FINANCE STATISTICS STUDY NOTES
TOPIC 1
THE BASIC IDEAS, SCOPE, AND TOOLS OF FINANCE
Introduction
Finance is the economics of allocating resources across time.
The market interest rate is the rate of exchange between present and future resources. Any
transaction that a participant might make by borrowing or lending at the market interest rate of
interest will produce a result that lies on the Financial Exchange Line.
The expression that moves us up or down the exchange line is:
CF(1) = CF(0) (1 + I)
Present Value
Present Value is defined as the amount of money you must invest or lend at the present time so as to
end up with a paticular amount of money in the future. Present value is also an accurate
representation of what the market does when it sets a price on a financial asset.
The present value of all present and future resources (cash flows) is known as present wealth. It is a
useful benchmark or standard to judge whether an individual will be better or worse off when
undertaking a proposed financial decision.
One cannot change present wealth merely bey transacting at the market rate. Borrowing and lending
at this rates simply moves you up and down the exchange line--so it impacts only time allocation.
(Only investing in real assets can increase present wealth. This produuces a parallel shift in the
financial exchange line.)
The amount of parallel shift in the exchange line can be calculated using the present value equation.
PV = CF(0) + CF(1)
(1 + i)
Net Present Value
PV (inflow - outflow) = - CF(0) + CF(1)
(1 + i)
The number produced by this expression is exactly equal to the change in present wealth.
Internal Rate of Return (IRR)
1) Average per period rate of return on the $$ invested.
2) The discount rate that equates the present values of investment’s cash inflows and outflows (rate
that causes NPV to equal 0).
NPV = 0 = -CF(0) + CF(1)
(1 + IRR)
Multiple Period Finance
PV = CF(2)
(1 + I(2))^2
PREPARED BY MR ANTONY AMBIA Page 1
,Compound Interest
CF(0) x [1+ (i/m)] ^mt
m = number of times/period compounding takes place
t = Number of periods
Continuous Compounding
CF(0) x (e^it)
Multiple Period Cash Flows
PV = CF(1) + CF(2) + CF(3)
(1+i(1)) (1+i(2))^2 (1+i(3))^3
NPV = CF(0) + CF(1) + CF(2) + CF(3)
( 1+i(1)) (1+i(2))^2 (1+i(3))^3
Finding IRR in these situations is more complicated, but still is found by setting NPV = 0, and solving
for discount rate. Usually found with good calculator, or by trial and error.
Math Summary:
PV = CF(t)
(1+ i(t))^t
From t=1 to T. Reduces to 1+i when the discount rate is the same across all periods.
Present Value Tables
Helpful when calculating far off periods. Find interest rate, and period. Multiply times actual
amount.
Perpetuity
PV = CF
i
When CF grows or declines at a constant rate:
PV = CF
(i - g)
Where g is the constant per period growth rate of the cash flow.
Interest Rates, Interest Future Rates, and Yields
The relationship between spot rates (those which begin at the present, and run to some future point),
forward rates (rates that begin at some point in the future), and and YTM (yield to maturity--the IRR
of a bond’s promised cash flows, total yield given term structure and rates), can be seen as follows:
Spot Rates:
$923 = $40 + $40 + $1040
PREPARED BY MR ANTONY AMBIA Page 2
, (1.05) (1.06)^2 (1.07)^3
Forward Rates:
$923 = $40 + $40 + $1040
(1.05) (1.05)(1.07) (1.05)(1.07)(1.09)
YTM:
$923 = $40 + $40 + $1040
(1.069) (1.069)^2 (1.069)^3
Summarizing:
(1 + i(2))^2 = (1 + 0f(1))(1 + 0f(2))
Example:
(1 + i(3))^3 = (1 + 0f(1)) (1 + 1f(2)) (1 + 2f(3))
(1.07)^3 = (1.05) (1.07) (1 + 2f(3))
2f(3)= 9%
Bond YTM
Yield’s can be different even if discount rates are the same due to the coupon effect on the yield to
maturity--that is, cash flow amounts that occur in different spot rate periods ipact the yield. YTM’s
reflect not only rates, but amounts invested.
Interest Future Rates
Financial markets allow you to guard against the risk of an interest rate change giving your
investment a negative NPV (whereas formerly, it was positive). One tactice would be to sell an
interest rate futures contract in the aproximate amounts and timings of the cash inflows of the
project.
Example:
Investment has following cash flows:
t(0) t(1) t(2)
-$1,700 $1000 $1000
i(1) = 10%
i(2) = 11%
NPV = -1700 + 1000 + 1000
(1.10) (1.11)^2
The forward rate implied by this term structure is :
(1 + i(2))^2 = (1+ i(1))(1 + 1f(2))
(1 + 1f(2)) = (1 + i(2))^2/ (1+ i(1))
(1 + 1f(2)) = (1.11)^2/(1.10)
1f(2) = 12.0009%
PREPARED BY MR ANTONY AMBIA Page 3
, Suppose the 1f(2) rate changes to 15%. Then, by plugging this into the equation above,
i(2) becomes 12.4722%. By doing an NPV, we get
NPV = -1700 + 1000 + 1000
(1.10) (1.124722)^2
=- $.40
The positive NPV became negative due to the change in interest rates. You could hedge by selling a
$1000 futures contract at a forward interest rate of 12.009%. (if the forward rate increases, the price
of your security will decline, but, since you have a contract to sell at a higher price, the value of your
contract will increase. The increase offsets the NPV of your investment.
Futures price = $1000
(1 + 1f(2) )
= $1000/1.12009 = $892.79
If forward rates gor to 15, then price of the cash flow is 1000/1.15=869.57. You sold at a higher price,
so your investment value increases by 892-869=21.11, offsetting the NPV decrease.
