Chapter 4: Interest rates

Question 1

Principal of a loan

Answer 1

Amount provided by a borrower when the loan is originated

Question 2

Simple interest:What is the accumulated value of deposit with this kind of interest?

Answer 2

Interest is only earned on the initial deposit.
Accumulated value = X(1+ti)

Question 3

Why don’t banks offer simple interest? When is it used?

Answer 3

It’s too simple. Investors could deposit funds then withdraw as soon as they get interest then deposit it back in.

Simple interest is sometimes used if accuracy is not very important. e.g the amount of money is small or time period is short. It might be sufficiently accurate.

Question 4

Accumulated Value of compound interest

Answer 4

Iinterest is earned on the initial deposit and on the interest.
It is analogous to using simple interest but the interest is credited to the account and the interest rate applies to the new balance.
AV=X(1+i)^t
Account experiences

Question 5

Compare the Accumulated Value of simple and compound interest

Answer 5

Simple:Linear
Compound:Geometric/Exponential growth
Compound is higher when t>1, equal at t=1

Question 6

annual effective interest rate

Answer 6

compound interest rate per year.
The default.

Question 7

accumulation factor / accumulated value factor

Answer 7

• Compound: A(t)=(1+i)^t
• Simple: A(t)=1+ti
• e^δt (if constant) else = e^[integral from 0 to t of δ(t)dt]
• =(1-d)^-t

Question 8

PV

Answer 8

• the present value of an amount.
• calculated by using the effective annual interest rate
• = X/(1+i)^t = X(1+i)^-t = Xv^t
• =v^t for X=1,
• =(1+ti)^-1 for simple interest
• (1-d)^t
• e^-δt (if constant) else =e^-[integral from 0 to t of δ(t)dt]

Question 9

v

Answer 9

1/(1+i)

Question 10

v^t

Answer 10

Discount factor

Question 11

d

Answer 11

• annual effective rate of discount
• the amount of interest payable at the start of the year, on a loan of £1 for one year
• d=1-v=i/(1+i)=iv
• the PV of a payment of i payable at the end of the year

Question 12

δ(t)

Answer 12

• force of interest
• continuously compounded interest rate
• instantaneous change in the account value, expressed as an annualised percentage of the current value.
• If annual effective interest rate is constant, just use δ, which is also constant
• [A(t)]’/A(t). Using d/dx a^x = a^x ln(a)
  [A(t)]’=d/dt X (1+i)^t= x(1+i)^t) ln(1+i)
  ==>
  δ=X(1+i)^t ln(1+i) / X(1+i)^t=ln(1+i)