咨询：基于Equations of Value Unknown interest rate的两笔不规则缴款评估问题

Hey there! I totally get how frustrating it is when you’re stuck on those equations of value problems with unknown interest rates and irregular payments—especially when you can’t find solid examples to work from. Let’s break down the two scenarios you mentioned step by step, with concrete examples to make it clear.

1. Scenario 1: Initial Deposit + One Future Deposit, Given Final Future Value #

Let’s use a concrete example to walk through this:

Suppose you deposit $1000 at time 0 (right now), then another $1500 one year from now. The total future value of these two deposits two years from now is $2700. What’s the annual interest rate r?

Step-by-Step Solution: #

1. Set up the equation of value by moving all cash flows to the final time point (year 2):
   • The $1000 earns interest for 2 years: 1000*(1 + r)^2
   • The $1500 earns interest for 1 year: 1500*(1 + r)
   • Their sum equals the given future value:

     1000*(1 + r)^2 + 1500*(1 + r) = 2700
     

2. Simplify using substitution to turn this into a quadratic equation. Let x = 1 + r (since we’re dealing with growth factors):

   1000x² + 1500x - 2700 = 0
   

   Divide all terms by 100 to simplify:

   10x² + 15x - 27 = 0
   

3. Solve the quadratic equation using the formula x = [-b ± √(b² - 4ac)]/(2a) (here, a=10, b=15, c=-27):
   • Calculate the discriminant: 15² - 4*10*(-27) = 225 + 1080 = 1305
   • Take the positive root (since x = 1 + r can’t be negative):

     x = [-15 + √1305]/20 ≈ (-15 + 36.12)/20 ≈ 21.12/20 ≈ 1.056
     

4. Find the interest rate: Since x = 1 + r, r = x - 1 ≈ 0.056 or 5.6%.

2. Scenario 2: Initial Deposit + Deposit Years Later, Given Final Future Value #

Let’s use another example here:

You deposit $3000 at time 0, then $2000 five years from now. The total future value 10 years from now is $8000. Find the annual interest rate r.

Step-by-Step Solution: #

1. Set up the equation of value aligned to year 10:
   • The $3000 earns interest for 10 years: 3000*(1 + r)^10
   • The $2000 earns interest for 5 years: 2000*(1 + r)^5
   • Sum equals the future value:

     3000*(1 + r)^10 + 2000*(1 + r)^5 = 8000
     

2. Use substitution to simplify the high-degree equation. Let y = (1 + r)^5:

   3000y² + 2000y - 8000 = 0
   

   Divide all terms by 1000:

   3y² + 2y - 8 = 0
   

3. Solve the quadratic equation (a=3, b=2, c=-8):
   • Discriminant: 2² - 4*3*(-8) = 4 + 96 = 100
   • Positive root:

     y = [-2 + √100]/6 = (-2 + 10)/6 = 8/6 ≈ 1.333
     

4. Find r from y: Since y = (1 + r)^5, take the 5th root of y:

   1 + r = y^(1/5) ≈ 1.333^(0.2) ≈ 1.0592
   

   So r ≈ 0.0592 or 5.92%.

Key Tips to Remember #

• Always align all cash flows to the same time point (we used the final future value here, but present value works too—just pick one and stick with it)
• Substitution is your best friend for these problems—it turns messy high-degree equations into solvable quadratics
• Never keep negative solutions for r—in practical finance, interest rates are positive (unless you’re dealing with rare edge cases like negative interest rates)

内容的提问来源于stack exchange，提问作者Zepalz