Question

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]

Answer 1

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