Time Value of Money II

Time-Varying Interest Rates

When the interest rate changes over time, each cash flow must be accumulated using the interest rate that applies to each period.

For a principal ​P exposed to rates ​i_1,i_2,\ldots,i_N,

F=P(1+i_1)(1+i_2)\cdots(1+i_N)

For multiple cash flows, first convert all of them to the same time point using their corresponding interest rates, then add them together.

In a debt-consolidation problem, first calculate the total outstanding balance at the consolidation date, then treat this balance as the present value of the new loan.

A=P_0(A/P,i,N)

The main difference from constant-rate compounding is that the growth factor changes from period to period.

Nominal and Effective Interest Rates

Compounding Frequency

The nominal annual interest rate is the stated annual rate, while the effective annual interest rate measures the actual annual growth after considering compounding frequency.

If the nominal annual rate is ​r^{(M)} and interest is compounded ​M times per year, then the rate per compounding period is

\frac{r^{(M)}}{M}

and the effective annual interest rate is

i=\left(1+\frac{r^{(M)}}{M}\right)^M-1

where ​M is the compounding frequency.

If ​M=1, then the nominal and effective annual rates are equal. If ​M>1, the effective annual rate is greater than the nominal annual rate because of intra-year compounding.

Continuous Compounding

Under continuous compounding, interest is calculated continuously over time.

Let ​B(t) be the balance at time ​t, with ​B(0)=P, and let ​r be the nominal annual interest rate.

The balance satisfies

dB(t)=B(t)r\,dt

with solution

B(t)=Pe^{rt}

Thus, under continuous compounding,

F=Pe^{rN}

and the effective annual interest rate is

i=e^r-1

Compared with discrete compounding, ​(1+i)^N is replaced by ​e^{rN}.

The time units of ​r and ​N must be consistent.

Loan Calculations

Loans and Monthly Payment

A secured loan requires collateral, while an unsecured loan does not. A mortgage is a secured loan, while a credit card is an example of an unsecured loan.

For a loan repaid with uniform monthly payments:

  • ​L: total amount borrowed
  • ​j: APR, or nominal annual interest rate
  • ​i: monthly interest rate
  • ​A: monthly payment
  • ​N: total number of monthly payments
i=\frac{j}{12}
N=\text{number of years}\times12

Since the monthly payments form an annuity whose present value equals the loan amount,

L=A(P/A,i,N)

so the monthly payment is

A=L(A/P,i,N)

or

A=L\frac{i}{1-(1+i)^{-N}}

Loan Amortization

For a loan repaid with uniform monthly payments:

  • ​I_n: interest paid in month ​n
  • ​P_n: principal repaid in month ​n
  • ​B_n: remaining loan balance at the end of month ​n

with

B_0=L

The interest payment in month ​n is

I_n=B_{n-1}i

The principal payment is

P_n=A-I_n

The remaining balance is

B_n=B_{n-1}-P_n

or equivalently,

B_n=B_{n-1}(1+i)-A

With a fixed monthly payment, the interest portion decreases over time while the principal portion increases.

Loan Balance: Retrospective and Prospective Methods

The remaining loan balance can be calculated without constructing the full amortization table.

Retrospective Method

The balance is calculated by looking at the history of payments.

B_n=L(F/P,i,n)-A(F/A,i,n)

The remaining balance equals the accumulated value of the original loan minus the accumulated value of payments already made.

Prospective Method

The balance is calculated by looking at future remaining payments.

B_n=A(P/A,i,N-n)

The remaining balance equals the present value of all future payments.

Both methods give the same loan balance.

Changing Loan Terms

When loan terms change, first calculate the remaining balance at the change date, then treat it as the principal of a new loan.

Change in Loan Period

After ​n payments, first calculate the remaining balance

B_n=L(F/P,i,n)-A(F/A,i,n)

If the new loan has ​N' remaining payments, the new payment is

A'=B_n(A/P,i,N')

Change in Interest Rate

First calculate the remaining balance using the original interest rate.

Then convert the new APR ​j' to the new periodic rate ​i' and calculate

A'=B_n(A/P,i',N-n)

A shorter repayment period or a higher interest rate generally increases the required periodic payment.

