Investments
Interest rates
Present value is the discounted value of future cash flows. We will relax the assumption of a
constant risk-free rate, meaning that the discount rate for the first year is different of the
second year, etc.
𝑇
𝐶𝐹 (𝑡) 𝐶𝐹 (1) 𝐶𝐹 (2)
𝑃𝑉 = ∑ = + +⋯
(𝑡) 𝑡 (1) 1 (2) 2
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟0 ) (1 + 𝑟0 )
(𝑡)
Notation: 𝑟0 means that these are rates today (0) and “t” is the maturity for this rate
(3)
Risk-free rates are also called spot rates: 𝑟0 is the 3-year spot rate
The spot rate in for example year 3 is ONLY valid for the cash flow occurring in year 3 (do not
(3)
use 𝑟0 to discount all the cash flows from today to year 3)
(3)
Spot rates are known today → I know today that I will use 𝑟0 to discount the cash flows in
year 3
Discount bonds
Discount bonds are bonds without a coupon. They are therefore only characterized by their
maturity and face value
(3)
Notation: 𝐵0 is the price today of a 3-year zero-coupon bond
Consider a discount bond with face amount $100 that will be paid in 5 years. Suppose that the
five-year spot rate is 3.3%. What should be the price of the discount bond?
(5) $100
𝐵0 = = $85 < $100
(1 + 3.3%)5
This means that the government will only receive $85 for this bond, in return of the promise
to pay back $100 at the maturity
General formula for discount bond pricing:
(𝑇) 𝐹
𝐵0 =
( ) 𝑇
(1 + 𝑟0 𝑇 )
1
,Coupon bonds
Coupon bonds return a face or principal amount after 𝑇 periods along with coupons at regular
periods in between.
Suppose that you are given the option to purchase a 5-year coupon bond with annual coupon
rate of 7% and a face amount of $100. Also, you know the following information for spot rates:
7 7 107
Price = 1
+ 2
+ ⋯+ = $117.1 > $100
(1 + 2.04%) (1 + 2.60%) (1 + 3.30%)5
Note that the coupon rate is only used for determining the cash flows (numerator!!). Don’t
use the coupon rates for discounting (denominator!!).
Can the price of a coupon bond be lower than its face value? Yes, if the spot rates are higher
than the coupon rate
𝑇
𝐶 𝐹
Price = ∑ ( 𝑡 )+
(𝑡) (𝑇) 𝑇
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟𝑜 )
Knowing that, for zero-coupon bonds:
(𝑡)
(𝑡) 𝐹 𝐵 1
𝐵0 = ⇔ 0 =
(𝑡) 𝑡 𝐹 (𝑡) 𝑡
(1 + 𝑟0 ) (1 + 𝑟0 )
We can compute the price of a coupon-paying bond with the price of a discount bond:
𝑇 𝑇 (𝑡) (𝑇)
1 1 𝐵 𝐵
Price = ∑ (𝐶 × )+𝐹× = ∑ (𝐶 × 0 ) + 𝐹 × 0
(𝑡) 𝑡 (𝑇) 𝑇 𝐹 𝐹
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟𝑜 ) 𝑡=1
Suppose that there are no discount bonds trading with exactly one and two years to maturity,
(1)
but there are coupon bonds with these maturities trading. Can we infer the spot rates 𝑟0 and
(2)
𝑟0 ?
Bond A: Years to maturity = 1; F = $100; Coupon rate = 5%; Price = $99.75
Bond B: Years to maturity = 2; F = $100; Coupon rate = 8%; Price = $104.80
Via bootstrapping:
5 100 (1) 105
Bond A: 99.75 = (1)
+ (1)
⇔ 𝑟0 = − 1 = 5.26%
1 + 𝑟0 1 + 𝑟0 99.75
8 8 100 (2) 108
Bond B: 104.80 = + + ⇔ 𝑟0 =√ −1
1 + 5.26% (2) 2 (2) 2 104.8 − 7.6
(1 + 𝑟0 ) (1 + 𝑟0 )
= 5.41%
2
,Note that we could not have calculated the 2 spot rates for bond B without calculating the
spot rate of bond A first. However, we could have calculated its Yield to maturity.
