Week 1
Discount rate
Reward that investors demand for tying up the investment in the asset
Simple interest
Interest earned or paid to the original balance of the deposit loan. After 𝑇 periods, the total
balance will be 𝑋! = 𝑋(1 + 𝑟𝑇)
Compound interest
Interest is paid on previously charged/earned interest. After 𝑇 periods, the total balance will
be 𝑋! = 𝑋(1 + 𝑟)! . This is also equal to the future value 𝐹𝑉(𝑋, 𝑟, 𝑇)
Annual percentage rate (APR)
The APR is based on simple interest, not compound
Effective annual rate (EAR)
If there is more than one payment per year, than the APR is not the EAR. For 𝑘 payments per
"#$ %
year: 𝐸𝐴𝑅 = 01 + % 1 − 1
Present value
The value of your money today if you would invest it with an interest rate of 𝑟, such that it
&
would be worth exactly 𝑋 in the future, is called the present value: 𝑃𝑉(𝑋, 𝑟, 𝑇) = (()*)! . The
(
discount factor is equal to (()*)!
Net present value (NPV)
Investments will typically involve multiple cash-flows, at different points in time. Here are
initial investments negative and future receipts positive. The net present value is the present
,-"
value of a stream of cash-flows: 𝑁𝑃𝑉 = ∑!./0 (()*) "
, where 𝐶𝐹. is the cash-flow at time 𝑡.
A positive NPV means invest, a negative NPV means do not invest
Annuities
Annuities are a common cash-flow stream, consisting of the payment of a fixed cash-flow,
once per period, for a known finite number of periods.
, , (
The present value: 𝑃𝑉 = ∑!./( "
= 01 −
(()*) * !
(()*)
1. Here you can solve 𝐶 to find the
annuity payment in a mortgage, 𝑇 to find how many periods you need to repay with fixed
repayment (payback period), 𝑟 with 𝑁𝑃𝑉 = 0 to find the internal rate of return (IRR).
Perpetuities
Perpetuities are a common cash-flow stream, consisting of the payment of a fixed cash-flow,
, ,
once per period, indefinitely. The present value: 𝑃𝑉 = ∑1
./( (()*)" = *
Law of one price
The law of one price: if equivalent investment opportunities trade simultaneously in
different competitive markets, then they must trade for the same price
Arbitrage
Arbitrage is the practice of selling and buying equivalent goods in different markets to take
advantage of price differences in these different markets
Week 2
Bond characteristics
Issuer agrees to make specified payments to the bondholder on specified dates. The
characteristics of a bond are
- Maturity, length of the contract
- Par value / face value / principal, the payment at the bond’s maturity
- Coupon rate, an interest payment per dollar of par value, this is periodic
, - Discount rate, typically differs from coupon rate
Bond pricing
The income stream of bonds is known, but lies in the future. Bond value = PV of coupons +
, - , ( -
PV of par value. So 𝑃 = ∑!./( "
(()*)
+ !
= 01 −
(()*) * !
1+
(()*) ! (()*)
Bond pricing variations
The different bond pricing variations are:
- Frequency of coupon payments
- ‘Callable’ bonds
- ‘Indexed’ bonds
Multiple coupon payments per year
If the coupon payment is paid 𝑘 times a year, then the 𝐶 and 𝑟 need to be divided by 𝑘 (if
they are given as the APR), and the number of periods needs to be multiplied with 𝑘
Indexed bonds
In indexed bonds, both the principal repayment and coupon payments get indexed by the
#$%
" #$%!
,× -×
! #$%& #$%&
inflation rate, so the price becomes 𝑃 = ∑./( +
(()*)" (()*)!
Callable bonds
Bond issuer has the right to buy the bond at a certain strike price 𝑆. The price of the bond
can never exceed the strike price around the call day
Characteristics of bond price
- If a bond comes closer to a coupon payment, the price rises. If the coupon payment has
been paid, the price of the bond drops
- If the discount rate is higher than the coupon rate, the initial price of the bond will lower
than the par value of the bond, because we need to be compensated for the difference
between the discount rate and coupon rate
Yield to maturity
The YTM is the constant, hypothetical discount rate that, when used to compute the PV of a
bond’s cashflows, gives you the bond’s market price as the answer, used to compare bonds:
, -
𝑃 = ∑!./( (()3!4)" + (()3!4)!
