Question 1
Assumptions
Risk adjusted rate of return which is necessary for calculating the intrinsic value is unknown. Therefore it is assumed that there is a risk adjusted rate of return of 10% per annum. The balance sheet and income statement below are randomly selected and can be used for the purpose of understanding the concept of cash flow from asset (CFFA). They may not represent ideal income statement for a firm in real market situation. The chosen company ABC is also fictitious.
ABC Company Balance Sheet ($ in Millions) 

Assets  2017  2018  Liabilities and Owners' Equity  2017  2018 
Current Assets  Current Liabilities  
Cash  12  13  Accounts Payable  8  9 
Accounts Receivable  25  26  Notes Payable  20  11 
Inventory  30  30  Total Current Liabilities  28  20 
Total Current Assets  67  69  LongTerm Liabilities  
LongTerm Debt  50  73  
Fixed Assets  Total LongTerm Liabilities  50  73  
Property, Plant, and Equipment  55  86  Owners' Equity  
Common Stock ($1 Par)  15  21  
Retained Earnings  19  41  
Total Owners' Equity  34  62  
Total Assets  112  155  Total Liabilities and Owners' Equity  112  155 
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ABC COMPANY Income Statement ($ in Millions) 

2018  
Sales  120 
Cost of Goods Sold  47 
Depreciation  8 
Earnings Before Interest and Taxes  65 
Interest Expense  5 
Taxable Income  60 
Taxes(27.5% of taxable income)  16.5 
Net Income  43.5 
Dividends  26.1 
Addition to Retained Earnings  17.4 
Tax expense in the income statement is found by applying 27.5% marginal tax rate on taxable income. 40% of net income is ploughed back to the investments so addition to retained earnings is 40% of net income while the remaining 60% of net income is paid as dividends.
Solution
Calculating cash flow from assets
Cash flow is simply net amount of money being transferred in and out a business ( Brealey et al., 2012) . Positive cash flows indicates that a business is getting more money than they are spending and this is good for business since the business is able to reinvest, pay debts, pay dividends and expenses. Earnings from sales are an example of cash inflow while spending $ 1000 to buy new office equipment is a cash outflow.
Cash flow from assets is the total cash flows that are related to assets of a given business entity. These include cash flow generated by operations, cash spent in acquiring fixed assets and change in working capital ( Faulkender et al., 2012) . Cash generated from operations are net income of a business plus all noncash expenses such as depreciation and amortization. These are cash inflows and therefore and therefore addition to cash flows from assets. An increase in working capital indicates the company is spending more cash to increase capita ( Brealey et al., 2012) . This is a cash outflow. Change in working capital is indicated by change in account payable, inventory and account receivable. An increase in fixed assets also indicates that the firm is spending more cash to purchase more assets and therefore this is a cash outflow.
Finding the cash flow involves finding the difference between the cash in and the cash out. In this case, the sum of (cash inflow) is taken less the sum of the amount spent (cash outflow). That is, net cash flow = total cash inflows – total cash outflows.
Calculating cash flow from assets (CFFA)
Operation cash flow
This measures ability of a firm to sell its products more than the cost of production. Calculation of this cash flow begins with earnings before interest and tax (EBIT). Operations do not depend on interest expense while depreciation expense is added back since it is a noncash expense which was subtracted in determining value of EBIT. Taxes that the firm pays during operations are also subtracted.
Therefore operation cash flow = EBIT + Depreciation – Tax Expense
For ABC company;
= 65 + 8 – 16.5
= 56.5
Capital spending on fixed assets
This is firm’s net investment in fixed assets. This can be calculate by finding the difference between ending net fixed assets and beginning net fixed assets. Depreciation is added to this difference since ending fixed assets in balance sheet was reduced by depreciation expense which was incurred during that period.
Therefore net capita spending = ending net fixed assets – beginning net fixed assets + Depreciation
For ABC Company, this is
Net capita spending = 8655+8 = 39
Change in net working capital
Net working capital is the difference between current assets and current liabilities ( Jensen, 2010) . To find change in networking capital for ABC, it is viable to find the difference between net working capital for 2017 and for 2018
Net working capital for 2017 = current assets – current liabilities
=67 – 28 = 39.
Net working capital for 2018 = current assets – current liabilities
= 69 – 20 = 49
Therefore change in net working capital = net working capital for 2018 – net working capital for 2017
= 49 – 39 = 10
Therefore CFFA = Total cash inflows – total cash outflows
Total cash inflows = operation cash flows = 56.5
Total cash out flows = capital spending + change in net working capital
= 39 + 10 = 49
Therefore CFFA = 56.5 – 49 = 7.5
But the values were in million dollars, so CFFA = 7,500,000 dollars.
Intrinsic Value
Since ABC is a constant growth perpetuity firm, the present value PV is given by;
PV = where r is the risk adjusted rate of return. Given our risk adjusted rate of return to be 10% and CFFA of 7,500,000 the intrinsic value of this firm is given by;
PV = = $ 75,000,000
 Question 2
The question is not stated correctly. Since perpetuity is annuity payable forever, it is the annuity that converges to perpetuity as time tends to infinity. So the equation should read, “ Let PV= converges to PV= as n tends to .”
