} = FV_{2} + i_{3} = FV_{2} + FV _{2}(i) = FV_{2}(1 + i)
= $121(1.10)=$133.10=PV(1 + i)^{2}(1 + i)=PV(1 + i)^{3}.
In general, we see that:
FV_{n} = PV(1 + i)^{n},
SO FV_{3} = $100(1.10)^{3} = $100(1.3310) = $133.10.
Note that this equation has 4 variables: FV_{n}, PV, i, and n. Here we know all except FV_{n}, so we solve for FV_{n}. We will, however, often solve for one of the other three variables. By far, the easiest way to work all time value problems is with a financial calculator. Just plug in any 3 of the four values and find the 4th.
Finding future values (moving to the right along the time line) is called compounding. Note that there are 3 ways of finding FV_{3}: using a regular calculator, financial calculator, or spreadsheets. For simple problems, we show only the regular calculator and financial calculator methods.
(1) regular calculator:
1. $100(1.10)(1.10)(1.10) = $133.10.
2. $100(1.10)^{3} = $133.10.
(2) financial calculator:
This is especially efficient for more complex problems, including exam problems. Input the following values: N = 3, I = 10, PV = 100, pmt = 0, and solve for FV = $133.10.
b. 2. What is the present value of $100 to be received in 3 years if the appropriate interest rate is 10 percent?
Answer: Finding present values, or discounting (moving to the left along the time line), is the reverse of compounding, and the basic present value equation is the reciprocal of the compounding equation:

0 1 2 3
   
PV = ? 100
FV_{n} = PV(1 + i)^{n} transforms to:
PV = = FV_{n} = FV_{n}(1 + i)^{n}
thus:
PV = $100 = $100(PVIF_{i,n}) = (0.7513) = $75.13.
The same methods used for finding future values are also used to find present values.
Using a financial calculator input N = 3, I = 10, pmt = 0, FV = 100, and then solve for PV = $75.13.
c. We sometimes need to find how long it will take a sum of money (or anything else) to grow to some specified amount. For example, if a company's sales are growing at a rate of 20 percent per year, how long will it take sales to double?
Answer: We have this situation in time line format:

0 1 2 3 3.8 4
     
12 2
Say we want to find out how long it will take us to double our money at an interest rate of 20%. We can use any numbers, say $1 and $2, with this equation:
FV_{n} = $2 = $1(1 + i)^{n} = $1(1.20)^{n}.
(1.2)^{n} = $2/$1 = 2
n LN(1.2) = LN(2)
n = LN(2)/LN(1.2)
n = 0.693/0.182 = 3.8.
Alternatively, we could use a financial calculator. We would plug I = 20, PV = 1, PMT = 0, and FV = 2 into our calculator, and then press the N button to find the number of years it would take 1 (or any other beginning amount) to double when growth occurs at a 20% rate. The answer is 3.8 years, but some calculators will round this value up to the next highest whole number. The graph also shows what is happening.
d. If you want an investment to double in three years, what interest rate must it earn?
Answer: 0 1 2 3
   
1 2
1(1 + i) 1(1 + i)^{2} 1(1 + i)^{3}
FV = $1(1 + i)^{3} = $2.
$1(1 + i)^{3} = $2.
(1 + i)^{3} = $2/$1 = 2.
1 + i = (2)^{1/3}
1 + i = 1.2599
i = 25.99%.
Use a financial calculator to solve: enter N = 3, PV = 1, PMT = 0, FV = 2, then press the I button to find I = 25.99%.
Calculators can find interest rates quite easily, even when periods and/or interest rates are not even numbers, and when uneven cash flow streams are involved. (With uneven cash flows, we must use the "CFLO" function, and the interest rate is called the IRR, or "internal rate of return;" we will use this feature in capital budgeting.)
e. What is the difference between an ordinary annuity and an annuity due? What type of annuity is shown below? How would you change it to the other type of annuity?
0 1 2 3
   
100 100 100
Answer: This is an ordinary annuityit has its payments at the end of each period; that is, the first payment is made 1 period from today. Conversely, an annuity due has its first payment today. In other words, anordinary annuity has endofperiod payments, while an annuity due has beginningofperiod payments.
The annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each payment to the left, so you end up with a payment under the 0 but none under the 3.
f. 1. What is the future value of a 3year ordinary annuity of $100 if the appropriate interest rate is 10 percent?

