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Difficulty: HardWeighted Averages in Applied Contexts

A commercial bank's loan portfolio consists of three categories of loans: small business loans, residential mortgages, and commercial real estate loans. Small business loans carry an average annual interest rate of 8.5%8.5\% and account for 30%30\% of the portfolio's total value. Residential mortgages carry an average annual interest rate of 5.0%5.0\%, and commercial real estate loans carry an average annual interest rate of 6.5%6.5\%. If the overall weighted average annual interest rate for the entire portfolio is 6.35%6.35\%, what percentage of the portfolio's total value is invested in commercial real estate loans?

  1. A
    15%15\%
  2. 20%20\%Answer
  3. C
    35%35\%
  4. D
    40%40\%
  5. E
    50%50\%

Answer

The commercial real estate loans account for 20%20\% of the portfolio's total value.
The overall portfolio rate of 6.35%6.35\% is a weighted average of small business loans (8.5%8.5\% rate, 30%30\% weight), residential mortgages (5.0%5.0\% rate, x%x\% weight), and commercial real estate loans (6.5%6.5\% rate, (70−x)%(70-x)\% weight). Solving 0.30(8.5)+0.05x+0.065(70−x)=6.350.30(8.5) + 0.05x + 0.065(70 - x) = 6.35 gives x=50%x = 50\%, which means commercial real estate loans constitute 70%−50%=20%70\% - 50\% = 20\% of the total portfolio.

Step-by-Step Solution

1
Define variables for the unknown portfolio weight components.
Let w1=0.30w_1 = 0.30 (small business loans), w2=xw_2 = x (residential mortgages), and w3=0.70−xw_3 = 0.70 - x (commercial real estate loans).
The sum of all weights across the three portfolio categories must equal 100%100\% (1.001.00). Since small business loans constitute 30%30\%, the remaining two categories must sum to 70%70\% (0.700.70).
2
Set up the weighted average formula for the overall portfolio interest rate.
0.30(8.5)+x(5.0)+(0.70−x)(6.5)=6.350.30(8.5) + x(5.0) + (0.70 - x)(6.5) = 6.35
The overall weighted average interest rate is the sum of each component's rate multiplied by its respective weight in the portfolio.
3
Expand and simplify the algebraic equation.
2.55+5.0x+4.55−6.5x=6.35  ⟹  7.10−1.5x=6.352.55 + 5.0x + 4.55 - 6.5x = 6.35 \implies 7.10 - 1.5x = 6.35
Multiplying out terms allows combining like terms to solve for xx.
4
Solve for xx to find the weight of residential mortgages, then find the weight of commercial real estate loans.
1.5x=0.75  ⟹  x=0.501.5x = 0.75 \implies x = 0.50 (50%50\%). Thus, commercial real estate weight =0.70−0.50=0.20= 0.70 - 0.50 = 0.20 (20%20\%).
Subtracting x=50%x = 50\% from the total remaining 70%70\% gives the exact share allocated to commercial real estate loans.

Key Concept

Weighted Averages in Multi-Component Portfolios
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Topics

Weighted Averages in Applied Contexts

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