PREPARED BY MR ANTONY AMBIA Page 4
TOPIC 1
THE BASIC IDEAS, SCOPE, AND TOOLS OF FINANCE
Introduction
Finance is the economics of allocating resources across time.
The market interest rate is the rate of exchange between present and future resources. Any
transaction that a participant might make by borrowing or lending at the market interest rate of
interest will produce a result that lies on the Financial Exchange Line.
The expression that moves us up or down the exchange line is:
CF(1) = CF(0) (1 + I)
Present Value
Present Value is defined as the amount of money you must invest or lend at the present time so as to
end up with a paticular amount of money in the future. Present value is also an accurate
representation of what the market does when it sets a price on a financial asset.
The present value of all present and future resources (cash flows) is known as present wealth. It is a
useful benchmark or standard to judge whether an individual will be better or worse off when
undertaking a proposed financial decision.
One cannot change present wealth merely bey transacting at the market rate. Borrowing and lending
at this rates simply moves you up and down the exchange line--so it impacts only time allocation.
(Only investing in real assets can increase present wealth. This produuces a parallel shift in the
financial exchange line.)
The amount of parallel shift in the exchange line can be calculated using the present value equation.
PV = CF(0) + CF(1)
(1 + i)
Net Present Value
PV (inflow - outflow) = - CF(0) + CF(1)
(1 + i)
The number produced by this expression is exactly equal to the change in present wealth.
Internal Rate of Return (IRR)
1) Average per period rate of return on the $$ invested.
2) The discount rate that equates the present values of investment’s cash inflows and outflows (rate
that causes NPV to equal 0).
NPV = 0 = -CF(0) + CF(1)
(1 + IRR)
Multiple Period Finance
PV = CF(2)
(1 + I(2))^2
PREPARED BY MR ANTONY AMBIA Page 1
,Compound Interest
CF(0) x [1+ (i/m)] ^mt
m = number of times/period compounding takes place
t = Number of periods
Continuous Compounding
CF(0) x (e^it)
Multiple Period Cash Flows
PV = CF(1) + CF(2) + CF(3)
(1+i(1)) (1+i(2))^2 (1+i(3))^3
NPV = CF(0) + CF(1) + CF(2) + CF(3)
( 1+i(1)) (1+i(2))^2 (1+i(3))^3
Finding IRR in these situations is more complicated, but still is found by setting NPV = 0, and solving
for discount rate. Usually found with good calculator, or by trial and error.
Math Summary:
PV = CF(t)
(1+ i(t))^t
From t=1 to T. Reduces to 1+i when the discount rate is the same across all periods.
Present Value Tables
Helpful when calculating far off periods. Find interest rate, and period. Multiply times actual
amount.
Perpetuity
PV = CF
i
When CF grows or declines at a constant rate:
PV = CF
(i - g)
Where g is the constant per period growth rate of the cash flow.
Interest Rates, Interest Future Rates, and Yields
The relationship between spot rates (those which begin at the present, and run to some future point),
forward rates (rates that begin at some point in the future), and and YTM (yield to maturity--the IRR
of a bond’s promised cash flows, total yield given term structure and rates), can be seen as follows:
Spot Rates:
$923 = $40 + $40 + $1040
PREPARED BY MR ANTONY AMBIA Page 2
, (1.05) (1.06)^2 (1.07)^3
Forward Rates:
$923 = $40 + $40 + $1040
(1.05) (1.05)(1.07) (1.05)(1.07)(1.09)
YTM:
$923 = $40 + $40 + $1040
(1.069) (1.069)^2 (1.069)^3
Summarizing:
(1 + i(2))^2 = (1 + 0f(1))(1 + 0f(2))
Example:
(1 + i(3))^3 = (1 + 0f(1)) (1 + 1f(2)) (1 + 2f(3))
(1.07)^3 = (1.05) (1.07) (1 + 2f(3))
2f(3)= 9%
Bond YTM
Yield’s can be different even if discount rates are the same due to the coupon effect on the yield to
maturity--that is, cash flow amounts that occur in different spot rate periods ipact the yield. YTM’s
reflect not only rates, but amounts invested.
Interest Future Rates
Financial markets allow you to guard against the risk of an interest rate change giving your
investment a negative NPV (whereas formerly, it was positive). One tactice would be to sell an
interest rate futures contract in the aproximate amounts and timings of the cash inflows of the
project.
Example:
Investment has following cash flows:
t(0) t(1) t(2)
-$1,700 $1000 $1000
i(1) = 10%
i(2) = 11%
NPV = -1700 + 1000 + 1000
(1.10) (1.11)^2
The forward rate implied by this term structure is :
(1 + i(2))^2 = (1+ i(1))(1 + 1f(2))
(1 + 1f(2)) = (1 + i(2))^2/ (1+ i(1))
(1 + 1f(2)) = (1.11)^2/(1.10)
1f(2) = 12.0009%
PREPARED BY MR ANTONY AMBIA Page 3
, Suppose the 1f(2) rate changes to 15%. Then, by plugging this into the equation above,
i(2) becomes 12.4722%. By doing an NPV, we get
NPV = -1700 + 1000 + 1000
(1.10) (1.124722)^2
=- $.40
The positive NPV became negative due to the change in interest rates. You could hedge by selling a
$1000 futures contract at a forward interest rate of 12.009%. (if the forward rate increases, the price
of your security will decline, but, since you have a contract to sell at a higher price, the value of your
contract will increase. The increase offsets the NPV of your investment.
Futures price = $1000
(1 + 1f(2) )
= $1000/1.12009 = $892.79
If forward rates gor to 15, then price of the cash flow is 1000/1.15=869.57. You sold at a higher price,
so your investment value increases by 892-869=21.11, offsetting the NPV decrease.
PREPARED BY MR ANTONY AMBIA Page 4
