A Summary on IFAS
Definition Deterministic Cash Flows
These cash flows are operating cash flows, cash flows from investments and cash flows from financing
𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 )
Here 𝐱 is a cashflow where 𝑥0 is the initial cash flow and 𝑥𝑇 the cash flow end of period 𝑡
Definition Simple Interest
The idea of simple interest is that the account grows linearly with time. Hence if we have investment
𝑃 and interest rate 𝑟 per year and 𝑡 ∈ ℝ then total value after 𝑡 years is:
𝐹 = (1 + 𝑟𝑡) ∙ 𝑃
Definition Compound Interest
The idea of compound interest is that it exhibits a geometric grows
If again we take investment 𝑃 and interest rate 𝑟 after one year and 𝑡 ∈ ℝ denotes the amount of
years then after 𝑡 years we have:
𝐹 = (1 + 𝑟)𝑡 ∙ 𝑃
Definition Discrete Compounding
Suppose we have interest rate 𝑟 per time period 𝑡 on investment 𝑃. If we divide the time period 𝑡 in
𝑚 even time periods than the interest rate after time period 𝑡 based on compounding in periods 𝑚 is:
𝑟 𝑚𝑡
𝐹 = (1 + ) 𝑃
𝑚
Definition Effective Interest Rate and Nominal Interest Rate
Let 𝑟 be the interest on investment 𝑃 after 1 year. Then 𝑟 is called the nominal interest rate. If we
decide to compound on the nominal interest rate, say in 𝑚 even discrete periods, then:
𝑟 𝑚
(1 + ) = 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝑚
Definition Continuous Compounding
If we compounded continuously, on interest rate 𝑟 obtained on investment 𝑃 after period 𝑡, in 𝑚
𝑟 𝑚
intervals then 𝑚 can become arbitrarily large hence lim (1 + 𝑚) = 𝑒 𝑟 .
𝑚→∞
𝑟 𝑚𝑡
Hence we have: 𝐹 = lim (1 + 𝑚
) 𝑃 = 𝑒 𝑟𝑡
𝑚→∞
,Definition Present and Future Value
Consider a cash flow stream given by 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ) . The future value 𝐹𝑉𝐱 of the cash flow is
the value of the cashflow on time 𝑇 hence
𝑇
𝑇 𝑇−1
𝐹𝑉𝐱 = 𝑥0 (1 + 𝑟) + 𝑥1 (1 + 𝑟) + ⋯ + 𝑥𝑇−1 (1 + 𝑟) + 𝑥𝑇 = ∑ 𝑥𝑖 (1 + 𝑟)𝑇−𝑖
1
𝑖=0
If we were to determine 𝐹𝑉𝐱 by simple interest. However this idea generalizes to interest types.
The present value 𝑃𝑉𝐱 is the value of cash flow 𝐱 is the value of the cash flow at 𝑡 = 0
𝑇
𝑥1 𝑥2 𝑥𝑇 𝑥𝑖
𝑃𝑉𝐱 = 𝑥0 + + + ⋯+ =∑
1 + 𝑟 (1 + 𝑟)2 (1 + 𝑟)𝑇 (1 + 𝑟)𝑖
𝑖=0
Let 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ) be a cash flow and 𝐲 = (𝑦0 , 𝑦1 , ⋯ , 𝑦𝑇 ) be the financing flow of the project.
So we borrow 𝑦0 initially and 𝑥0 = 𝑦0 . We also use that PV𝐲 = 𝑦0 + PV(𝑦1 ,⋯,𝑦𝑇 ) = 0
Definition Net Present Value
𝑁𝑃𝑉 = PV𝐱 − PV𝐲 = PV𝐱
This is because PV𝐲 = 0.