Yield to maturity (YTM)
The YTM is the IRR of a bond (the rate that makes the NPV equal to 0)
YTM on a discount bond:
(𝑇) 𝐹 𝐹 𝐹
𝐵0 = = =
(𝑇) 𝑇 (1 + 𝐼𝑅𝑅)𝑇 (𝑇) 𝑇
(1 + 𝑟0 ) (1 + 𝑌𝑇𝑀0 )
In other words, the YTM on a zero-coupon bond is the same as its spot rate.
YTM on coupon bonds:
Retake the previous example of a 5-year coupon bond with annual coupon rate of 7%
and a face amount of $100 and price of $117.1.
7 7 107
$117.1 = + + ⋯ + ⇔ 𝑌𝑇𝑀 = 3.23%
(1 + 𝑌𝑇𝑀) (1 + 𝑌𝑇𝑀)2 (1 + 𝑌𝑇𝑀)5
This rate is lower than the coupon rate
The YTM on a coupon bond gives us no information about its spot rates!!
Careful about terminology: we talk about YTM for both the discount and the coupon bond.
However:
(𝑇)
- 𝑌𝑇𝑀0 on a coupon bond ≠ spot rate
(𝑇) (𝑇)
- 𝑌𝑇𝑀0 on a zero-coupon bond ≡ spot rate 𝑟0 for T years
To calculate the YTM on a coupon bond, we don’t need the spot rates. We also cannot go back
from the YTM to calculate the individual spot rates
In order to get YTM via the spot
rate, we need to go through 2
steps (calculate the price first)
(𝑇)
If a coupon bond is trading at par, its coupon rate is equal to its YTM: 𝑃0 = 𝐹 ⇔ 𝑐 = 𝑌𝑇𝑀𝑜
(𝑇)
If a coupon bond is trading at discount: 𝑃0 < 𝐹 ⇔ 𝑐 < 𝑌𝑇𝑀𝑜
(𝑇)
If a coupon bond is trading at premium: 𝑃0 > 𝐹 ⇔ 𝑐 > 𝑌𝑇𝑀𝑜
3
, In case of perpetual bonds (𝑇 → ∞):
𝑐×𝐹 (∞) 𝑐𝐹
𝑃0 = ∞ ⇔ 𝑌𝑇𝑀0 =
𝑌𝑇𝑀0 𝑃0
𝑐𝐹
The ratio 𝑃 is called the current yield. If the coupon bond is trading at par, the current yield
0
is equal to the YTM
Holding period returns (HPR)
HPR is not to be confused with YTM.
(1) (2)
If 𝑟0 = 2.04% and 𝑟0 = 2.60%, then the price of a 2-year 5% coupon bond is:
(2) 5 105
𝑃0 = + = $104.65
1.0204 1.02602
The holding period return depends on the next year’s 1-year spot rate:
(1)
(2) (1) 𝑃̃1
Next year rate 𝑟̃1 Next year price 𝑃̃1 ̃1 =
𝐻𝑃𝑅 −1
(2)
𝑃0
2.04% 102.90 3.11%
2.60% 102.34 2.57%
3.16% 101.78 2.04%
3.26% 101.69 1.95%
YTM HPR
Rate of return over a particular investment
Average return if the bond is held to maturity
period
Depends on the coupon rate, the maturity, and Depends on the bond’s price at the end of the
the par value holding period, an unknown future value
Readily observable Can only be forecasted
Yield curve
YTMs on zero-coupon bonds with different maturities is called the term structure of (spot)
interest rates or yield curve. It is a plot of interest rates vs. different maturities
This is the yield curve for today and only today → the yield curve tomorrow is unknown today
It’s the market who determines these rates
Why does it slope up?
- There is a risk that the prices of government bonds, even those not subject to default
risk, will change (because of changes in interest rates)
- This risk is higher for long-maturity bonds than that for short-maturity bonds
4
Interest rates
Present value is the discounted value of future cash flows. We will relax the assumption of a
constant risk-free rate, meaning that the discount rate for the first year is different of the
second year, etc.
