Mathematics of Finance(§5.1-5.4)
§5.1: Compound Interest
Main formula for compound interest is on p.209
P=principal (the amount invested)
r=periodic interest rate (amount of interest paid per compounded period)
n=number of compounding periods (number of time the interest is paid)
-if interest is paid every month it would be 12
We’re often given an APR (Annual % Rate of interest) and a frequency (number of times interest
is paid per year, number of compoundings per year).
k= frequency a= APR
∴ r= when t is # of years, we have that n=kt
∴ We have this expanded compound and formula
S=P(1+ ) kt
Ex1) You invest $10K @ 3.05% APR compounding monthly for 5 years.
a) Find the compound amount, S.
b) Compound interest= S-P
c) How long to reach $15K We solve for t in the following equation:
15000=10000(1+ )2t ln key (log key)
ln(1.5)=ln[(1+ ) ]
∴ t= ≈13.31 years
So, we need 13 years and 4 months. ▓
(P>0), (a>0, a decimal),∈IN
Ex2) [Example on “Doubling time”, p.210 ex.2&6] We invest $P at an APR compounding k
times a year. Solve the compound amount formula for the time to increase m times, m∈IN.
So we solve for t where S=mP
mP=P(1+ ) , t=years
ln(m)=ln[(1+ ) ]kt
∴t= Note: this does not depend on P! (I.e. Independent of P) ▓
The Effective Rate Concept (p.211)
Consider a period of 1 year. Invest $P @ an APR of r compounding n times for that year.
0 1 year
n=4, S=P(1+ ) n
The effective rate for interest iser , is the simple rate that gives the same compound amount that
would occur above.
re=(1+ ) -1 How did we get this? n n
P(1+r e=P(1+ ) P(1+r e=P(1+ )
simple compound Cancel P
re=(1+ ) -1 We expect that r er because r hes no intermediate compounding.
Ex3) Effective rate is often used to compare interest scheme.
Scheme A: 2.95% APR, semi annual
Scheme B: 2.93% APR, bi-weekly (a=26)
Which is better? Take the larger effective rate! A is better.▓
§5.2: Present and Future Value
PV= present value (or past value)
FV= future value=P(1+ ) P(NOW) t years in the future
PV= s(1+ ) = present (or past) value (negative means going back in time)
PV is the amount to be unvested t years ago so that in t years we have S
We use PV and FV in Equations of Value (p.214-215)
An equation of value is an equation that describes payments and debts, both under the
assumption of an interest.
KEY CONCEPT: At all times, (value of all payments) = (value of all payments)
Ex4) You owe $32K and this must be paid in 6 years. You pay 10K now and the balance @ the
end of 6 years. Interest is 3% APR, quarterly. Find the ending payment.
At time 6 years, (value of all pay)=(value of all debt)=32,000
X≈20,035.86 (Sept 18, Week 2) Equations of Value (cont..)
Example: At the end of 7 years, $40,000 debt is to be paid off. We made 3 payments: $10,000 @
the end of year 1, $8,000 @ the end of year 4, and the final payment is made @ the end of year 5.
Interest is 2%, compounded semi-annually. Calculate amount of final payment.
Let X rep the amount of final payment.
(10,000) (8,000) (X)
0 1 2 3 4 5 6 (40,000)