Introduction to Rates
Present Value, Interest Rate Conventions and Bonds
Abdelaziz Baihi
MATH-UA 250 Mathematics of Finance
Spring 2024
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
1 / 19
Outline
1 Interest Rates: Basic Concepts
Credit and Debt
Interest Rates
Day Count Convention
Interest Rates Compounding Conventions
Present Value and The Discount Factor
The Risk-Free Rate
2 Bonds
Bond Types
Bond Valuation
Bond Risks
3 References and Further Reading
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
2 / 19
Table of Contents
1 Interest Rates: Basic Concepts
Credit and Debt
Interest Rates
Day Count Convention
Interest Rates Compounding Conventions
Present Value and The Discount Factor
The Risk-Free Rate
2 Bonds
Bond Types
Bond Valuation
Bond Risks
3 References and Further Reading
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
3 / 19
Credit and Debt
Access to credit is getting access to extra buying power.
Credit is not free: the buying power is granted for a promise to pay it back. Debt is the
technical name of this promise.
The moral question of good or bad: it is clear that giving the ability to access extra funds is a
good thing and not providing credit can be a bad thing. Examples: credit to fund research,
infrastructure investments, health care etc.
The problem of bad debt arises when the debt service becomes intolerable i.e. the debt was
not used in a productive way to generate sucient income to service the debt.
This course will not tackle the macroeconomic aspects of credit/debt but we will talk about
usual debt instruments that market participants use for various goals: investing, taking a
directional bet, hedging risks etc.
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
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Interest Rates
When cash is deposited into a bank account, the account is generally getting remunerated an
extra-amount of money called the interest.
The interest can be computed and paid following di↵erent conventions (usually documented in
term sheets).
Interest rates are usually quoted in annual rate, i.e. as if the operation would take 1 year.
The interest paid are proportional to the amount of cash invested. This amount is usually
referred to as the principal or notional.
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MATH-UA 250 Mathematics of Finance Spring 2024
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Day Count Convention
The day count fraction is used to measure the amount of time cash has been held between
two dates in order to compute the corresponding interest amount
The calculation of the day count fraction is done using a day count convention. The most
common ones are:
I ACT/360: ”actual over three-sixty” counts the number of calendar days (including weekends and
holidays) between two dates and divide that number by 360, as if a year was of 12 months each of
which was of 30 days. Mathematically, the accrual factor between days t1 and t2 is:
T =
d2 d1
360
Where d2 d1 is the number of actual days between t1 and t2.
I ACT/365: same as the previous one, but dividing by 365.
I ACT/ACT:
T =
Days in a non-leap year
365
+
Days in a leap year
366
I BUS/252: count the number of business days between 2 dates and divide that number by 252.
The corresponding banking holiday calendar (or union of calendars) is needed to know the
holidays to exclude.
Day count conventions are given by the market (in case of listed products) or specific
term-sheets in case of Over The Counter (OTC) products.
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
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Interest Rates Compounding Conventions
Compounding frequencies are usually needed when interest rates are involved over
multiple periods (from reinvestment or simply from the product definition).
Compounding frequencies can be daily (or overnight), weekly, monthly, quarterly,
semi-annually, annually or of type ”bullet” (a.k.a. zero coupon) where the latter means that
there is only one payment at the very end of the period.
When investing a notional of N dollars from t0 and over a schedule of t1 < t2 < ... < tN , and
assuming a rate ri is applied for the period between ti1 and ti, the interest amount at the end
of the schedule is:
I Linear compounding convention: N · (1 + r1t1 + r2t2 + ...rNtN )
I Periodic compounding convention:
N · (1 + r1t1) · (1 + r2t2)...(1 + rNtN )
I Power compounding convention
N · (1 + r1)t1 · (1 + r2)t2 ...(1 + rN )tN
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
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Continuous Compounding
If:
I the interest rate is constant equal to r
I and ti is constant and represent a year fraction, say ti = 1/n
I and the investment period is over m years i.e that we got N = n⇥m periods,
I then the amount of interest earned over at the end is
(1 + r1t1)(1 + r2t2)...(1 + rNtN ) =
⇣
1 +
r
n
⌘nm
I as the frequency increases, formally n! +1 (in practice, the frequency becomes daily), then⇣
1 +
r
n
⌘nm ! erm
Recall that interest rates are usually expressed in year, and we can clearly see above that after
m years, the rate r is ”applied” m times.
if r is small then er ⇡ 1 + r and erm ⇡ (1 + r)...(1 + r) multiplied m times, which indeed
represents a growth of the yearly interest rate r over m years.
For a monthly frequency n = 12, bi-monthly n = 6, quarterly m = 4, semi-annually n = 2 and
annually n = 1.
We define the continuously compounded interest rate to be: er1t1+r2t2+...+rNtN = e
R tN
t0
r(u)du
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
8 / 19
Why various various compounding conventions?
Di↵erent markets operate using di↵erent schedules (holidays etc.), date conventions and
compounding. At the end of the day, the local culture prevails.
What really matters is the cash amount of the interest amount, i.e. that is paid.
It means that for a corresponding compounding convention, for instance the linear 1 + rt vs
the power (1 +R)t, the rates would be di↵erent for the same interest amount and computing
their relationship can be done using simple algebra, for example in this case
r =
1
t
⇥
(1 +R)t 1⇤
and
R = (1 + rt)
1
t 1
Of course, the same rate applied with di↵erent compounding conventions will yield di↵erent
interest amounts.
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
9 / 19
Present Value and The Discount Factor
If we apply the old adage ”A bird in the hand is worth two in the bush” to money, we would
say ”a dollar now is better than a dollar in a year”.