Solution
Annuities are series of guaranteed constant payments that are payable at the end of a given period of time. Example of annuities include payments of $ 2 paid at the end of each year for the next five years or a payment of $ 5 in five years, $10 in 10 years,…, $50 in 50 years. The first example indicates annuity certain with constant payments while the second indicates annuity with increasing payments. Annuity can be fixed, increasing or decreasing.
The present value of annuity is the total value of all series of payments one unit of time before first statement.
PV = where CF is the regular payments, r is the risk adjusted rate of return and n is the period of payments. For example an insurance paying annuity of $1 per year for a period of two years at a rate of return of 10% has a present value of;
PV = = = 0.9 + 0.8 = 1.7
When annuity is payable forever, n becomes too big that it tends to infinity. This kind of annuity payment is known as perpetuity. The present value of perpetuity is given by
= = . As n approaches infinity, the value of converges to .
For example a perpetuity paying $1000 per year at a rate of return of 10% has a present value of
PV = = $10,000.
This can be verified by finding present value of the annuity at n=5, n=50, n=100 and n= 200
Inputs  5 periods  10%  $1000  
n  Interest per year  Present value  payments  Future value  
Output  $3790.79 
Inputs  50 periods  10%  $1000  
n  Interest per year  Present value  payments  Future value  
$9914.81 
Inputs  100 periods  10%  $1000  
n  Interest per year  Present value  payments  Future value  
$9999.27 
Inputs  200 periods  10%  $1000  
n  Interest per year  Present value  payments  Future value  
$9999.95 
It can be clearly seen that as periods increase, the present value of annuity converges to a $10,000 which is the value of . This is a proof that as n approaches the present value of annuity paying CF per year at a rate of r converges to .
Therefore converges to as n approaches .
 Question3
A b ond is a fixed income investment while its values change over time . Bonds are issued by investors or business entities to raise capital to finance various projects and activities. Price at which bonds are issued is set at a par value or face value usually $100 or $ 1000 per unit bond ( Jensen, 2010) . Actual value of bond depends on various factors such as credit quality, length until expiration, interest rates and coupon rates. Bonds pay a regular cash flow or coupons. A bond that doesn’t pay coupon is known as zerocoupon bond. The value of bond over different period of time is determined by calculating the present value of the bond.
Over a given period of investment, a bond value can go up or down depending on a change in interest rate and change in quality of credit. Bonds issued at a fixed rate of interest have a decreasing coupon values as they approach maturity. The cash flow of a coupon paying bond is described as a regular payment of coupons at end of each period and a final payment of coupon and par value at the end of investment. For example, a bond with a coupon rate of 7% per annum, par value of $1000 and a maturity period of 5 years will be paying * 1000 = 70 coupons per year and a final payment of 1000 + 70= 1070 at maturity. Therefore the cash flow such bond is given by;
Period  1  2  3  4  6 
Cash flow  70  70  70  70  1070 
If the rate of return in the above example is given by 5% per annum, then it is possible to find the present value of the bond.
Period  1  2  3  4  6 
Cash flow  70  70  70  70  1070 
Present value at 5% rate of return  66.67  63.49  60.47  57.59  838.37 
The present value of the bond = 66.67 + 63.49 + 60.47 +57.59 + 838.37 = 1086.37
Calculating present value at rate of return of 10%
Period  1  2  3  4  6 
Cash flow  70  70  70  70  1070 
Present value at 10% rate of return  63.63  57.85  52.63  47.81  664.60 
From the tables, it can be seen that that value of bond decreases with increase in interest rates. The bond value calculated at 5% rate of return is higher compared to value calculated at 10% rate of return. Similarly, the older the cash flow, the lower its present value ( Faulkender et al., 2012) . Calculating the first example at 5% rate of return for a period of 4 years, one period less, the value of the bond would be 1070.92. This indicates that a bond with maturity period has a higher value compared to bonds with fewer periods ( Damodaran, 2016) . Despite their high value, long term bonds are too risky as there is possibility of a change in interest rates.
 Question 4
The present value of constant growth perpetuity firm with a constant growth rate g, rate of return r and cash flows of CFFA is given by;
PV = + + + …
This is a geometric series with a common ratio of .
So the summation of series becomes . Thus this is the formula for present value of a constant growth perpetuity firm.
Given the present value of the comparable firm, cash flow and rate of growth, it is possible to find the market’s rate of return.
PV = but PV = 50,000,000, g= 0.03 and CFFA = 2,750,000
Therefore, 50,000,000 =
r0.03 = = 0.055
r = 0.055 + 0.03 = 0.085
Since the value of r is known , it is possible to find the value of subject firm
PV of subject firm =
Picking CFFA of 3,000,000, r =0.085 and g = 0.02
PV of subject firm = = = $ 46,153,846
Therefore the value of the subject firm is $ 46,153,846
References
Brealey, R. A., Myers, S. C., Allen, F., & Mohanty, P. (2012). Principles of corporate finance . Tata McGrawHill Education.
Damodaran, A. (2016). Damodaran on valuation: security analysis for investment and corporate finance (Vol. 324). John Wiley & Sons.
Faulkender, M., Flannery, M. J., Hankins, K. W., & Smith, J. M. (2012). Cash flows and leverage adjustments. Journal of Financial Economics , 103 (3), 632646.
Jensen, M. C. (2010). Value maximization, stakeholder theory, and the corporate objective function. Journal of applied corporate finance , 22 (1), 3242.