Solving Outside Discount and Inflation Math Problems

Financial Mathematics: Outside Discount and Inflation

This set of solved exercises covers key concepts in financial mathematics, focusing on outside discount operations, nominal and real interest rates, and the effects of inflation on purchasing power.

Exercise 1: Outside Discount Calculation

A company has a discount operation with a banking financial institution with a deadline of 23 days. The bank operates with an effective interest rate of 45.76% p.a. discount rate to determine the "outside" discount to be used in the operation.

• Effective Annual Rate (e): 45.76% p.a.
• Term (n): 23 days
• Effective Rate for 23 days (I_f): (1 + 0.4576)^{23 / 360} - 1 = 0.0243648 or 2.43648% per 23 days

To find the equivalent "outside" discount rate (d) for the period:

d = I_f / (1 + I_f)
d = 0.02436 / (1 + 0.02436) = 0.023780
d = 2.3780% for 23 days

Exercise 2: Determining the Face Value of a Title

A bank credits a client's account with a sum of 27,000.00 from the discount of a title made 80 days prior to maturity. Given a 2.85% monthly discount rate and a 1.5% administrative fee charged by the bank, determine the nominal value (denomination) of this title.

• Net Amount Received (PV): 27,000.00
• Monthly Discount Rate (d): 2.85% a.m.
• Term (n): 80 days
• Administrative Fee (t): 1.5%
• Nominal Value (N): ?

Formula and calculation:

N = PV / [1 - (d * n + t)]
N = 27,000 / [1 - (0.0285 * (80 / 30) + 0.015)]
N = 27,000 / [1 - (0.076 + 0.015)]
N = 27,000 / 0.909
N = 29,702.97

Exercise 3: Nominal Monthly Equivalent Interest Rate

A person takes out a loan to be repaid at the end of four months, paying a real interest rate of 20% per annum. Determine the nominal monthly equivalent rate of interest of this operation, given the following monthly inflation rates: 1.5%, 1.2%, 2.2%, and 1.7%.

1. Calculate the real monthly interest rate (r):
   r = (1 + 0.20)^{1 / 12} - 1 = 0.0153094 or 1.530946% per month
2. Calculate the cumulative inflation (I_f) and average monthly inflation:
   Inflation rates: i₁ = 1.5%, i₂ = 1.2%, i₃ = 2.2%, i₄ = 1.7%
   Cumulative Inflation = (1 + 0.015) * (1 + 0.012) * (1 + 0.022) * (1 + 0.017) - 1 = 0.0676241 (6.76241%)
   Average Monthly Inflation (I_fm) = (1 + 0.0676241)^{1 / 4} - 1 = 0.016493 or 1.6493% a.m.
3. Calculate the nominal monthly interest rate (i):
   (1 + i) = (1 + r) * (1 + I_fm)
   1 + i = (1 + 0.0153094) * (1 + 0.016493)
   i = 0.0320548
   i = 3.20548% a.m.

Exercise 4: Nominal and Real Annual Return

A property was purchased for 3,000.00 and sold for 30,000.00 four years later. Given an annual inflation rate of 100%, determine the nominal and real annual return of this operation.

• Purchase Price (PV): 3,000.00
• Sale Price (FV): 30,000.00
• Term (n): 4 years
• Annual Inflation (I): 100% p.a.

Nominal Annual Return (i):

30,000 = 3,000 * (1 + i)⁴
10 = (1 + i)⁴
1 + i = 10^{1 / 4}
1 + i = 1.778279
i = 0.778279
i = 77.8279% p.a.

Real Annual Return (r):

Since the nominal return (77.83%) is lower than the inflation rate (100%), the real return is negative, indicating a loss in real purchasing power.

Exercise 5: Loss in Purchasing Power

In a semester where inflation reached 15%, wages were adjusted by 11.5%. Determine the actual loss in purchasing power of the employees.

• Inflation (I): 15%
• Nominal Wage Adjustment (t): 11.5%
• Real Adjustment (r): ?

Formula and calculation:

1 + r = (1 + t) / (1 + I)
r = (1.115 / 1.15) - 1
r = -0.03043
r = -3.04%

The employees suffered a 3.04% loss in purchasing power.