Two Types of Mortgages

There are two major types of mortgages:

Fixed Rate Mortgage (FRM)

In a fixed rate mortgage, the APR remains constant over the loan period.

As long as the loan term and payment schedule remain unchanged, the periodic payment is generally fixed.

Adjustable Rate Mortgage (ARM)

In an adjustable rate mortgage, the APR may change over the loan period.

In Hong Kong, the APR is often determined by

APR=\min(\text{HIBOR}+\text{Spread}_1,\ \text{Prime Rate}-\text{Spread}_2)

HIBOR is the average interest rate banks charge each other for borrowing and changes over time.

Prime Rate is the rate banks charge their most trustworthy customers and typically remains unchanged for a relatively long period.

​\text{Spread}_1 and ​\text{Spread}_2 are fixed.

When the APR changes, the remaining loan balance is recalculated under the new rate, causing the periodic payment to change.

Inflation and the Real Interest Rate

Inflation and Deflation

Inflation is an increase in the general price level of goods and services in an economy.

Deflation is the opposite of inflation, meaning a decrease in the general price level.

High or hyperinflation is undesirable because it reduces purchasing power and may lower quality of life.

Moderate inflation is generally considered acceptable, and most central banks aim for a low and stable inflation rate.

Deflation is generally regarded as undesirable because it is often associated with economic recession.

Inflation Measures by Price Index

A price index is a dimensionless numerical measure of the price level of a group of goods and services.

Common price indexes include:

  • CPI (Consumer Price Index): measures prices faced by consumers.
  • PPI (Producer Price Index): measures prices faced by producers.

For a single good or service, choose a base time and assign an index value of 100.

index_k=\frac{price_k}{price_b}\times100

For multiple goods and services,

index=\sum_{i=1}^{N}w_i\,index_i

where ​w_i is the weight of item ​i.

The inflation rate between two consecutive periods is

\text{Inflation Rate}_k = \frac{index_k-index_{k-1}}{index_{k-1}}

A negative inflation rate indicates deflation.

Purchasing Power

Purchasing power is the amount of goods and services that can be purchased with a given amount of money.

If the price level increases due to inflation, the same amount of money can purchase fewer goods and services.

For example, if the inflation rate is 2%, the price of one unit of goods increases from ​1 to 1.02, so the purchasing power of $1 becomes

\frac{1}{1.02}\approx0.98

units of goods and services.

Inflation reduces the purchasing power of money.

Actual Dollars and Real Dollars

Choose a base time ​b before performing inflation analysis.

Actual dollars ​A are the dollar amount of a cash flow at the time it occurs. They are also called nominal dollars or current dollars.

Real dollars ​R are the equivalent amount expressed in the purchasing power of the base time.

To convert actual dollars at time ​k into real dollars,

R=\frac{A}{(1+f)^{k-b}}

or

R=A(P/F,f,k-b)

To convert real dollars into actual dollars,

A=R(1+f)^{k-b}

or

A=R(F/P,f,k-b)

The conversion between actual and real dollars is similar to discounting and compounding, with the inflation rate ​f playing the role of the interest rate.

Market Interest Rate vs. Real Interest Rate

The market interest rate ​i_m measures the return in actual dollars, while the real interest rate ​i_r measures the return in terms of purchasing power.

Most interest rates observed in practice are market interest rates.

Let ​f be the inflation rate. The relationship between market and real interest rates is

1+i_m=(1+i_r)(1+f)

Therefore,

i_m=i_r+f+i_rf

and

i_r=\frac{i_m-f}{1+f}

When ​f is small,

i_r\approx i_m-f

This approximation is called the Fisher equation, but the exact formula should be used in calculations.

Present Value and Future Value under Inflation

Set time 0 as the base time.

For actual-dollar cash flows, use the market interest rate ​i_m.

For real-dollar cash flows, use the real interest rate ​i_r.

Both methods give the same present value if the cash flows and interest rates are matched correctly.

For future value:

  • actual-dollar cash flows compounded at ​i_m give FV in actual dollars;
  • real-dollar cash flows compounded at ​i_r give FV in real dollars.

The type of cash flow and the interest rate must be consistent.