Zero-coupon bonds
A bond without coupon payments, so we only obtain repayment of the par value at
-
maturity. 𝐶 = 0, and hence 𝑃56*7 = (()*)! . In this case, 𝑌𝑇𝑀 = 𝑟
Default risk
There is a promise of cash-flows, but the company can go bankrupt. However, Dutch
government bonds are essentially a risk-free asset. There is virtually no risk of no
repayment. We call the associated rate the ‘risk-free rate’, 𝑟8 . If the probability of default.
i.e. non-repayment, is higher, investors require a higher YTM. For risk-neutral investors this
means 𝐸(𝑌𝑇𝑀) = 𝑟8 , and for risk-averse investors 𝐸(𝑌𝑇𝑀) > 𝑟8 . We call the difference the
default premium
Holding period return
Default risk changes over time, but YTM assumes constant compound return. Then you can
look at the holding period return. It is the rate of return over a particular investment period.
It depends on the bond’s price at the end of the holding period.
It can be calculated as 𝐻𝑃𝑅09: = 𝑃: /𝑃0 − 1, where 𝑖 is the holding period, 𝑃: is the par
value discounted with the YTM in period 𝑖
Yield curve
Yields depend on lots of different features, they are not set in stone and may differ across
time. Moreover, it is ‘risky’ to buy a long-maturity bond, a longer horizon means more
, uncertainty about economic states and default risk. The relationship between YTM and
maturity is called the yield curve
Implications yield curve
A payment in a bond with a coupon payment should be discounted with the YTM of a zero-
bond in that specific year
Yield curve under uncertainty
Suppose that you can buy a 2-year zero with a YTM of 6%, or a 1-year zero with a YTM of 4%
and reinvest after one year. Than these two strategies should yield the same (expected)
return. So, 1.06; should be equal to 1.04 times the YTM in the second year of a 1-year zero
bond (the short/forward rate)
Yield curve under uncertainty
The rate one year from now may actually not be the same as its expectation. The spot rate
reflects market expectations of the future rate. The actual rate will almost surely be lower.
The liquidity premium on longer-term bonds compensates short-term investors for
uncertainty about future prices
Expectations hypothesis theory
The expectations hypothesis theory says that the forward rate equals market consensus of
future short interest rate, so 𝑓; = 𝐸(𝑟; ). If this theory holds true and if there are only zero
prices given of bonds, then the forward rate 1 + 𝑓: = 𝑃:9( /𝑃:
Liquidity preference theory
The liquidity preference theory says that long-term bonds are riskier, so 𝑓; > 𝐸(𝑟; ). Excess
is the liquidity premium. The yield curve has an upward bias built into long-term rates
Interpreting yield curves
An upward sloping yield curve could mean that rates are expected to rise, or large liquidity
premiums. An inverted yield curve could mean that interest rates are expected to fall, a
recession signal
Duration
,- (()3!4)'(
Define the importance of the cash-flow at time 𝑡 as 𝑤. = " # . The duration is
!
defined as duration = ∑./( 𝑡𝑤. . The higher the coupon rate, the lower the duration
Week 3
Book value
The book value is the accounting value of the firm’s equity as reflected on the company’s
balance sheet, it is backward looking
Market value
The market value is the price of the firm’s equity as resulting from market transactions.