If 𝑁𝑃𝑉 > 0 we accept project if 𝑁𝑃𝑉 ≤ 0 we reject project
Definition Internal Rate/ Return
The internal rate or internal return is the interest rate 𝑟 at which PV𝐱 = 0 hence
𝑟 > 𝐼𝑅𝑅 ⇒ 𝑁𝑃𝑉 < 0 𝑎𝑛𝑑 𝑟 < 𝐼𝑅𝑅 ⇒ 𝑁𝑃𝑉 > 0
The 𝐼𝑅𝑅 rule has the following pittfalls:
- Delayed investments. (+1, −0.5, −0.5, −0.5)
- Multiple 𝐼𝑅𝑅s. (+0.55, −0.5, −0.5, −0.5, +1)
- Non-existent 𝐼𝑅𝑅 (+0.75, −0.5, −0.5, −0.5, +1)
If we make decisions on investment projects we accept if 𝑁𝑃𝑉 > 0 and 𝐼𝑅𝑅 > 𝑟
Theorem The IRR rule
If for a project 𝐼𝑅𝑅 > 𝑟 then we accept the project
Important note: the IRR rule only applies to a stand-alone project if all of the project’s negative cash
flows precede its positive cash flows
,Comparing projects of different lengths
Suppose the cash flow consists of 𝑡 periodic payments of amount 𝐴. The present value of this stream
relative to the interest rate 𝑟 per period is given by:
𝑡
𝑡
𝐴 1 𝑖
𝑃=∑ = 𝐴 ∑ ( )
(1 + 𝑟)𝑖 1+𝑟
𝑖=1
𝑖=1
Now note that this is a summable geometric series, therefore if 𝑡 → ∞ we can define the sum in a
formula. However 𝑡 is defined hence we subtract a subsequence of the infinite sum to still obtain 𝑃.
𝑛 𝑛
𝑖
1 1 𝑖
𝑃 = 𝐴 (∑ ( ) −∑( ))
1+𝑟 1+𝑟
𝑖=1 𝑖=𝑡+1
𝑛 𝑛
𝑖−1 𝑡
1 1 1 1 𝑖−1
= 𝐴( ) (∑ ( ) −( ) ∑( ) )=
1+𝑟 1+𝑟 1+𝑟 1+𝑟
𝑖=1 𝑖=1
1 𝑛 𝑡 1+( 1 )
𝑛
𝑡
1 1 + ( )
lim 𝐴 ( )( 1+𝑟 −( 1 ) ∙ 1 + 𝑟 ) = 𝐴 ( 1 ) (1 + 𝑟 − ( 1 ) ∙ 1 + 𝑟 )
𝑛→∞ 1+𝑟 1 1+𝑟 1 1+𝑟 𝑟 1+𝑟 𝑟
1−1+𝑟 1−1+𝑟
1+𝑟 1 𝑡 𝐴 1
=𝐴 (1 − ( ) ) = ∙ (1 − )
𝑟(1 + 𝑟) 1+𝑟 𝑟 (1 + 𝑟)𝑡
So for any cashflow 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ), we have:
𝐴 1
NPV𝐱 ≡ 𝑃 = ∙ (1 − )
𝑟 (1 + 𝑟)𝑡
Now to determine 𝐴 we rewrite this equation.
