Summary of "Present Value 3 | Interest and debt | Finance & Capital Markets | Khan Academy"

Summary of “Present Value 3 | Interest and debt | Finance & Capital Markets | Khan Academy”

This video continues the discussion on present value (PV) calculations for different payment timing options under varying discount rates, emphasizing how changes in the interest rate affect the valuation of future payments.

Main Ideas and Concepts

Review of Present Value and Discounting:
Present value is the current worth of future payments discounted at a risk-free interest rate.
Compounding forward (multiplying by 1 + interest rate) and discounting backward (dividing by 1 + interest rate) are inverse operations.
Example: $110 received two years from now discounted at 5% is calculated as [ \frac{110}{(1.05)^2} = 99.77 ]
Three Payment Timing Choices Compared at 5% Discount Rate: 1. $100 today → PV = $100 2. $110 in two years → PV = $99.77 3. $20 today + $50 in one year + $35 in two years → [ 20 + \frac{50}{1.05} + \frac{35}{(1.05)^2} = 99.36 ]
Ranking at 5% discount rate: Choice 1 > Choice 2 > Choice 3.
Effect of Lowering the Discount Rate to 2%:
Lower discount rates increase the present value of future payments.
At 2%, the PV of $110 in two years becomes [ \frac{110}{(1.02)^2} = 105.72 ]
PV of choice 3 becomes [ 20 + \frac{50}{1.02} + \frac{35}{(1.02)^2} = 102.66 ]
Ranking at 2% discount rate: Choice 2 > Choice 3 > Choice 1.
Choice 2’s PV increased by about $6, while choice 3’s increased by less than $3.
Why Different Payments React Differently to Interest Rate Changes:
Payments further in the future benefit more from a decrease in discount rate because the discount factor is raised to a higher power.
Choice 2’s payment is fully two years out, so it gains the most.
Choice 3’s payments are spread over different times, so the benefit is less pronounced.
Implications:
The choice of discount rate significantly affects which payment stream is most valuable.
Understanding timing and discounting is crucial for bond valuation and other financial decisions.
The next video will explore varying discount rates for different time periods.

Methodology: Step-by-Step Present Value Calculation

Identify the payment amounts and their timing (years from now).
Choose an appropriate discount rate (risk-free rate or treasury rate).
Calculate present value for each payment using the formula: [ PV = \frac{\text{Future Payment}}{(1 + r)^t} ] where: - ( r ) = discount rate (e.g., 0.05 for 5%) - ( t ) = number of years until payment
Sum the present values of all payments to get the total present value of the payment stream.
Compare total present values across different payment options to determine which is most valuable.
Analyze how changes in discount rate affect the present value and ranking of payment options.