0 1 2 3
   
100 100 100
110
121
$331
Go through the following discussion. One approach would be to treat each annuity flow as a lump sum. Here we have
FVA_{n} = $100(1) + $100(1.10) + $100(1.10)^{2}
= $100[1 + (1.10) + (1.10)^{2}] = $100(3.3100) = $331.00.
Using a financial calculator, N = 3, I = 10, PV = 0, PMT = 100. This gives FV = $331.00.
f. 2. What is the present value of the annuity?

0 1 2 3
   
100 100 100
90.91
82.64
75.13
$248.68
The present value of the annuity is $248.68. Using a financial calculator, input N = 3, I = 10, PMT = 100, FV = 0, and press the PV button.
Spreadsheets are useful for time lines with multiple cash flows.
The following spreadsheet shows this problem:

A 
B 
C 
D 
1 
0 
1 
2 
3 
2 
100 
100 
100 

3 
248.69 
The excel formula in cell A3 is = NPV(10%,B2:D2). This gives a result of 248.69. Note that the interest rate can be either 10% or 0.10, not just 10. Also, note that the range does not include any cash flow at time zero.
Excel also has special functions for annuities. For ordinary annuities, the excel formula is = PV(interest rate, number of periods, payment). In this problem, = PV(10%,3,100), gives a result of 248.96. For the future value, it would be = FV(10%,3,100), with a result of 331.
f. 3. What would the future and present values be if the annuity were an annuity due?
Answer: If the annuity were an annuity due, each payment would be shifted to the left, so each payment is compounded over an additional period or discounted back over one less period.
To find the future value of an annuity due use the following formula:
FVA_{n}(Annuity Due) = FVA_{n}(1 + i).
In our situation, the future value of the annuity due is $364.10:
FVA_{3}(Annuity Due) = $331.00(1.10)^{1} = $364.10.
This same result could be obtained by using the time line: $133.10 + $121.00 + $110.00 = $364.10.
The best way to work annuity due problems is to switch your calculator to "beg" or beginning or "due" mode, and go through the normal process. Note that it's critical to remember to change back to "end" mode after working an annuity due problem with your calculator.
This formula could be used to find the present value of an annuity due:
PVA_{n}(Annuity Due) = PVA_{n}(1 + i) = PMT(PVIFA _{i,n})(1 + i).
In our situation, the present value of the annuity due is $273.56:
PVA_{3}(Annuity Due) = $248.69(1.10)^{1} = $273.56.
The Excel function is = PV(10%,3,100,0,1). The fourth term, 0, tells Excel there are no additional cash flows. The fifth term, 1, tells Excel it is an annuity due. The result is $273.56.
A similar modification gives the future value: = FV(10%,3,100,0,1), with a result of 364.10.
g. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10 percent, compounded annually.
0 1 2 3 4 years
    
0 100 300 300 50
Answer: Here we have an uneven cash flow stream. The most straightforward approach is to find the PVs of each cash flow and then sum them as shown below:

0 1 2 3 4 years
    
100 300 300 50
90.91
225.39
(34.15 )
530.08
Note (1) that the $50 year 4 outflow remains an outflow even when discounted. There are numerous ways of finding the present value of an uneven cash flow stream. But by far the easiest way to deal with uneven cash flow streams is with a financial calculator or a spreadsheet. Calculators have a function which on the HP 17B is called "CFLO," for "cash flow." other calculators could use other designations such as cf _{0} and CF_{i}, but they explain how to use them in the manual. You would input the cash flows, so they are in the calculator's memory, then input the interest rate, I, and then press the NPV or PV button to find the present value.
Spreadsheets are especially useful for uneven cash flows. The following spreadsheet shows this problem:
A 
B 
C 
D 
E 

