IB 1320 Notes
Lecture 1: The firm: Public firms – shares traded at a stock exchange;
market value = the share price x number of shares outstanding;
limited liability: at most, shareholders lose value of shares;
Separation between ownership and management – shareholders elect board of directors, 1
share = 1 vote; BoD elects and monitors CEO, sets executive remuneration; Managers are
legally distinct from owners, pay their own taxes
Management’s objective should always be to maximise shareholder’s values.
Investments as projects: project: a set of cash flows in the present and at different points in
the future. Projects usually entail immediate outflows (costs/expenses)… followed by a
series of future inflows (payoffs/revenues).
Firm Decisions: Capital Decisions: How can firms choose the best investments when they
have multiple investment opportunities?
Capital Structure: After deciding which investments to undertake, how should the firm pay
for the investments?
Payout decision: After the investments have generated cash, how should we return the cash
to the shareholders?
Project valuation: Valuation: assigning a project a monetary value today. Valuation is
forward looking. Valuation is relative to alternatives:
Law of one price: identical projects have identical prices (under the assumption that markets
are perfect). This means therefore that value in finance is defined in relative terms.
Central to LOP-pricing is the opportunity cost of capital – the cost of foregoing alternative
projects/investments. Example: You consider investing in Project X. Similar projects earn a
return of w%. Project X has opportunity costs w%.
Debt and Equity: Bonds = debts, stocks = equity.
You can think of buying a bond as lending money to the issuer.
Lecture 2: Present Value: Time value of money: £10 present value is more valuable than
£10 future value due to an interest rate of r if you invest today. This notion is known as the
time value of money.
How we represent different time periods: 0 = today. 1 = next period (day, year, etc). t =
some future period. T = the final time period (if finite duration)
How we represent cash flows (-ve or +ve): C0 = cash amount today. C1 = cash amount in the
next period.
C1−C0 C1
Rates of return: from investing C0 and receiving C1: r 0,1 =r 1= = −1
C0 C0
C o (1+r )=C 1
Formula 1 is important as it gives the percent change in our initial investment (when x100)
t
In general, for t periods: C t=C 0 ( 1+ r )
,Two important simplifying assumptions: Perfect capital markets which means: we pay no
taxes, no transaction costs, etc; we have no information asymmetries; individual buyers and
sellers are small.
No risk or uncertainty, which means: we have perfect foresight about the future, we know
for certain the cash flows of each investment.
Future Values of Money: Holding Period Return:
172.8−100
The three year return of the project is: r 3 = =72.8 %
100
C t −C 0 C t
= −1=r t
C0 C0
The return rt is the holding period return for a t-period project. It reflects compounding, and
relates to 1-period returns as follows: (1+72.8%) = (1+20%)(1+20%)(1+20%) = (1+20%) 3
t
In general for t periods: ( 1+r t )=( 1+r )( 1+r ) … ( 1+r )= ( 1+ r )
Manipulations of the Formula: Finding Period Returns: What constant 1 year interest rate r
would give us a 2 year holding period return of 50%?
( 1+r )2=1+50 % ⇔ r =❑2 √ 1.50−1=22.47 %
In general r = √t 1+r t −1
Manipulations of the Formula: Fractional Periods: The formula also works for fractional
periods. Say the interest rate is 5% per year and we want to find out the rate of return
within 6 months. Because (1+r0.5)2 = (1+r1) we compute:
0.5
( 1+r 0.5 )=( 1+0.05 ) =1.02469 so r0.5 = 2.469%
We can also use the formula to determine how long it will take for us to recoup a specific
return, say 100%, given a specific interest rate, say 3% per year.
log (1+100 %) log ( 2 )
( 1+3 % )x =( 1+100 % ) → x= = =23.5
log (1+3 %) log (1.03)
Present Value of Money: An amount tomorrow is worth less today, because money can
earn interest. Use discounting to get present values from future values.
C1 C2 C3
The equation for the PV is: C 0= + + +… .
1+ r ( 1+r ) ( 1+r )3
2
Here r, reflects the time value of money and is commonly called the opportunity cost of
capital (OCC) – that is, the rate at which our money can grow if we invest in other similar
projects. r is the risk-free rate.
1 t
The term ( )
1+r 1
is called the discount factor and it translates future cash flows into money
today.
Opportunity Cost of Capital: Bonds: Zero-coupon government bonds cost C0 today, pay
CT = (1 + r)C0 in the future.
The economy wide interest rate is the cost of capital for bonds. So the prices of bonds varies
inversely with the interest rate.
A bond promises to pay £1000 in one year at the prevailing interest rate of 5%. This bond
has a price equal to PV = 1000/1.05 = 952.38
, Net Present Value: A Verbal Explanation: Investment projects entail cash in and outflows
over time. Outflows arise from startup costs, cleanup costs etc.