𝑇
𝐶𝐹 (𝑡) 𝐶𝐹 (1) 𝐶𝐹 (2)
𝑃𝑉 = ∑ = + +⋯
(𝑡) 𝑡 (1) 1 (2) 2
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟0 ) (1 + 𝑟0 )
(𝑡)
Notation: 𝑟0 means that these are rates today (0) and “t” is the maturity for this rate
(3)
Risk-free rates are also called spot rates: 𝑟0 is the 3-year spot rate
The spot rate in for example year 3 is ONLY valid for the cash flow occurring in year 3 (do not
(3)
use 𝑟0 to discount all the cash flows from today to year 3)
(3)
Spot rates are known today → I know today that I will use 𝑟0 to discount the cash flows in
year 3
Discount bonds
Discount bonds are bonds without a coupon. They are therefore only characterized by their
maturity and face value
(3)
Notation: 𝐵0 is the price today of a 3-year zero-coupon bond
Consider a discount bond with face amount $100 that will be paid in 5 years. Suppose that the
five-year spot rate is 3.3%. What should be the price of the discount bond?
(5) $100
𝐵0 = = $85 < $100
(1 + 3.3%)5
This means that the government will only receive $85 for this bond, in return of the promise
to pay back $100 at the maturity
General formula for discount bond pricing:
(𝑇) 𝐹
𝐵0 =
( ) 𝑇
(1 + 𝑟0 𝑇 )
1
,Coupon bonds
Coupon bonds return a face or principal amount after 𝑇 periods along with coupons at regular
periods in between.
Suppose that you are given the option to purchase a 5-year coupon bond with annual coupon
rate of 7% and a face amount of $100. Also, you know the following information for spot rates:
7 7 107
Price = 1
+ 2
+ ⋯+ = $117.1 > $100
(1 + 2.04%) (1 + 2.60%) (1 + 3.30%)5
Note that the coupon rate is only used for determining the cash flows (numerator!!). Don’t
use the coupon rates for discounting (denominator!!).
Can the price of a coupon bond be lower than its face value? Yes, if the spot rates are higher
than the coupon rate
𝑇
𝐶 𝐹
Price = ∑ ( 𝑡 )+
(𝑡) (𝑇) 𝑇
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟𝑜 )
Knowing that, for zero-coupon bonds:
(𝑡)
(𝑡) 𝐹 𝐵 1
𝐵0 = ⇔ 0 =
(𝑡) 𝑡 𝐹 (𝑡) 𝑡
(1 + 𝑟0 ) (1 + 𝑟0 )
We can compute the price of a coupon-paying bond with the price of a discount bond:
𝑇 𝑇 (𝑡) (𝑇)
1 1 𝐵 𝐵
Price = ∑ (𝐶 × )+𝐹× = ∑ (𝐶 × 0 ) + 𝐹 × 0
(𝑡) 𝑡 (𝑇) 𝑇 𝐹 𝐹
𝑡=1 (1 + 𝑟0 ) (1 + 𝑟𝑜 ) 𝑡=1
Suppose that there are no discount bonds trading with exactly one and two years to maturity,
(1)
but there are coupon bonds with these maturities trading. Can we infer the spot rates 𝑟0 and
(2)
𝑟0 ?
Bond A: Years to maturity = 1; F = $100; Coupon rate = 5%; Price = $99.75
Bond B: Years to maturity = 2; F = $100; Coupon rate = 8%; Price = $104.80
Via bootstrapping:
5 100 (1) 105
Bond A: 99.75 = (1)
+ (1)
⇔ 𝑟0 = − 1 = 5.26%
1 + 𝑟0 1 + 𝑟0 99.75
8 8 100 (2) 108
Bond B: 104.80 = + + ⇔ 𝑟0 =√ −1
1 + 5.26% (2) 2 (2) 2 104.8 − 7.6
(1 + 𝑟0 ) (1 + 𝑟0 )
= 5.41%
2
,Note that we could not have calculated the 2 spot rates for bond B without calculating the
spot rate of bond A first. However, we could have calculated its Yield to maturity.