𝐴 1 𝑃𝑟 𝑃𝑟(1 + 𝑟)𝑡
𝑃= ∙ (1 − ) ⇔ 𝐴 = ⇔ 𝐴 =
𝑟 (1 + 𝑟)𝑡 1 (1 + 𝑟)𝑡 − 1
(1 − 𝑡 )
(1 + 𝑟)
Now if we want to compare projects we prefer the projects for which the 𝐴 is the largest. NOTE THAT
P IS THE PRESENT VALUE OF THE PROJECT
Now suppose we want to compound those 𝑡 periods into 𝑚 evenly distributed subperiods. Then the
Present value of the cash flow, subject to the interest rate 𝑟 per time period 𝑡 is:
𝑡
𝑡 𝑖∙𝑚
𝐴 1
𝑃=∑ 𝑟 𝑖∙𝑚 = 𝐴 ∑ ( )
1 + 𝑟/𝑚
𝑖=1 (1 + 𝑚)
𝑖=1
Now this is still summable hence we have
𝑛 𝑛
𝑖∙𝑚 𝑖∙𝑚
1 1 𝐴∙𝑚 1
𝑃 = 𝐴 ∑( 𝑟) − ∑( 𝑟) =⋯= ∙ (1 − )
1+𝑚 1+𝑚 𝑟 𝑟 𝑡∙𝑚
(1 + 𝑚)
( 𝑖=1 𝑖=𝑡+1 )
Hence:
𝑃∙𝑟 𝑟 𝑡∙𝑚
(1 + )
𝐴= 𝑚 𝑚
𝑟 𝑡∙𝑚
(1 + 𝑚) − 1
,Mortgages in the Netherlands
𝑟 𝑟 𝑟
𝐱 = (+𝑀, − ( ) × 𝑀, ⋯ , − ( ) × 𝑀, − ( ) × 𝑀 − 𝑀) 𝐱 ∈ ℝ(𝑻×𝒎)+𝟏
𝑚 𝑚 𝑚
The cash flow above is an interest-only mortgage, which has the following structure:
- You borrow 𝑀 today
- You pay interest during 𝑇 yeas, 𝑚 times a year
𝑟 𝑇∙𝑚
- Nominal interest rate is 𝑟 hence effective interest rate is (1 + 𝑚)
- At the end of 𝑇 years you repay 𝑀
Definition Bonds
A bond is an obligations by the bond issuer (government, corporation, etc.) to pay money to the bond
holder according to the rule specified at the time the bond is issued.
Typically included is the Face value or Par value which is the payment at maturity and the coupon
payments which are periodic payments up to maturity.
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝑄𝑢𝑜𝑡𝑒𝑑 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒 + 𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
Bonds are subject to default because issuers sometimes go bankrupt. This is not the case for
government bonds which have no default risk
The image on the right displays the change in
value of a bond. The clean price is the face
value or Par value of the bond and remains the
same until maturity. This is also called the
quoted bond price.
Suppose we have a bond with a quoted bond
price/ face price 𝑀. Suppose the coupon rate of
this bond is 𝑟% after time period 𝑡 and the
bond matures after 𝑚 times 𝑡 periods. Then the
accrued interest (𝐴𝐼) is:
𝑡𝑖𝑚𝑒 𝑢𝑛𝑖𝑡𝑠 𝑠𝑐𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛
𝐴𝐼 = ( )∙𝑟∙𝑀
𝑡
Bond Characteristics
A bond has bond price 𝑃, face value 𝐹 at maturity which is 𝑇 years from issuing. Coupon Payments
𝐶
are 𝑚 times per year, each time 𝑚 where 𝐶 = 𝑐 × 𝐹 and 𝑐 is the coupon rate. Also 𝑃 = 𝑝 × 𝐹 and
here 𝑝 represents the percentage of the bond price with resp to the face value of the bond.
A cash flow corresponding to buying a bond looks as following:
𝐶 𝐶 𝐶 𝑐 𝑐 𝑐
𝐱 = (−𝑃, , ⋯ , , + 𝐹) = 𝐹 × (−𝑝, , ⋯ , , + 1)
𝑚 𝑚 𝑚 𝑚 𝑚 𝑚
, Bond Quality Rating
Bonds are investment grade if they’re either high or medium grade.
Bonds are junk bonds if they’re in or below the speculative category.