The NPV of a project is the present value of cash inflows minus the present value of cash
outflows. To apply the NPV rule follow these steps: 1. Translate inflows into £’s today and
sum to PV (inflows). 2. Translate outflows into £’s today and sum to PV (outflows). 3.
Subtract PV(outflows) from PV(inflows).
Note: most commonly projects have one initial outflow at time 0, i.e. the cost of the
investment.
T
Ct
Present value of future cash flows: PV =∑
t =1 ( 1+r )t
T T
Ct Ct
Net Present Value of the project: NPV =C 0+ ∑ t
=∑
( 1+r )
t =1 t =0 (1+r )t
NPV tells us how much a specific project is worth today
Capital budgeting rule: In a perfect market we should take all positive NPV projects. A
positive NPV project increases your wealth by more than what other similar projects in the
economy can.
Lecture 3: Perpetuities and Annuities:
Perpetuities: A perpetuity is an asset that pays C per period forever. With interest rate r,
cash flow C, and present value PV:
∞
C C1
PV =∑ t =
t =1 ( 1+r ) r
Time subscript “1” reminds you that the stream of cash flows start from the next period, not
the current one. Note: that we have assumed C and r are the same every period.
Growing Perpetuities: Cash flows are CF1, CF2 = CF1 (1 + g)1, CF3 = CF1 (1 + g)2….
Goes up by growth rate g each year.
∞ t −1
C F 1 ( 1+ g ) C F1
PV =∑ =
t =1 ( 1+ r ) t
r−g
The growth rate g effectively “reduces” the interest rate r.
How can a growing infinite sum be less than infinite? Because the growth is not fast enough.
This is only the case if r > g. The formula makes no sense if r < g.
Note again the subscript is 1. So you need to have your starting value as the income in the
first period (not immediately). If there is any immediate payments you need to add that on
at the end.
The Gordon Growth Model: Share Price Valuation: We can use the growing perpetuity
formula to derive a firm’s share price, because in principle firms can exist forever.
Example: A firm pays dividends per share that start at £10 next year and grow at g = 5%
perpetually. Capital costs 10%. Using the growing perpetuity formula, the firm’s share price
10
is: Stock Price P Today = =200
0.10−0.05
Lecture 1: The firm: Public firms – shares traded at a stock exchange;
market value = the share price x number of shares outstanding;
limited liability: at most, shareholders lose value of shares;
Separation between ownership and management – shareholders elect board of directors, 1
share = 1 vote; BoD elects and monitors CEO, sets executive remuneration; Managers are
legally distinct from owners, pay their own taxes
Management’s objective should always be to maximise shareholder’s values.
Investments as projects: project: a set of cash flows in the present and at different points in
the future. Projects usually entail immediate outflows (costs/expenses)… followed by a
series of future inflows (payoffs/revenues).
Firm Decisions: Capital Decisions: How can firms choose the best investments when they
have multiple investment opportunities?
Capital Structure: After deciding which investments to undertake, how should the firm pay
for the investments?
Payout decision: After the investments have generated cash, how should we return the cash
to the shareholders?
Project valuation: Valuation: assigning a project a monetary value today. Valuation is
forward looking. Valuation is relative to alternatives:
Law of one price: identical projects have identical prices (under the assumption that markets
are perfect). This means therefore that value in finance is defined in relative terms.
Central to LOP-pricing is the opportunity cost of capital – the cost of foregoing alternative
projects/investments. Example: You consider investing in Project X. Similar projects earn a
return of w%. Project X has opportunity costs w%.
Debt and Equity: Bonds = debts, stocks = equity.
You can think of buying a bond as lending money to the issuer.
Lecture 2: Present Value: Time value of money: £10 present value is more valuable than
£10 future value due to an interest rate of r if you invest today. This notion is known as the
time value of money.
How we represent different time periods: 0 = today. 1 = next period (day, year, etc). t =
some future period. T = the final time period (if finite duration)
How we represent cash flows (-ve or +ve): C0 = cash amount today. C1 = cash amount in the
next period.
C1−C0 C1
Rates of return: from investing C0 and receiving C1: r 0,1 =r 1= = −1
C0 C0
C o (1+r )=C 1
Formula 1 is important as it gives the percent change in our initial investment (when x100)
t
In general, for t periods: C t=C 0 ( 1+ r )
,Two important simplifying assumptions: Perfect capital markets which means: we pay no
taxes, no transaction costs, etc; we have no information asymmetries; individual buyers and
sellers are small.
