Pertanyaan :

Barry heard in his Personal Finance class that he should start investing as soon as possible. He had always thought that it would be smart to start investing after he finishes college and when his salary is high enough to pay the bills and to have money left over. He projects that will be 5–10 years from now. Barry wants to compare the difference between investing now and investing later. A financial planner who spoke to the class suggested that a Roth IRA (Individual Retirement Account) would be a more profitable investment over the long term than a regular IRA, so Barry wants to seriously consider the Roth IRA.

When table values do not include the information you need, use the formula where R is the period rate and N is the number of periods.

• If Barry purchases a $2,000 Roth IRA when he is 25 years old and expects to earn an average of 6% per year compounded annually over 35 years (until he is 60), how much will he accumulate in the investment?
• If Barry doesn’t put the money in the IRA until he is 35 years old, how much money will accumulate in the account by the time he is 60 years old? How much less will he earn because he invested 10 years later?
• Interest rate is critical to the speed at which your investment grows. If $1 is invested at 2% compounded annually, it takes approximately 34.9 years to double. If $1 is invested at 5% compounded annually, it takes approximately 14.2 years to double. Use Table 13-1 to determine how many years it takes $1 to double if invested at 10% compounded annually; at 12% compounded annually.
• At what interest rate would you need to invest to have your money double in 10 years if it is compounded annually?

  Status : 100%
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  Jawab ;

1) Barry membeli $2.000 pada usia 25, bunga 6% per tahun, dikompaun tahunan selama 35 tahun (sampai 60)

Rumus nilai masa depan (future value) untuk jumlah tunggal:

FV=PV×(1+R)NFV = PV \times (1+R)^NFV=PV×(1+R)N

Di sini $PV=2000,R=0.06,N=35$.

Hitungan antara:

(1+0.06)35=(1.06)35≈7.6860867923(1+0.06)^{35} = (1.06)^{35} \approx 7{.}6860867923(1+0.06)35=(1.06)35≈7.6860867923 $$FV=2000\times 7.6860867923\approx 15,372.17$$

Jawaban: Barry akan memiliki sekitar $15,372.17 pada usia 60.

2) Barry menunda sampai umur 35 (investasi 10 tahun terlambat) — berapa pada umur 60?

Sekarang $PV=2000,R=0.06,N=25$ (dari 35 sampai 60 = 25 tahun).

(1.06)25≈4.2918707197(1.06)^{25} \approx 4{.}2918707197(1.06)25≈4.2918707197 $$FV=2000\times 4.2918707197\approx 8,583.74$$

Jawaban: Jika diinvestasikan pada usia 35, jumlahnya akan menjadi sekitar $8,583.74 pada usia 60.

Berapa lebih sedikit karena terlambat 10 tahun?

$$\text{Selisih}=15,372.17-8,583.74=6,788.43$$

Jadi Barry akan memperoleh sekitar $6,788.43 lebih sedikit karena menunda investasi 10 tahun.

3) Berapa lama $1 menggandakan diri pada 10% dan 12% (kompaun tahunan)?

Gunakan rumus gandaan: (1+r)N=2⇒N=ln⁡2ln⁡(1+r)(1+r)^N = 2 \Rightarrow N = \dfrac{\ln 2}{\ln(1+r)}(1+r)N=2⇒N=ln(1+r)ln2​.

• Untuk $r=10\%=0.10$:

  N=ln⁡2ln⁡1.10≈7.27 tahunN = \frac{\ln 2}{\ln 1.10} \approx 7.27 \text{ tahun}N=ln1.10ln2​≈7.27 tahun

• Untuk $r=12\%=0.12$:

  N=ln⁡2ln⁡1.12≈6.12 tahunN = \frac{\ln 2}{\ln 1.12} \approx 6.12 \text{ tahun}N=ln1.12ln2​≈6.12 tahun

Jawaban: Pada 10% → sekitar 7.27 tahun; pada 12% → sekitar 6.12 tahun.

(Perbandingan: yang diberikan di soal: 2% → ~34.9 tahun, 5% → ~14.2 tahun — konsisten: tingkat bunga lebih tinggi → waktu gandakan lebih singkat.)

4) Tingkat bunga yang diperlukan agar uang menggandakan dalam 10 tahun (kompaun tahunan)

Kita ingin (1+r)10=2(1+r)^{10} = 2(1+r)10=2. Maka

1+r=21/10⇒r=21/10−11+r = 2^{1/10} \quad\Rightarrow\quad r = 2^{1/10} – 11+r=21/10⇒r=21/10−1 $$r\approx 0.0717734625=7.17734625\%$$

Jawaban: Diperlukan tingkat bunga sekitar 7.18% per tahun (dibulatkan) agar uang menggandakan dalam 10 tahun dengan kompaun tahunan.