Theoretically, this coincides with the present value of discounted expected dividends, it is
forward looking
Secondary market
The secondary market is a market, often a stock exchange, in which previously issued shares
are traded amongst investors
Dividend
The dividend is the residual part of earnings from the firm’s operation that are paid out to
shareholders, proportional to the number of stocks owned
Expected return
Suppose you buy a share in a corporation. At time 𝑡, the price is known, and equal to 𝑃. . At
time 𝑡 + 1, you received a dividend 𝐷.)( , and the price will be 𝑃.)( . These two quantities
involve uncertainty. The estimated one-period percentage return 𝑟.)( from holding the stock
<(=")( ) < (# 9# )
is 𝐸. (𝑟.)( ) = #"
+ " ")( " (= dividend yield + capital gain)
#"
Discount rate
Reward that investors demand for tying up the investment in the asset
Simple interest
Interest earned or paid to the original balance of the deposit loan. After 𝑇 periods, the total
balance will be 𝑋! = 𝑋(1 + 𝑟𝑇)
Compound interest
Interest is paid on previously charged/earned interest. After 𝑇 periods, the total balance will
be 𝑋! = 𝑋(1 + 𝑟)! . This is also equal to the future value 𝐹𝑉(𝑋, 𝑟, 𝑇)
Annual percentage rate (APR)
The APR is based on simple interest, not compound
Effective annual rate (EAR)
If there is more than one payment per year, than the APR is not the EAR. For 𝑘 payments per
"#$ %
year: 𝐸𝐴𝑅 = 01 + % 1 − 1
Present value
The value of your money today if you would invest it with an interest rate of 𝑟, such that it
&
would be worth exactly 𝑋 in the future, is called the present value: 𝑃𝑉(𝑋, 𝑟, 𝑇) = (()*)! . The
(
discount factor is equal to (()*)!
Net present value (NPV)
Investments will typically involve multiple cash-flows, at different points in time. Here are
initial investments negative and future receipts positive. The net present value is the present
,-"
value of a stream of cash-flows: 𝑁𝑃𝑉 = ∑!./0 (()*) "
, where 𝐶𝐹. is the cash-flow at time 𝑡.
A positive NPV means invest, a negative NPV means do not invest
Annuities
Annuities are a common cash-flow stream, consisting of the payment of a fixed cash-flow,
once per period, for a known finite number of periods.
, , (
The present value: 𝑃𝑉 = ∑!./( "
= 01 −
(()*) * !
(()*)
1. Here you can solve 𝐶 to find the
annuity payment in a mortgage, 𝑇 to find how many periods you need to repay with fixed
repayment (payback period), 𝑟 with 𝑁𝑃𝑉 = 0 to find the internal rate of return (IRR).
Perpetuities
Perpetuities are a common cash-flow stream, consisting of the payment of a fixed cash-flow,
, ,
once per period, indefinitely. The present value: 𝑃𝑉 = ∑1
./( (()*)" = *
Law of one price
The law of one price: if equivalent investment opportunities trade simultaneously in
different competitive markets, then they must trade for the same price
Arbitrage
Arbitrage is the practice of selling and buying equivalent goods in different markets to take
advantage of price differences in these different markets
Week 2
Bond characteristics
Issuer agrees to make specified payments to the bondholder on specified dates. The
characteristics of a bond are
- Maturity, length of the contract
- Par value / face value / principal, the payment at the bond’s maturity
- Coupon rate, an interest payment per dollar of par value, this is periodic
, - Discount rate, typically differs from coupon rate
Bond pricing
The income stream of bonds is known, but lies in the future. Bond value = PV of coupons +
, - , ( -
PV of par value. So 𝑃 = ∑!./( "
(()*)
+ !
= 01 −
(()*) * !
1+
(()*) ! (()*)
Bond pricing variations
The different bond pricing variations are:
- Frequency of coupon payments
- ‘Callable’ bonds
- ‘Indexed’ bonds
Multiple coupon payments per year
If the coupon payment is paid 𝑘 times a year, then the 𝐶 and 𝑟 need to be divided by 𝑘 (if
they are given as the APR), and the number of periods needs to be multiplied with 𝑘
Indexed bonds
In indexed bonds, both the principal repayment and coupon payments get indexed by the
#$%
" #$%!
,× -×
! #$%& #$%&
inflation rate, so the price becomes 𝑃 = ∑./( +
(()*)" (()*)!
Callable bonds
Bond issuer has the right to buy the bond at a certain strike price 𝑆. The price of the bond
can never exceed the strike price around the call day
Characteristics of bond price
- If a bond comes closer to a coupon payment, the price rises. If the coupon payment has
been paid, the price of the bond drops
- If the discount rate is higher than the coupon rate, the initial price of the bond will lower
than the par value of the bond, because we need to be compensated for the difference
between the discount rate and coupon rate
Yield to maturity
The YTM is the constant, hypothetical discount rate that, when used to compute the PV of a
bond’s cashflows, gives you the bond’s market price as the answer, used to compare bonds:
, -
𝑃 = ∑!./( (()3!4)" + (()3!4)!
