question archive A sinking fund is set up so that level annual payments will accumulate, at i %p
A sinking fund is set up so that level annual payments will accumulate, at i %p
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A sinking fund is set up so that level annual payments will accumulate, at i %p.a compounding annually, to $50,000 in 10 years. Assuming the payments made are at each year-end, after exactly 5 years the sinking fund will be: [Note: a sinking fund allows for money to be set aside by periodical deposits in order to accumalte an amout to e.g replace equipment or repay debt due at a future date] Select one: a. Less than $25,000 b. Equal to $25,000 c. Greater than $25,000
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Option A
| Assuming the payments made are at each year-end, after exactly 5 years the sinking fund will be |
| the future value of the equally set-aside sum,ie. annuity amt. |
| at r= i% p.a. |
| for n=5 |
| using the formula for future value of ordinary annuity |
| FV(OA)=Pmt.*((1+r)^n-1)/r |
| ie. FV(OA)=Pmt.*((1+i)^5-1)/i |
| For example, |
| suppose i=10% |
| given that the FV of the fund =50000 |
| at end of n= 10 yrs. |
| we can find the annual pmt. Into the fund , |
| using the above formula, |
| 50000=Pmt.*((1+0.1)^10-1)/0.1 |
| th eannual payment into the fund will be |
| 50000/(((1+0.1)^10-1)/0.1) |
| 3137.27 |
| So, after 5 yrs.,ie. At end of 5 yrs., |
| the amt. accumulated in the fund will be: |
| 3137.27*((1+0.1)^5-1)/0.1)= |
| 19153.35 |
| In the above question, 2 variables are unknown,ie.i & the pmt. |
| in which case , it can only be stated as a formula. |
| Hence the example. |
| Similarly, if the annual payments/deposits into the fund are given, we can calculate the i% . |
| Suppose the annual deposit is $ 3500 |
| for n=10 yrs. |
| we can find i% at which it |
| becomes $ 50000 |
| 50000=3500*((1+r)^10-1)/r)= |
| Solving for r, we get |
| r=7.71% |
| With this, we can find the amt. accumulated in the fund at end of 5 yrs. |
| Amt. at end of 5 yrs.=3500*((1+0.0771)^5-1)/0.0771)= |
| 20414.70 |