Definition Yield to Maturity (𝒀𝑻𝑴)
The YTM is the same as the 𝐼𝑅𝑅 but then for bonds and is denoted by 𝜆 which is found by solving:
𝑇×𝑚 𝐶 𝑇×𝑚 𝑐
𝑚 𝐹 𝑚 1
−𝑃 + ∑ + 𝑚×𝑇
= −𝑝 + ∑ + =0
(1 + 𝜆/𝑚) 𝑖 (1 + 𝜆/𝑚) (1 + 𝜆/𝑚) 𝑖 (1 + 𝜆/𝑚)𝑚×𝑇
𝑖=1 𝑖=1
Elaboration: We buy bond at bond price 𝑃 and we receive coupon rate 𝑐 × 𝐹 and on period 𝑚 × 𝑇 we
also receive face value 𝐹. Now 𝑁𝑃𝑉 = 0 hence the formula above
Now to make the 𝑌𝑇𝑀 formula easier we use the same concept for the 𝐼𝑅𝑅 and we obtain:
𝑇×𝑚 𝑐
𝑚 1
∑ + =𝑝
(1 + 𝜆/𝑚)𝑖 (1 + 𝜆/𝑚)𝑚×𝑇
𝑖=1
𝑐 1 1 1 1
⇔ − 𝑇𝑚 ∙ + =𝑝
𝑚 1− 1 𝜆 1 −
1 𝜆 𝑇𝑚
𝜆 (1 + 𝑚 ) 𝜆 (1 + 𝑚 )
[ 1+𝑚 1 + 𝑚]
𝑐 𝑚 1 1
⇔ ∙ ∙ [1 − 𝑇𝑚 ] + =𝑝
𝑚 𝜆 𝜆 𝜆 𝑇𝑚
(1 + 𝑚) (1 + 𝑚)
𝑐 1 1
⇔ ∙ [1 − 𝑇𝑚 ] + =𝑝
𝜆 𝜆 𝜆 𝑇𝑚
(1 + 𝑚) (1 + 𝑚)
This is the bond price formula.
From this formula we can conclude that price and yield have a negative relation, because if 𝜆 ↑ it
follows that 𝑝 ↓. Hence we get convex Price-Yield curves such as in the following graphs
Definition Deterministic Cash Flows
These cash flows are operating cash flows, cash flows from investments and cash flows from financing
𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 )
Here 𝐱 is a cashflow where 𝑥0 is the initial cash flow and 𝑥𝑇 the cash flow end of period 𝑡
Definition Simple Interest
The idea of simple interest is that the account grows linearly with time. Hence if we have investment
𝑃 and interest rate 𝑟 per year and 𝑡 ∈ ℝ then total value after 𝑡 years is:
𝐹 = (1 + 𝑟𝑡) ∙ 𝑃
Definition Compound Interest
The idea of compound interest is that it exhibits a geometric grows
If again we take investment 𝑃 and interest rate 𝑟 after one year and 𝑡 ∈ ℝ denotes the amount of
years then after 𝑡 years we have:
𝐹 = (1 + 𝑟)𝑡 ∙ 𝑃
Definition Discrete Compounding
Suppose we have interest rate 𝑟 per time period 𝑡 on investment 𝑃. If we divide the time period 𝑡 in
𝑚 even time periods than the interest rate after time period 𝑡 based on compounding in periods 𝑚 is:
𝑟 𝑚𝑡
𝐹 = (1 + ) 𝑃
𝑚
Definition Effective Interest Rate and Nominal Interest Rate
Let 𝑟 be the interest on investment 𝑃 after 1 year. Then 𝑟 is called the nominal interest rate. If we
decide to compound on the nominal interest rate, say in 𝑚 even discrete periods, then:
𝑟 𝑚
(1 + ) = 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝑚
Definition Continuous Compounding
If we compounded continuously, on interest rate 𝑟 obtained on investment 𝑃 after period 𝑡, in 𝑚
𝑟 𝑚
intervals then 𝑚 can become arbitrarily large hence lim (1 + 𝑚) = 𝑒 𝑟 .