1 
0 
1 
2 
3 
4 
2 
100 
300 
300 
50 

3 
530.09 
The Excel formula in cell A3 is = NPV(10%,B2:E2), with a result of 530.09.
h. 1. Define (a) the stated, or quoted, or nominal rate, (i _{Nom}), and (b) the periodic rate (i_{Per}).
ANSWER: The quoted, or nominal, rate is merely the quoted percentage rate of return. The periodic rate is the rate charged by a lender or paid by a borrower each period (periodic rate = i_{nom}/m).
h. 2. Will the future value be larger or smaller if we compound an initial amount more often than annually, for example, every 6 months, or semiannually, holding the stated interest rate constant? Why?
Answer: Accounts that pay interest more frequently than once a year, for example, semiannually, quarterly, or daily, have future values that are higher because interest is earned on interest more often. Virtually all banks now pay interest daily on passbook and money fund accounts, so they use daily compounding.
h. 3. What is the future value of $100 after 5 years under 12 percent annual compounding? Semiannual compounding? Quarterly compounding? Monthly compounding? Daily compounding
Answer: Under annual compounding, the $100 is compounded over 5 annual periods at a 12.0 percent periodic rate:
i_{Nom} = 12%.
FV_{n} = = $100 = $100(1.12)^{5} = $176.23.
Under semiannual compounding, the $100 is compounded over 10 semiannual periods at a 6.0 percent periodic rate:
i_{Nom} = 12%.
FV_{n} = = $100 = $100(1.06)^{10} = $179.08.
quarterly: FV_{n} = $100(1.03)^{20} = $180.61.
monthly: FV_{n} = $100(1.01)^{60} = $181.67.
daily: FV_{n} = $100(1+ 0.12/365)^{365*5} = $182.19.
h. 4. What is the effective annual rate (EAR)? What is the ear for a nominal rate of 12 percent, compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily?
Answer: The effective annual rate is the annual rate that causes the PV to grow to the same FV as under multiperiod compounding. For 12 percent semiannual compounding, the ear is 12.36 percent:
EAR = Effective Annual Rate =
IF i_{Nom} = 12% and interest is compounded semiannually, then:
EAR = = (1.06)^{2} – 1.0 = 1.1236 – 1.0 = 0.1236 = 12,36%.
For quarterly compounding, the effective annual rate is:
(1.03)^{4}  1.0 = 12.55%.
For monthly compounding, the effective annual rate is:
(1.01)^{12}  1.0 = 12.55%.
For daily compounding, the effective annual rate is:
(1 + 0.12/365)^{365}  1.0 = 12.75%.
i. Will the effective annual rate ever be equal to the nominal (quoted) rate?
Answer: If annual compounding is used, then the nominal rate will be equal to the effective annual rate. If more frequent compounding is used, the effective annual rate will be above the nominal rate.
j. 1. Construct an amortization schedule for a $1,000, 10 percent annual rate loan with 3 equal installments.
2. What is the annual interest expense for the borrower, and the annual interest income for the lender, during year 2?
Answer: To begin, note that the face amount of the loan, $1,000, is the present value of a 3year annuity at a 10 percent rate:

0 1 2 3
   
1,000 PMT PMT PMT
PVA_{3} = PMT + PMT + PMT
$1,000 = PMT(1 + i)^{1} + PMT(1 + i)^{2} + PMT(1 + i) ^{3}
= PMT(1.10)^{1} + PMT(1.10)^{2} + PMT(1.10)^{3}.
We have an equation with only one unknown, so we can solve it to find PMT. The easy way is with a financial calculator. Input n = 3, i = 10, PV = 1,000, FV = 0, and then press the PMT button to get PMT = 402.1148036, rounded to $402.11.
Now make the following points regarding the amortization schedule:
· The $402.11 annual payment includes both interest and principal. Interest in the first year is calculated as follows:
1st year interest = i ´ beginning balance = 0.1 ´ $1,000 = $100.
· The repayment of principal is the difference between the $402.11 annual payment and the interest payment:
1st year principal repayment = $402.11  $100 = $302.11.
· The loan balance at the end of the first year is:
1st year ending balance = beginning balance – principal repayment
= $1,000  $302.11 = $697.89.
· We would continue these steps in the following years.
· Notice that the interest each year declines because the beginning loan balance is declining. Since the payment is constant, but the interest component is declining, the principal repayment portion is increasing each year.
· The interest component is an expense which is deductible to a business or a homeowner, and it is taxable income to the lender. If you buy a house, you will get a schedule constructed like ours, but longer, with 30 ´ 12 = 360 monthly payments if you get a 30year, fixed rate mortgage.
· The payment may have to be increased by a few cents in the final year to take care of rounding errors and make the final payment produce a zero ending balance.
· The lender received a 10% rate of interest on the average amount of money that was invested each year, and the $1,000 loan was paid off. This is what amortization schedules are designed to do.
· Most financial calculators have amortization functions built in.
k. Suppose on January 1 you deposit $100 in an account that pays a nominal, or quoted, interest rate of 11.33463 percent, with interest added (compounded) daily. How much will you have in your account on October 1, or after 9 months?
Answer: The daily periodic interest rate is r_{Per} = 11.3346%/365 = 0.031054%. There are 273 days between January 1 and October 1. Calculate FV as follows:
FV_{273} = $100(1.00031054)^{273}
= $108.85.
Using a financial calculator, input n = 273, i = 0.031054, PV = 100, and PMT = 0. Pressing FV gives $108.85.
An alternative approach would be to first determine the effective annual rate of interest, with daily compounding, using the formula:
EAR =  1 = 0.12 = 12.0%.
(Some calculators, e.g., the hp 10b and 17b, have this equation built in under the ICNV [interest conversion] function.)
Thus, if you left your money on deposit for an entire year, you would earn $12 of interest, and you would end up with $112. The question, though, is this: how much will be in your account on October 1, 2002?
Here you will be leaving the money on deposit for 9/12 = 3/4 = 0.75 of a year.

0 0.75 1
  
100 FV = ? 112
You would use the regular setup, but with a fractional exponent:
FV_{0.75} = $100(1.12)^{0.75} = $100(1.088713) = $108.87.
This is slightly different from our earlier answer, because n is actually 273/365 = 0.7479 rather than 0.75.
Fractional time periods
Thus far all of our examples have dealt with full years. Now we are going to look at the situation when we are dealing with fractional years, such as 9 months, or 10 years. In these situations, proceed as follows:
· As always, start by drawing a time line so you can visualize the situation.
· Then think about the interest ratethe nominal rate, the compounding periods per year, and the effective annual rate. If you have been given a nominal rate, you may have to convert to the ear, using this formula:
EAR = .
· If you have the effective annual rateeither because it was given to you or after you calculated it with the formulathen you can find the PV of a lump sum by applying this equation:
PV = FV_{t} .
· Here t can be a fraction of a year, such as 0.75, if you need to find the PV of $1,000 due in 9 months, or 450/365 = 1.2328767 if the payment is due in 450 days.
· If you have an annuity with payments different from once a year, say every month, you can always work it out as a series of lump sums. That procedure always works. We can also use annuity formulas and calculator functions, but you have to be careful.
l. 1. What is the value at the end of year 3 of the following cash flow stream if the quoted interest rate is 10 percent, compounded semiannually?
0 1 2 3 YEARS
      
100 100 100
Answer:

0 1 2 3
      
100 100 100
110.25 = 100(1.05)^{2}
121.55 = 100(1.05)^{4}
331.80
Here we have a different situation. The payments occur annually, but compounding occurs each 6 months. Thus, we cannot use normal annuity valuation techniques. There are two approaches that can be applied: (1) treat the cash flows as lump sums, as was done above, or (2) treat the cash flows as an ordinary annuity, but use the effective annual rate:
Now we have this 3period annuity:
FVA_{3} = $100(1.1025)^{2} + $100(1.1025)^{1} + $100 = $331.80.
You can plug in n = 3, I = 10.25, PV = 0, and PMT = 100, and then press the FV button to find FV = $331.80.
l. 2. What is the PV of the same stream?