No risk or uncertainty, which means: we have perfect foresight about the future, we know
for certain the cash flows of each investment.
Future Values of Money: Holding Period Return:
172.8−100
The three year return of the project is: r 3 = =72.8 %
100
C t −C 0 C t
= −1=r t
C0 C0
The return rt is the holding period return for a t-period project. It reflects compounding, and
relates to 1-period returns as follows: (1+72.8%) = (1+20%)(1+20%)(1+20%) = (1+20%) 3
t
In general for t periods: ( 1+r t )=( 1+r )( 1+r ) … ( 1+r )= ( 1+ r )
Manipulations of the Formula: Finding Period Returns: What constant 1 year interest rate r
would give us a 2 year holding period return of 50%?
( 1+r )2=1+50 % ⇔ r =❑2 √ 1.50−1=22.47 %
In general r = √t 1+r t −1
Manipulations of the Formula: Fractional Periods: The formula also works for fractional
periods. Say the interest rate is 5% per year and we want to find out the rate of return
within 6 months. Because (1+r0.5)2 = (1+r1) we compute:
0.5
( 1+r 0.5 )=( 1+0.05 ) =1.02469 so r0.5 = 2.469%
We can also use the formula to determine how long it will take for us to recoup a specific
return, say 100%, given a specific interest rate, say 3% per year.
log (1+100 %) log ( 2 )
( 1+3 % )x =( 1+100 % ) → x= = =23.5
log (1+3 %) log (1.03)
Present Value of Money: An amount tomorrow is worth less today, because money can
earn interest. Use discounting to get present values from future values.
C1 C2 C3
The equation for the PV is: C 0= + + +… .
1+ r ( 1+r ) ( 1+r )3
2
Here r, reflects the time value of money and is commonly called the opportunity cost of
capital (OCC) – that is, the rate at which our money can grow if we invest in other similar
projects. r is the risk-free rate.
1 t
The term ( )
1+r 1
is called the discount factor and it translates future cash flows into money
today.
Opportunity Cost of Capital: Bonds: Zero-coupon government bonds cost C0 today, pay
CT = (1 + r)C0 in the future.
The economy wide interest rate is the cost of capital for bonds. So the prices of bonds varies
inversely with the interest rate.
A bond promises to pay £1000 in one year at the prevailing interest rate of 5%. This bond
has a price equal to PV = 1000/1.05 = 952.38
, Net Present Value: A Verbal Explanation: Investment projects entail cash in and outflows
over time. Outflows arise from startup costs, cleanup costs etc.
The NPV of a project is the present value of cash inflows minus the present value of cash
outflows. To apply the NPV rule follow these steps: 1. Translate inflows into £’s today and
sum to PV (inflows). 2. Translate outflows into £’s today and sum to PV (outflows). 3.
Subtract PV(outflows) from PV(inflows).
Note: most commonly projects have one initial outflow at time 0, i.e. the cost of the
investment.
T
Ct
Present value of future cash flows: PV =∑
t =1 ( 1+r )t
T T
Ct Ct
Net Present Value of the project: NPV =C 0+ ∑ t
=∑
( 1+r )
t =1 t =0 (1+r )t
NPV tells us how much a specific project is worth today
Capital budgeting rule: In a perfect market we should take all positive NPV projects. A
positive NPV project increases your wealth by more than what other similar projects in the
economy can.
Lecture 3: Perpetuities and Annuities:
Perpetuities: A perpetuity is an asset that pays C per period forever. With interest rate r,
cash flow C, and present value PV:
∞
C C1
PV =∑ t =
t =1 ( 1+r ) r
Time subscript “1” reminds you that the stream of cash flows start from the next period, not
the current one. Note: that we have assumed C and r are the same every period.
Growing Perpetuities: Cash flows are CF1, CF2 = CF1 (1 + g)1, CF3 = CF1 (1 + g)2….
Goes up by growth rate g each year.
∞ t −1
C F 1 ( 1+ g ) C F1
PV =∑ =
t =1 ( 1+ r ) t
r−g
The growth rate g effectively “reduces” the interest rate r.
How can a growing infinite sum be less than infinite? Because the growth is not fast enough.
This is only the case if r > g. The formula makes no sense if r < g.
Note again the subscript is 1. So you need to have your starting value as the income in the
first period (not immediately). If there is any immediate payments you need to add that on
at the end.
The Gordon Growth Model: Share Price Valuation: We can use the growing perpetuity
formula to derive a firm’s share price, because in principle firms can exist forever.
Example: A firm pays dividends per share that start at £10 next year and grow at g = 5%
perpetually. Capital costs 10%. Using the growing perpetuity formula, the firm’s share price
10
is: Stock Price P Today = =200
0.10−0.05