𝑚→∞
𝑟 𝑚𝑡
Hence we have: 𝐹 = lim (1 + 𝑚
) 𝑃 = 𝑒 𝑟𝑡
𝑚→∞
,Definition Present and Future Value
Consider a cash flow stream given by 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ) . The future value 𝐹𝑉𝐱 of the cash flow is
the value of the cashflow on time 𝑇 hence
𝑇
𝑇 𝑇−1
𝐹𝑉𝐱 = 𝑥0 (1 + 𝑟) + 𝑥1 (1 + 𝑟) + ⋯ + 𝑥𝑇−1 (1 + 𝑟) + 𝑥𝑇 = ∑ 𝑥𝑖 (1 + 𝑟)𝑇−𝑖
1
𝑖=0
If we were to determine 𝐹𝑉𝐱 by simple interest. However this idea generalizes to interest types.
The present value 𝑃𝑉𝐱 is the value of cash flow 𝐱 is the value of the cash flow at 𝑡 = 0
𝑇
𝑥1 𝑥2 𝑥𝑇 𝑥𝑖
𝑃𝑉𝐱 = 𝑥0 + + + ⋯+ =∑
1 + 𝑟 (1 + 𝑟)2 (1 + 𝑟)𝑇 (1 + 𝑟)𝑖
𝑖=0
Let 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ) be a cash flow and 𝐲 = (𝑦0 , 𝑦1 , ⋯ , 𝑦𝑇 ) be the financing flow of the project.
So we borrow 𝑦0 initially and 𝑥0 = 𝑦0 . We also use that PV𝐲 = 𝑦0 + PV(𝑦1 ,⋯,𝑦𝑇 ) = 0
Definition Net Present Value
𝑁𝑃𝑉 = PV𝐱 − PV𝐲 = PV𝐱
This is because PV𝐲 = 0.
If 𝑁𝑃𝑉 > 0 we accept project if 𝑁𝑃𝑉 ≤ 0 we reject project
Definition Internal Rate/ Return
The internal rate or internal return is the interest rate 𝑟 at which PV𝐱 = 0 hence
𝑟 > 𝐼𝑅𝑅 ⇒ 𝑁𝑃𝑉 < 0 𝑎𝑛𝑑 𝑟 < 𝐼𝑅𝑅 ⇒ 𝑁𝑃𝑉 > 0
The 𝐼𝑅𝑅 rule has the following pittfalls:
- Delayed investments. (+1, −0.5, −0.5, −0.5)
- Multiple 𝐼𝑅𝑅s. (+0.55, −0.5, −0.5, −0.5, +1)
- Non-existent 𝐼𝑅𝑅 (+0.75, −0.5, −0.5, −0.5, +1)
If we make decisions on investment projects we accept if 𝑁𝑃𝑉 > 0 and 𝐼𝑅𝑅 > 𝑟
Theorem The IRR rule
If for a project 𝐼𝑅𝑅 > 𝑟 then we accept the project
Important note: the IRR rule only applies to a stand-alone project if all of the project’s negative cash
flows precede its positive cash flows
,Comparing projects of different lengths
Suppose the cash flow consists of 𝑡 periodic payments of amount 𝐴. The present value of this stream
relative to the interest rate 𝑟 per period is given by:
𝑡
𝑡
𝐴 1 𝑖
𝑃=∑ = 𝐴 ∑ ( )
(1 + 𝑟)𝑖 1+𝑟
𝑖=1
𝑖=1
Now note that this is a summable geometric series, therefore if 𝑡 → ∞ we can define the sum in a
formula. However 𝑡 is defined hence we subtract a subsequence of the infinite sum to still obtain 𝑃.