0 1 2 3
      
100 100 100
90.70
82.27 _{ }PV = 100(1.05)^{4}
74.62
247.59
PV = $100(2.4759) = $247.59 AT 10.25%.
To use a financial calculator, input N = 3, I = 10.25, PMT = 100, FV = 0, and then press the PV key to find PV = $247.59.
l. 3. Is the stream an annuity?
Answer: The payment stream is an annuity in the sense of constant amounts at regular intervals, but the intervals do not correspond with the compounding periods. This kind of situation occurs often. In this situation the interest is compounded semiannually, so with a quoted rate of 10%, the ear will be 10.25%. Here we could find the effective rate and then treat it as an annuity. Enter N = 3, I = 10.25, PMT = 100, and FV = 0. Now press PV to get $247.59.
l. 4. An important rule is that you should never show a nominal rate on a time line or use it in calculations unless what condition holds? (Hint: think of annual compounding, when i _{Nom} = EAR = i_{Per}.) What would be wrong with your answer to questions l(1) and l(2) if you used the nominal rate (10%) rather than the periodic rate (i_{Nom }/2 = 10%/2 = 5%)?
Answer: i_{Nom} can only be used in the calculations when annual compounding occurs. If the nominal rate of 10% was used to discount the payment stream the present value would be overstated by $272.32  $247.59 = $24.73.
m. Suppose someone offered to sell you a note calling for the payment of $1,000 15 months from today. They offer to sell it to you for $850. You have $850 in a bank time deposit which pays a 6.76649 percent nominal rate with daily compounding, which is a 7 percent effective annual interest rate, and you plan to leave the money in the bank unless you buy the note. The note is not riskyyou are sure it will be paid on schedule. Should you buy the note? Check the decision in three ways:
(1) by comparing your future value if you buy the note versus leaving your money in the bank,
(2) by comparing the PV of the note with your current bank account, and
(3) by comparing the ear on the note versus that of the bank account.
Answer: You can solve this problem in three ways(1) by compounding the $850 now in the bank for 15 months and comparing that FV with the $1,000 the note will pay, (2) by finding the PV of the note and then comparing it with the $850 cost, and (3) finding the effective annual rate of return on the note and comparing that rate with the 7% you are now earning, which is your opportunity cost of capital. All three procedures lead to the same conclusion. Here is the time line:

0 1 1.25
  
850 1,000
(1) FV = $850(1.07)^{1.25} = $925.01 = amount in bank after 15 months versus $1,000 if you buy the note. (Again, you can find this value with a financial calculator. Note that certain calculators like the hp 12c perform a straightline interpolation for values in a fractional time period analysis rather than an effective interest rate interpolation. The value that the hp 12c calculates is $925.42.) This procedure indicates that you should buy the note.
Alternatively, 15 months = (1.25 years)(365 days per year) = 456.25 456 days.
FV_{456} = $850(1.00018538)^{456}
= $924.97.
The slight difference is due to using n = 456 rather than n = 456.25.
(2) PV = $1,000/(1.07)^{1.25} = $918.90. Since the present value of the note is greater than the $850 cost, it is a good deal. You should buy it.
Alternatively, PV = $1000/(1.00018538)^{456} = $918.95.
(3) FV_{n} = PV(1 + i)^{n}, SO $1,000 = $850(1 + i) ^{1.25} = $1,000. Since we have an equation with one unknown, we can solve it for i. You will get a value of i = 13.88%. The easy way is to plug values into your calculator. Since this return is greater than your 7% opportunity cost, you should buy the note. This action will raise the rate of return on your asset portfolio.
Alternatively, we could solve the following equation:
$1,000 = $850(1 + i)^{456} for a daily i = 0.00035646,
With a result of EAR = EFF% = (1.00035646)^{365 } 1 = 13.89%.
Chapter 5 – Bonds, Bond Valuation and Interest Rates
Problems
51
The current price is the PV of interest and principal. We use the PV function to calculate the current price
Maturity = 12 years
Par value = 1,000
Annual Interest = 80
YTM = 9%
Current Price = PV (9%, 12, 80, 1000) = $928.39
= 80 * 7.1607 + 1000 * 0.3555 = $928.356
52
100+1000850/12/1000+850/2 = 112.5/925 = .1216 or 12.16%
53
Calculator solution:
N = 7
I/Y = 8
PMT = 90
FV = 1000
CPT PV = 1052.06
Current yield = coupon/current price
= 90 / 1052.06
= 8.55%
54
55
YTMLiquidityRisk free = default risk premium...
YTM = 9%
Liquidity = 0.5%
Risk free = 6%
9%0.5%6% = 2.5%
56
R = R* + IP + DRP + LP + MRP
6.3% = 3% + 3% + 0% + 0% + MRP
6.3%  6% = MRP
0.3% = MRP
57
Answer: FV =1,000, PMT= 50, N= 16, R= 4.25%, PV=?
Present Value = $1,085.80
58
Given:
TTM = 10 years Face Value = $1,000 C = 8% ($40 semiannual) Pricecallable = $1,050 TTC = 5 years YTM = solve
Price = $1,100 YTC = solve
Using Finance Functions: (For Yield to maturity) n
= 20 PMT = 40 PV = 1100 FV = 1000
i/2
= 3.3085%
i = 6.617%
YTM = 6.62%
Using Finance Functions: (For Yield to call)
n
= 9 PMT = 40 PV = 1100 FV = 1050
Note:
Instead of having a tenth payment, the exdividend assumption accounts for
the tenth cashflow
in addition to the final value. i/2 = 3.1924%
i = 6.3849%
YTC = 6.38%
59
Bond L:
Face Value = $1,000, Coupon Payment = $100, maturity = 15
Bond S:
Face Value = $1,000, Coupon Payment = $100, maturity = 1
1) Interest Rate = 5%
Bond L:
= ($100 x 10.3797) + ($1,000 x 0.481)
= $1,037.97 + $481.017
= $1,518.98
Bond S:
= ($100 x 0.952381) + ($1,000 x 0.952381)
= $95.2381 + $952.381
= $1,047.62
As you can see, the bond sells at a premium (Above Face Value) since the coupon rate is higher than the market rate. The coupon rate is 10%, while the market rate is 5%.
1) 5% Interest: 

