To find the maximum amount you would pay for an asset generating an income of $100,000 at the end of each of five years with an opportunity cost of 8%, you calculate the present value (PV) of these future cash flows discounted at 8%. The formula for the present value of an annuity (equal payments for a fixed number of periods) is:
PV=∑t=15100,000(1+0.08)tPV=\sum_{t=1}^{5}\frac{100,000}{(1+0.08)^t}PV=t=1∑5(1+0.08)t100,000
Calculating each term:
- Year 1: 100,0001.081=92,592.59\frac{100,000}{1.08^1}=92,592.591.081100,000=92,592.59
- Year 2: 100,0001.082=85,735.03\frac{100,000}{1.08^2}=85,735.031.082100,000=85,735.03
- Year 3: 100,0001.083=79,383.70\frac{100,000}{1.08^3}=79,383.701.083100,000=79,383.70
- Year 4: 100,0001.084=73,503.43\frac{100,000}{1.08^4}=73,503.431.084100,000=73,503.43
- Year 5: 100,0001.085=68,059.29\frac{100,000}{1.08^5}=68,059.291.085100,000=68,059.29
Summing these:
92,592.59+85,735.03+79,383.70+73,503.43+68,059.29=399,274.0492,592.59+85,735.03+79,383.70+73,503.43+68,059.29=399,274.0492,592.59+85,735.03+79,383.70+73,503.43+68,059.29=399,274.04
So, the maximum amount you would pay for the asset is approximately $399,274
. Alternatively, using the present value of an annuity formula:
PV=P×1−(1+r)−nrPV=P\times \frac{1-(1+r)^{-n}}{r}PV=P×r1−(1+r)−n
Where P=100,000P=100,000P=100,000, r=0.08r=0.08r=0.08, and n=5n=5n=5:
PV=100,000×1−(1.08)−50.08≈100,000×3.993=399,300PV=100,000\times \frac{1-(1.08)^{-5}}{0.08}\approx 100,000\times 3.993=399,300PV=100,000×0.081−(1.08)−5≈100,000×3.993=399,300
This confirms the above calculation. Therefore, the maximum price to pay is about $399,274 to $399,300 given the 8% opportunity cost of funds.