Yield to maturity (YTM)
The YTM is the IRR of a bond (the rate that makes the NPV equal to 0)
YTM on a discount bond:
(𝑇) 𝐹 𝐹 𝐹
𝐵0 = = =
(𝑇) 𝑇 (1 + 𝐼𝑅𝑅)𝑇 (𝑇) 𝑇
(1 + 𝑟0 ) (1 + 𝑌𝑇𝑀0 )
In other words, the YTM on a zero-coupon bond is the same as its spot rate.
YTM on coupon bonds:
Retake the previous example of a 5-year coupon bond with annual coupon rate of 7%
and a face amount of $100 and price of $117.1.
7 7 107
$117.1 = + + ⋯ + ⇔ 𝑌𝑇𝑀 = 3.23%
(1 + 𝑌𝑇𝑀) (1 + 𝑌𝑇𝑀)2 (1 + 𝑌𝑇𝑀)5
This rate is lower than the coupon rate
The YTM on a coupon bond gives us no information about its spot rates!!
Careful about terminology: we talk about YTM for both the discount and the coupon bond.
However:
(𝑇)
- 𝑌𝑇𝑀0 on a coupon bond ≠ spot rate
(𝑇) (𝑇)
- 𝑌𝑇𝑀0 on a zero-coupon bond ≡ spot rate 𝑟0 for T years
To calculate the YTM on a coupon bond, we don’t need the spot rates. We also cannot go back
from the YTM to calculate the individual spot rates
In order to get YTM via the spot
rate, we need to go through 2
steps (calculate the price first)
(𝑇)
If a coupon bond is trading at par, its coupon rate is equal to its YTM: 𝑃0 = 𝐹 ⇔ 𝑐 = 𝑌𝑇𝑀𝑜
(𝑇)
If a coupon bond is trading at discount: 𝑃0 < 𝐹 ⇔ 𝑐 < 𝑌𝑇𝑀𝑜
(𝑇)
If a coupon bond is trading at premium: 𝑃0 > 𝐹 ⇔ 𝑐 > 𝑌𝑇𝑀𝑜
3
, In case of perpetual bonds (𝑇 → ∞):
𝑐×𝐹 (∞) 𝑐𝐹
𝑃0 = ∞ ⇔ 𝑌𝑇𝑀0 =
𝑌𝑇𝑀0 𝑃0
𝑐𝐹
The ratio 𝑃 is called the current yield. If the coupon bond is trading at par, the current yield
0
is equal to the YTM
Holding period returns (HPR)
HPR is not to be confused with YTM.
(1) (2)
If 𝑟0 = 2.04% and 𝑟0 = 2.60%, then the price of a 2-year 5% coupon bond is:
(2) 5 105
𝑃0 = + = $104.65
1.0204 1.02602
The holding period return depends on the next year’s 1-year spot rate:
(1)
(2) (1) 𝑃̃1
Next year rate 𝑟̃1 Next year price 𝑃̃1 ̃1 =
𝐻𝑃𝑅 −1
(2)
𝑃0
2.04% 102.90 3.11%
2.60% 102.34 2.57%
3.16% 101.78 2.04%
3.26% 101.69 1.95%
YTM HPR
Rate of return over a particular investment
Average return if the bond is held to maturity
period
Depends on the coupon rate, the maturity, and Depends on the bond’s price at the end of the
the par value holding period, an unknown future value
Readily observable Can only be forecasted
Yield curve
YTMs on zero-coupon bonds with different maturities is called the term structure of (spot)
interest rates or yield curve. It is a plot of interest rates vs. different maturities
This is the yield curve for today and only today → the yield curve tomorrow is unknown today
It’s the market who determines these rates
Why does it slope up?
- There is a risk that the prices of government bonds, even those not subject to default
risk, will change (because of changes in interest rates)
- This risk is higher for long-maturity bonds than that for short-maturity bonds
4
