On Becca’s 1st birthday, a savings account was opened for her containing $10,000
ID: 2775312 • Letter: O
Question
On Becca’s 1st birthday, a savings account was opened for her containing $10,000. By her 5th birthday, the account had grown to $13,500 and $1000 was added. By her 13th birthday the account had grown to $17,000 and $2000 was added. On her 16th birthday, she withdrew $5,000 to buy a car, leaving a balance of $15,000. On her 18th birthday, the account contained $16,500 which she will use for college. No other deposits or withdrawals were made. Use the time-weighted method to calculate the effective annual yield rate over the seventeen-year period. (Answer: 3.63%)
Explanation / Answer
Suppose that we make investments in a fund over time and we know the outstanding balance before each deposit or withdrawal occurs. Let B0 be the initial balance in the fund. Let Bj be the balance in the fund immediately before time tj , for 1 j n. Let Wj be the amount of each deposit or withdrawal at time tj . Wj > 0 for deposits and Wj < 0 for withdrawal. In a table, we have:
Time
0
t1
t2
t3
tn
Account balance(Before Deposits or withdrawls)
B1
B2
B3
Bn
Deposits/Withdrawals
W1
W2
W3
Wn
Total Balance
B0
B1+W1
B2+W2
B3+W3
Bn+Wn
Therefore Time weighted annual rate of returns, i , is the solution of this equation:
(1+i)n = B1/B0 x B2/(B1+W1) x B3/(B2+W2) x ….x Bn/(Bn-1 + Wn-1)
So by using this formula we get i as time weighted return.
Now as per the problem we can make this investment chart like this:
Investment Account
Date
1st Birthday
5th Birthday
13th Birthday
16th Birthday
18th Birthday
Account balance(Before Deposits or withdrawls)
10000
13500
17000
20000
16500
Deposits
1000
2000
Withdrawals
5000
Total Balance
10000
14500
19000
15000
16500
And for the equation:
(1+i)17 = (13500/10000) x (17000/14500) x (20000/15000) x
(16500/15000)
(1+i)17 = 1.35 x 1.1724 x 1.0526 x 1.1 = 1.83259
(1+i) = (1.83259)1/17 = 1.03627359
And , i = 0.03627359 = 3.627359%
Hence annual yield by time weighted method = 3.63%
Time
0
t1
t2
t3
tn
Account balance(Before Deposits or withdrawls)
B1
B2
B3
Bn
Deposits/Withdrawals
W1
W2
W3
Wn
Total Balance
B0
B1+W1
B2+W2
B3+W3
Bn+Wn
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