Advanced Time Value of
Money
Yield to Maturity
Print version of the lectures in MA270 Financial Mathematics I presented in Winter, 2021
by c Adam Metzler from The Department of Mathematics at Wilfrid Laurier University
Advanced Time Value
Motivating Example
• The City of Guelph has an outstanding bond that matures on July 11, 2030, paying a
semi-annual coupon of 2.45% on a face value of $100.
– Assume today is January 11, 2021 and that today’s coupon was just paid.
Relationship Between Price and Yield for the Guelph Bond
140
120
Price ($)
100
80
60
40
20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Per-Period Yield
Advanced Time Value
Buying the Guelph Bond
• I’m interested in buying the bond, and my broker tells me I can purchase it for
$106.30.
– This price is non-negotiable.
• What is the corresponding yield?
1
MA270 Financial Mathematics I (Advanced Time Value of Money)
c Adam Metzler, Page
20212
Relationship Between Price and Yield for the Guelph Bond
140
120
Price ($)
100
80
60
40
20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Per-Period Yield
Advanced Time Value
Who Cares?
• In practice, bonds are bought and sold in the secondary market at market prices.
– Individual investors and issuers have essentially no control over these prices.
• Given the market price of a bond, the corresponding yield is of great importance to
both investors and issuers.
– Gives investors an indication of what they might earn on their investment, if they
were to purchase the bond.
– Gives issuers an indication of the interest rate they would have to pay, if they
decided to borrow money through the bond market.
• Yield is so much more than a number we plug into a formula to get a price.
– Bond yields have significant economic content.
Advanced Time Value
Formal Definition
• Consider a bond with price function P .
– Since P is strictly decreasing, it is one-to-one.
• Let $B be an observed market price for the bond.
• Then there is a unique number Y T M such that P (Y T M ) = B.
– i.e. Y T M = P −1 (B).
• We call Y T M the bonds yield to maturity.
– Of considerable importance to both investors and issuers.
Advanced Time Value
MA270 Financial Mathematics I (Advanced Time Value of Money)
c Adam Metzler, Page
20213
Yield as a Function of Price
• In the last video we looked at price as a function of yield.
– i.e. we considered the function P .
• Today we are thinking about yield as function of price.
– i.e. we are considering the function P −1 .
• Unfortunately there is no formula for P −1 .
– How do we compute something if we don’t have a formula for it?
– We use numerical methods to approximate yields.
• That being said, there are some things we can say about yields, without a formula for
them.
Advanced Time Value
Yield is a Decreasing and Convex Function of Price
• So Yield = P −1 (Price).
• From calculus, we know how to differentiate inverse functions.
–
dYield
1
1
= 0 −1
= 0
.
dPrice
P (P (Price))
P (Yield)
–
P 00 (P −1 (Price))
d2 Yield
P 00 (Yield)
=
−
=
−
.
[P 0 (P −1 (Price))]3
[P 0 (Yield)]3
dPrice2
• So yield is a decreasing and convex function of price.
Advanced Time Value
Example 1
Concordia University has an outstanding bond that matures in 3.5 years and pays a semiannual coupon of 4.4% on a face value of $100. The bond is currently trading at $107.11.
Prove that its yield does not exceed 3%.
Advanced Time Value
Example 2
Sketch the yield on the Guelph bond, as a function of its price.
Advanced Time Value
Computing Yields
• There are a small number of special cases where we can compute yields explicitly.
• If price equals face value, then yield equals
– This is because
.
.
MA270 Financial Mathematics I (Advanced Time Value of Money)
c Adam Metzler, Page
20214
• If price equals dollar value of remaining cash flows, then yield equals
– This is because
.
.
• If there are two or fewer coupons remaining, yield can be computed using the quadratic
formula.
Example 3
Nova Scotia Power Inc. has an outstanding bond maturing on January 10, 2020, and
paying a semi-annual coupon of 9.90% on a face value of $100. The current market price
of the bond is $106.77. What rate is the bond market charging NSP to borrow money for
one year?
Advanced Time Value
Advanced Time Value
Final Thought - Terminology
• If the market price of a bond is greater than its face value, we say it is trading at a
premium.
– Occurs if, and only if, coupon exceeds yield.
• If the market price of a bond is less than its its face value, we say it is trading at a
discount.
– Occurs if, and only if, yield exceeds coupon.
• If the market price of a bond is equal to its face value, we say it is trading at par.
– Occurs if, and only if, yield equals coupon.
Advanced Time Value