Zero-coupon bonds
A bond without coupon payments, so we only obtain repayment of the par value at
-
maturity. 𝐶 = 0, and hence 𝑃56*7 = (()*)! . In this case, 𝑌𝑇𝑀 = 𝑟
Default risk
There is a promise of cash-flows, but the company can go bankrupt. However, Dutch
government bonds are essentially a risk-free asset. There is virtually no risk of no
repayment. We call the associated rate the ‘risk-free rate’, 𝑟8 . If the probability of default.
i.e. non-repayment, is higher, investors require a higher YTM. For risk-neutral investors this
means 𝐸(𝑌𝑇𝑀) = 𝑟8 , and for risk-averse investors 𝐸(𝑌𝑇𝑀) > 𝑟8 . We call the difference the
default premium
Holding period return
Default risk changes over time, but YTM assumes constant compound return. Then you can
look at the holding period return. It is the rate of return over a particular investment period.
It depends on the bond’s price at the end of the holding period.
It can be calculated as 𝐻𝑃𝑅09: = 𝑃: /𝑃0 − 1, where 𝑖 is the holding period, 𝑃: is the par
value discounted with the YTM in period 𝑖
Yield curve
Yields depend on lots of different features, they are not set in stone and may differ across
time. Moreover, it is ‘risky’ to buy a long-maturity bond, a longer horizon means more
, uncertainty about economic states and default risk. The relationship between YTM and
maturity is called the yield curve
Implications yield curve
A payment in a bond with a coupon payment should be discounted with the YTM of a zero-
bond in that specific year
Yield curve under uncertainty
Suppose that you can buy a 2-year zero with a YTM of 6%, or a 1-year zero with a YTM of 4%
and reinvest after one year. Than these two strategies should yield the same (expected)
return. So, 1.06; should be equal to 1.04 times the YTM in the second year of a 1-year zero
bond (the short/forward rate)
Yield curve under uncertainty
The rate one year from now may actually not be the same as its expectation. The spot rate
reflects market expectations of the future rate. The actual rate will almost surely be lower.
The liquidity premium on longer-term bonds compensates short-term investors for
uncertainty about future prices
Expectations hypothesis theory
The expectations hypothesis theory says that the forward rate equals market consensus of
future short interest rate, so 𝑓; = 𝐸(𝑟; ). If this theory holds true and if there are only zero
prices given of bonds, then the forward rate 1 + 𝑓: = 𝑃:9( /𝑃:
Liquidity preference theory
The liquidity preference theory says that long-term bonds are riskier, so 𝑓; > 𝐸(𝑟; ). Excess
is the liquidity premium. The yield curve has an upward bias built into long-term rates
Interpreting yield curves
An upward sloping yield curve could mean that rates are expected to rise, or large liquidity
premiums. An inverted yield curve could mean that interest rates are expected to fall, a
recession signal
Duration
,- (()3!4)'(
Define the importance of the cash-flow at time 𝑡 as 𝑤. = " # . The duration is
!
defined as duration = ∑./( 𝑡𝑤. . The higher the coupon rate, the lower the duration
Week 3
Book value
The book value is the accounting value of the firm’s equity as reflected on the company’s
balance sheet, it is backward looking
Market value
The market value is the price of the firm’s equity as resulting from market transactions.
Theoretically, this coincides with the present value of discounted expected dividends, it is
forward looking
Secondary market
The secondary market is a market, often a stock exchange, in which previously issued shares
are traded amongst investors
Dividend
The dividend is the residual part of earnings from the firm’s operation that are paid out to
shareholders, proportional to the number of stocks owned
Expected return
Suppose you buy a share in a corporation. At time 𝑡, the price is known, and equal to 𝑃. . At
time 𝑡 + 1, you received a dividend 𝐷.)( , and the price will be 𝑃.)( . These two quantities
involve uncertainty. The estimated one-period percentage return 𝑟.)( from holding the stock
<(=")( ) < (# 9# )
is 𝐸. (𝑟.)( ) = #"
+ " ")( " (= dividend yield + capital gain)
#"