Bond L: 
Bond S: 

Settlement Date 
1/1/00 
Settlement Date 
1/1/00 

Maturity Date 
1/1/15 
Maturity Date 
1/1/01 

Annual Coupon Rate 
0.1 
Annual Coupon Rate 
0.1 

Yield to Maturity 
0.05 
Yield to Maturity 
0.05 

Face Value (% of Par) 
100 
Face Value (% of Par) 
100 

Coupons per Year 
1 
Coupons per Year 
1 

Bond Price (% of par) 
151.8983 
Bond Price (% of par) 
104.7619048 

Bond Price 
$1,518.98 
Bond Price 
$1,047.62 
2) 8% Interest: 

Bond L: 
Bond S: 

Settlement Date 
1/1/00 
Settlement Date 
1/1/00 

Maturity Date 
1/1/15 
Maturity Date 
1/1/01 

Annual Coupon Rate 
0.1 
Annual Coupon Rate 
0.1 

Yield to Maturity 
0.08 
Yield to Maturity 
0.08 

Face Value (% of Par) 
100 
Face Value (% of Par) 
100 

Coupons per Year 
1 
Coupons per Year 
1 

Bond Price (% of par) 
117.119 
Bond Price (% of par) 
101.8518519 

Bond Price 
$1,171.19 

Bond Price 
$1,018.52 
3) 12% Interest: 

Bond L: 
Bond S: 

Settlement Date 
1/1/00 
Settlement Date 
1/1/00 

Maturity Date 
1/1/15 
Maturity Date 
1/1/01 

Annual Coupon Rate 
0.1 
Annual Coupon Rate 
0.1 

Yield to Maturity 
0.12 
Yield to Maturity 
0.12 

Face Value (% of Par) 
100 
Face Value (% of Par) 
100 

Coupons per Year 
1 
Coupons per Year 
1 

Bond Price (% of par) 
86.37827 
Bond Price (% of par) 
98.21428571 

Bond Price 
$863.78 

Bond Price 
$982.14 
The risk that arises for bond owners from fluctuating interest rates is called Interest Rate Risk. How much interest rate risk a bond has depends on how sensitive its price is to interest rate changes.
The reason that longerterm bonds have greater interest rate sensitivity is that a large portion of the bond’s value comes from the $1,000 face amount. The present value of this amount is not greatly affected by a small change in interest rates if the amount to be received in one year. Even a small change in interest rates, however, once it is compounded for 15 years, can have a significant effect on the present value.
510
511
512
513
Current price = 80/.0821 = 974
YTM = 80+1000974/5/1000+974/2 = 85.2/987 = .0863 or 8.63%
514
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