2  Compound Interest and Discount

2.1 Compound Interest

Definition: We say that the interest rate is a compound interest rate when it has the property that the interest earned at the end of one period is automatically invested in the next period to earn additional interest.

Example: Consider a loan where the interest earned over each unit of time is added to the balance of the loan. Since the original principal of a loan is equal to the PV of the loan, let’s use \(PV_0\) ot denote the original loan amount:

• At time 1, the accumulated value is the original principal plus interest: \[AV_1 = PV_0 \times (1 + i)\]
• At time 2, the accumulated value is the time 1 value plus interest: \[AV_2 = AV_1 \times (1 + i) = PV_0 \times (1+i)^2\]
• At time 3, the accumulated is the time 2 value plus interest: \[AV_3 = AV_2 \times (1 + i) = PV_0 \times (1+i)^3\]
• At time \(t\), the accumulated value is the time \((t-1)\) value plus interest: \[AV_t = PV_0 \times (1+i)^t\]

Definition: When an annual interest rate is compounded annually it is known as an annual effective interest rate. Under compound interest, the accumulation function is:

\[ a(t) = \frac{\text{Accumulated Value}}{\text{Present Value}} = (1+i)^t \]

2.1.1 PV and AV under Compound Interest

The accumulated value is the present value times the accumulation function: \[ AV_t =PV_0(1 +i)^t \]

Or, a more general form of the formula above: \[ AV_{t_2} =PV_{t_1}(1+i)^{t_2-t_1} \qquad \text{where } t_2 > t_1 \]

Example: Smith deposits 1000 into an account on January 1, 2011. The account credits interest at an annual effective interest rate of 5% every December 31. Smith withdraws 200 on January 1, 2013, deposits 10 on January 1, 2014, and withdraws 250 on January 1, 2016. What is the balance in the account just after interest is credited on December 31, 2017?

Solution: We can accumulate each transaction to the December 31, 2017 date of valuation and combine all accumulated values, adding deposits and subtracting withdrawals. Then we have \[ 1000(1.05)^7 - 200(1.05)^5 + 100(1.05)^4 - 250(1.05)^2 = 997.77 \]

for the balance on December 31, 2017.

When considering the equation \(X(1+i)^t =Y\) given any three of the four variables \(X,Y,i,t\), it is possible to find the fourth. If the unknown variable is \(t\), then solving for \(t\) gives us \(t=\frac{\ln(Y/X)}{\ln(1+i)}\). If the unknown variable is the interest rate \(i\), then solving for \(i\) results in \(i = (\frac{Y}{X})^{1/t} - 1\). Financial calculators have functions that allow you to enter three of the variables and calculate the fourth.

2.1.2 Compound vs Simple Interest

Even though it might seem that the accumulated value is always higher under compound interest, but it might not be the case. In fact, for the same interest rate, for \(t \in (0,1)\), \(1 + it > (1+i)^t\).

2.1.3 Current Value under Compound Interest

The one-year accumulation factor is \((1+i)\), and the one-year discount factor is its inverse. This discount factor appears frequently, so for convenience we shorten it to \(v = (1+i)^{-1}\). We can discount a cash flow for \(t\) years by multiplying the cash flow by \(v^t\).

The current values under compound interest are related as follows:

\[CV_t = AV_t(\text{payments occuring before t}) + Pmt_t + PV_t(\text{payments occuring after t})\]

2.2 Compound Discount

The rate of interest is defined as a measure of the interest paid at the end of the period. On the other hand, the rate of discount, denoted by \(d\), is a measure of interest where the interest is paid at the beginning of the period.

For example, when \(\$1\) is borrowed at a discount rate of \(d\), the borrower will have to pay \(\$d\) in order to receive the use of \(\$1\). Therefore, instead of the borrower having the use of \(\$1\) at the beginning of a period he will only have the use of \(\$(1 − d)\).

Compound discount applies a discount factor of \((1-d)\) to the accumulated value to find the value one unit of time earlier. This can be done repeatedly to obtain the discounted value at time \(0\) of the accumulated value at time \(t\).

Since \((1-d)\) is the one-year discount factor, it is equal to \(v\): \(v=1-d\)

2.2.1 PV and AV under Compound Discount

The present value is the accumulated value times the discount factor: \[ PV_0 = AV_t\times (1-d)^t \]

Or, a more general form of the formula above: \[ PV_{t_1} =AV_{t_2}(1-d)^{t_2-t_1} \qquad \text{where } t_2 > t_1 \]

Even more generally, it can be written in terms of current values: \[ CV_{t_1} = CV_{t_2}(1-d)^{t_2-t_1} \]

2.3 Equivalent Compound Interest and Discount Rates

Two rates of interest or discount are equivalent if they produce the same present values and accumulated values. The relationship between equivalent \(i\) and \(d\):

\[ 1+i = \frac{1}{1-d} \implies \begin{cases} i = \frac{d}{1-d}\\ d= \frac{i}{1+i} = iv \end{cases} \] In this case, notice that \(d < i\).