𝑛 𝑛
𝑖
1 1 𝑖
𝑃 = 𝐴 (∑ ( ) −∑( ))
1+𝑟 1+𝑟
𝑖=1 𝑖=𝑡+1
𝑛 𝑛
𝑖−1 𝑡
1 1 1 1 𝑖−1
= 𝐴( ) (∑ ( ) −( ) ∑( ) )=
1+𝑟 1+𝑟 1+𝑟 1+𝑟
𝑖=1 𝑖=1
1 𝑛 𝑡 1+( 1 )
𝑛
𝑡
1 1 + ( )
lim 𝐴 ( )( 1+𝑟 −( 1 ) ∙ 1 + 𝑟 ) = 𝐴 ( 1 ) (1 + 𝑟 − ( 1 ) ∙ 1 + 𝑟 )
𝑛→∞ 1+𝑟 1 1+𝑟 1 1+𝑟 𝑟 1+𝑟 𝑟
1−1+𝑟 1−1+𝑟
1+𝑟 1 𝑡 𝐴 1
=𝐴 (1 − ( ) ) = ∙ (1 − )
𝑟(1 + 𝑟) 1+𝑟 𝑟 (1 + 𝑟)𝑡
So for any cashflow 𝐱 = (𝑥0 , 𝑥1 , ⋯ , 𝑥𝑇 ), we have:
𝐴 1
NPV𝐱 ≡ 𝑃 = ∙ (1 − )
𝑟 (1 + 𝑟)𝑡
Now to determine 𝐴 we rewrite this equation.
𝐴 1 𝑃𝑟 𝑃𝑟(1 + 𝑟)𝑡
𝑃= ∙ (1 − ) ⇔ 𝐴 = ⇔ 𝐴 =
𝑟 (1 + 𝑟)𝑡 1 (1 + 𝑟)𝑡 − 1
(1 − 𝑡 )
(1 + 𝑟)
Now if we want to compare projects we prefer the projects for which the 𝐴 is the largest. NOTE THAT
P IS THE PRESENT VALUE OF THE PROJECT
Now suppose we want to compound those 𝑡 periods into 𝑚 evenly distributed subperiods. Then the
Present value of the cash flow, subject to the interest rate 𝑟 per time period 𝑡 is:
𝑡
𝑡 𝑖∙𝑚
𝐴 1
𝑃=∑ 𝑟 𝑖∙𝑚 = 𝐴 ∑ ( )
1 + 𝑟/𝑚
𝑖=1 (1 + 𝑚)
𝑖=1
Now this is still summable hence we have
𝑛 𝑛
𝑖∙𝑚 𝑖∙𝑚
1 1 𝐴∙𝑚 1
𝑃 = 𝐴 ∑( 𝑟) − ∑( 𝑟) =⋯= ∙ (1 − )
1+𝑚 1+𝑚 𝑟 𝑟 𝑡∙𝑚
(1 + 𝑚)
( 𝑖=1 𝑖=𝑡+1 )
Hence:
𝑃∙𝑟 𝑟 𝑡∙𝑚
(1 + )
𝐴= 𝑚 𝑚
𝑟 𝑡∙𝑚
(1 + 𝑚) − 1
,Mortgages in the Netherlands
𝑟 𝑟 𝑟
𝐱 = (+𝑀, − ( ) × 𝑀, ⋯ , − ( ) × 𝑀, − ( ) × 𝑀 − 𝑀) 𝐱 ∈ ℝ(𝑻×𝒎)+𝟏
𝑚 𝑚 𝑚
The cash flow above is an interest-only mortgage, which has the following structure:
- You borrow 𝑀 today
- You pay interest during 𝑇 yeas, 𝑚 times a year
𝑟 𝑇∙𝑚
- Nominal interest rate is 𝑟 hence effective interest rate is (1 + 𝑚)
- At the end of 𝑇 years you repay 𝑀
Definition Bonds
A bond is an obligations by the bond issuer (government, corporation, etc.) to pay money to the bond
holder according to the rule specified at the time the bond is issued.
Typically included is the Face value or Par value which is the payment at maturity and the coupon
payments which are periodic payments up to maturity.