Market participants have a preference for money now as opposed to the future. This concept is
known at time value of money.
We can quantify this by using the funding rate i.e. the interest rates that the market is
willing to charge them to access credit
Note that the funding rate is dependent on the market participant’s ability and credibility to
pay back their debts.
1 dollar invested at the funding rate r now will yield in one year 1 + r. Hence we define the
present value of 1 dollar in 1 year the value: 11+r
More generally, a discount factor of maturity T is the cash amount today that if invested at
the funding rate would lead to a value of 1 dollar at time T .
If we use continuous rates (an arbitrary choice), we can define the discount factor to be:
DF (T ) =
1
er·T
= er·T
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
10 / 19
Present Value and The Discount Factor
Present Value
If the funding rate has a term structure, the discount factor would look like (using continuous
rates):
DF (T ) = e
R T
0 r(t) dt
The present value (PV) of a portfolio V at time t is the dollar amount of the portfolio valued
as of time t. Example: the PV of a dollar received in 1 year is the 1 year discount factor, the
PV of a fixed amount N received in 10 year is N · DF (T = 1o years).
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
11 / 19
The Risk-Free Rate
So far, we have ignored the fact that lending happens only in exchange of a promise of paying
back (with interest). We have not discussed the possible scenario of not fulfilling that promise.
When a market participant fails the promise of a payment (whether it is an interest amount or
some part of the principal amount), we call this a credit event or default event.
In many textbooks and academic papers, there is often a mention of the concept of risk-free
rate, i.e. credit risk-free rate, e.g. a loan (or investment) without any credit risk.
The risk-free rate is a pure theoretical concept although many argue that US treasury bonds
(or any sovereign bonds in the local currency of the country issuing them) are 100% secure
debt. This is still debatable.
Some argue some overnight rates are a good proxy for the risk-free rate.
In this course, we will not really dwell on this. The rate we would use is the rate that is o↵ered
to us by the market i.e. our funding rate.
The discount theory under credit risk, collateral agreements etc. is interesting and di↵ers from
the above. It is outside the scope of this course.
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
12 / 19
Table of Contents
1 Interest Rates: Basic Concepts
Credit and Debt
Interest Rates
Day Count Convention
Interest Rates Compounding Conventions
Present Value and The Discount Factor
The Risk-Free Rate
2 Bonds
Bond Types
Bond Valuation
Bond Risks
3 References and Further Reading
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
13 / 19
Bonds
A bond is a debt instrument that represents a loan made by an investor to a borrower
(corporations, governments etc.).
A Zero Coupon Bond (ZCB) is a bond that will have only a final payment of the notional
(in this case called face amount without any interest coupon. For instance, if you buy a $100
1-Y zero coupon bond for $95 dollars this e↵ectively means that you pay $95 dollars now to
get $100 in one year from now.
Treasury bonds are bonds issued by a goverment’s treasury department to borrow money in
its currency. Treasury bills (referred to as T-bills in the U.S.) are short term (usually less
than 1Y in the U.S.) zero coupon bonds. A Treasury note is a treasury bond with maturity
between one and ten years.
Municipal bonds, usually referred to as Munis, are bonds issued by municipalities, i.e. cities
or states.
Corporate bonds are bonds issued by (private) companies.
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
14 / 19
Bond Valuation
Typically, most bonds pay coupons to the holder with a predetermined frequency with a final
payment of the notional at the very end.
Bond Yield: it is the constant rate that if applied to all the bond’s cashflows would give a
present value equal to the market price of the bond. For example, using continuous rates, the
bond yield y is:X
i
Notional · coupon ratei · ti · ey·ti +Notional · ey·T = Market Price of the bond
Par coupon (a.k.a par yield): It is the coupon C that causes the bond price to equal its par
value (i.e. notional) for a specifc discount curve. Mathemtically:X
i
Notional · C · ti · DF (ti) + Notional · DF (T ) = Notional
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
15 / 19
Bond Risks
The duration of a bond is a measure of the sensitivity of the bond’s price P to its yield y.
Mathematically, it is defined as:
D = 1
P
@P
@y
Remember, the bond price as a function of its yield is given by:
P =
NX
i=1
cie
yti
where: N is the last period in which the notional payment takes place, ci = Cti for i < N and
cN = 1 + CtN .
Then, the analytical formula of the duration is:
D = 1
P
@P
@y
=
1
P
NX
i=1
citie
yti =
NX
i=1
witi
where wi =
ci
P · eyti
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
16 / 19
Bond Risks
Since the weights wi represent the proportion of the discounted cash flow i to the bond price P , we
see that the duration D can be expressed as a weighted average of the cash flow payment times.
The duration is then a measure of how long the bond holder would have to wait before getting the
bulk of the bond’s cash payments.
The convexity of a bond is defined as:
C =
1
P
@2P
@y2
=
1
P
NX
i=1
cit
2
i e
yti
A second order Taylor expansion shows that
P
P
= Dy + 1
2
Cy2 + o(y2)
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
17 / 19
Table of Contents
1 Interest Rates: Basic Concepts
Credit and Debt
Interest Rates
Day Count Convention
Interest Rates Compounding Conventions
Present Value and The Discount Factor
The Risk-Free Rate
2 Bonds
Bond Types
Bond Valuation
Bond Risks
3 References and Further Reading
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024
18 / 19
References and Further Reading
J. Hull Chapter 4
Abdelaziz Baihi Introduction to Rates
MATH-UA 250 Mathematics of Finance Spring 2024