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 = 𝑄𝑢𝑜𝑡𝑒𝑑 𝐵𝑜𝑛𝑑 𝑃𝑟𝑖𝑐𝑒 + 𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
Bonds are subject to default because issuers sometimes go bankrupt. This is not the case for
government bonds which have no default risk
The image on the right displays the change in
value of a bond. The clean price is the face
value or Par value of the bond and remains the
same until maturity. This is also called the
quoted bond price.
Suppose we have a bond with a quoted bond
price/ face price 𝑀. Suppose the coupon rate of
this bond is 𝑟% after time period 𝑡 and the
bond matures after 𝑚 times 𝑡 periods. Then the
accrued interest (𝐴𝐼) is:
𝑡𝑖𝑚𝑒 𝑢𝑛𝑖𝑡𝑠 𝑠𝑐𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛
𝐴𝐼 = ( )∙𝑟∙𝑀
𝑡
Bond Characteristics
A bond has bond price 𝑃, face value 𝐹 at maturity which is 𝑇 years from issuing. Coupon Payments
𝐶
are 𝑚 times per year, each time 𝑚 where 𝐶 = 𝑐 × 𝐹 and 𝑐 is the coupon rate. Also 𝑃 = 𝑝 × 𝐹 and
here 𝑝 represents the percentage of the bond price with resp to the face value of the bond.
A cash flow corresponding to buying a bond looks as following:
𝐶 𝐶 𝐶 𝑐 𝑐 𝑐
𝐱 = (−𝑃, , ⋯ , , + 𝐹) = 𝐹 × (−𝑝, , ⋯ , , + 1)
𝑚 𝑚 𝑚 𝑚 𝑚 𝑚
, Bond Quality Rating
Bonds are investment grade if they’re either high or medium grade.
Bonds are junk bonds if they’re in or below the speculative category.
Definition Yield to Maturity (𝒀𝑻𝑴)
The YTM is the same as the 𝐼𝑅𝑅 but then for bonds and is denoted by 𝜆 which is found by solving:
𝑇×𝑚 𝐶 𝑇×𝑚 𝑐
𝑚 𝐹 𝑚 1
−𝑃 + ∑ + 𝑚×𝑇
= −𝑝 + ∑ + =0
(1 + 𝜆/𝑚) 𝑖 (1 + 𝜆/𝑚) (1 + 𝜆/𝑚) 𝑖 (1 + 𝜆/𝑚)𝑚×𝑇
𝑖=1 𝑖=1
Elaboration: We buy bond at bond price 𝑃 and we receive coupon rate 𝑐 × 𝐹 and on period 𝑚 × 𝑇 we
also receive face value 𝐹. Now 𝑁𝑃𝑉 = 0 hence the formula above
Now to make the 𝑌𝑇𝑀 formula easier we use the same concept for the 𝐼𝑅𝑅 and we obtain:
𝑇×𝑚 𝑐
𝑚 1
∑ + =𝑝
(1 + 𝜆/𝑚)𝑖 (1 + 𝜆/𝑚)𝑚×𝑇
𝑖=1
𝑐 1 1 1 1
⇔ − 𝑇𝑚 ∙ + =𝑝
𝑚 1− 1 𝜆 1 −
1 𝜆 𝑇𝑚
𝜆 (1 + 𝑚 ) 𝜆 (1 + 𝑚 )
[ 1+𝑚 1 + 𝑚]
𝑐 𝑚 1 1
⇔ ∙ ∙ [1 − 𝑇𝑚 ] + =𝑝
𝑚 𝜆 𝜆 𝜆 𝑇𝑚
(1 + 𝑚) (1 + 𝑚)
𝑐 1 1
⇔ ∙ [1 − 𝑇𝑚 ] + =𝑝
𝜆 𝜆 𝜆 𝑇𝑚
(1 + 𝑚) (1 + 𝑚)
This is the bond price formula.
From this formula we can conclude that price and yield have a negative relation, because if 𝜆 ↑ it
follows that 𝑝 ↓. Hence we get convex Price-Yield curves such as in the following graphs
