Exam FM sample questions, with solutions and BA II Plus keystrokes

1. Question 1Time Value of Money

   Ana deposits 5,000 into an account today and 3,000 at the end of year 2. The account earns a nominal rate of 6% a year, convertible monthly.

   Calculate the account balance at the end of year 5.

   A9,744B10,126C10,264D10,334E10,791

   Show the solution

   1. The monthly rate is $6\% \div 12$ $= 0.5\%$.
   2. The 5,000 grows for 60 months: $5{,}000(1.005)^{60}$ $= 6{,}744.25$.
   3. The 3,000 grows for 36 months: $3{,}000(1.005)^{36}$ $= 3{,}590.04$.
   4. Balance: $6{,}744.25 + 3{,}590.04$ $= 10{,}334.29$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 60 N 0.5 I/Y 5000 +/− PV 0 PMT CPT FV → 6,744.25
   3. 36 N 3000 +/− PV CPT FV → 3,590.04
   4. 6744.25 + 3590.04 = → 10,334.29
2. Question 2Annuities

   Twenty annual payments of 1,500 are to be made, the first one 3 years from today.

   At an annual effective interest rate of 5%, calculate the present value of the payments.

   A16,148B16,955C17,803D18,693E19,628

   Show the solution

   1. Valued as an annuity-immediate, the 20 payments are worth $1{,}500\,a_{\overline{20}|}$ $= 1{,}500 \times 12.46221$ $= 18{,}693.32$ at time 2, one year before the first payment.
   2. Discount 2 years to today: $18{,}693.32 \div 1.05^2$ $= 16{,}955.39$.

   On a BA II Plus

   1. 2nd CLR TVM, with END showing (no BGN on the screen)
   2. 20 N 5 I/Y 1500 PMT 0 FV CPT PV → −18,693.32
   3. 2 N 0 PMT 18693.32 FV CPT PV → −16,955.39
3. Question 3Annuities

   Payments of 100, 200, 300, …, 1,000 are made at the ends of years 1 to 10.

   At an annual effective interest rate of 4%, calculate the present value of the payments.

   A4,199B4,367C4,433D4,723E5,500

   Show the solution

   1. This is 100 times an increasing annuity: $(Ia)_{\overline{10}|}$ $= \dfrac{\ddot a_{\overline{10}|} - 10v^{10}}{i}$.
   2. $a_{\overline{10}|}$ $= 8.110896$, so $\ddot a_{\overline{10}|}$ $= 1.04 \times 8.110896$ $= 8.435332$, and $10v^{10}$ $= 6.755642$.
   3. $(Ia)_{\overline{10}|}$ $= \dfrac{8.435332 - 6.755642}{0.04}$ $= 41.99225$.
   4. Present value: $100 \times 41.99225$ $= 4{,}199.22$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 2nd BGN 2nd SET so BGN shows, then 2nd QUIT
   3. 10 N 4 I/Y 1 PMT 0 FV CPT PV → −8.435332, the annuity-due
   4. 2nd BGN 2nd SET back to END, 2nd QUIT
   5. 0 PMT 10 FV CPT PV → −6.755642, which is 10v¹⁰
   6. 8.435332 − 6.755642 = ÷ 0.04 = × 100 = → 4,199.22
4. Question 4Annuities

   An annuity pays 2,000 at the end of year 1, and each later payment is 3% larger than the one before. There are 25 payments.

   At an annual effective interest rate of 7%, calculate the present value.

   A23,307B30,711C31,244D31,632E50,000

   Show the solution

   1. Geometric annuity: $PV$ $= 2{,}000 \times \dfrac{1 - (1.03/1.07)^{25}}{0.07 - 0.03}$.
   2. $(1.03/1.07)^{25}$ $= 0.385777$, so $PV$ $= 2{,}000 \times \dfrac{0.614223}{0.04}$ $= 30{,}711.16$.

   On a BA II Plus

   1. The calculator has no growth key, so value it as a level annuity at the rate j = 1.07 ÷ 1.03 − 1 = 3.883495%, with level payment 2,000 ÷ 1.03 = 1,941.7476.
   2. 2nd CLR TVM
   3. 25 N 3.883495 I/Y 1941.7476 PMT 0 FV CPT PV → −30,711.16
5. Question 5Loans

   A 250,000 mortgage is repaid with level monthly payments over 30 years, at a nominal rate of 4.8% a year convertible monthly.

   Calculate the outstanding balance just after the 120th payment.

   A92,600B166,667C202,119D202,927E314,799

   Show the solution

   1. The monthly rate is $0.4\%$ and there are 360 payments: $P$ $= 250{,}000 \div a_{\overline{360}|0.004}$ $= 1{,}311.66$.
   2. Just after payment 120, the balance is the value of the 240 payments still to come: $1{,}311.66\,a_{\overline{240}|0.004}$ $= 202{,}118.54$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 360 N 0.4 I/Y 250000 PV 0 FV CPT PMT → −1,311.66
   3. 120 N CPT FV → −202,118.54 (still owed after 120 payments)
6. Question 6Loans

   A loan of 20,000 is repaid with 8 level payments at the end of each year, at an annual effective rate of 6%.

   Calculate the principal repaid in the 5th payment.

   A2,021B2,407C2,500D2,551E2,704

   Show the solution

   1. Payment: $20{,}000 \div a_{\overline{8}|0.06}$ $= 20{,}000 \div 6.209794$ $= 3{,}220.72$.
   2. The principal in payment $k$ is $P\,v^{\,n-k+1}$. For $k$ $= 5$ and $n$ $= 8$: $3{,}220.72 \times 1.06^{-4}$ $= 2{,}551.11$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 8 N 6 I/Y 20000 PV 0 FV CPT PMT → −3,220.72
   3. 2nd AMORT 5 ENTER ↓ 5 ENTER ↓ BAL ↓ PRN → −2,551.11
7. Question 7Bonds

   A 10-year bond with face amount 1,000 pays coupons at 7% a year, convertible semiannually, and will be redeemed at 1,050.

   Calculate its price to yield 6% a year, convertible semiannually.

   A729B1,025C1,050D1,074E1,102

   Show the solution

   1. Per half-year: coupon $1{,}000 \times 3.5\%$ $= 35$, yield $3\%$, and 20 periods.
   2. $P$ $= 35\,a_{\overline{20}|0.03}$${}+ 1{,}050\,v^{20}$ $= 35(14.877475)$${}+ 1{,}050(0.553676)$ $= 520.71 + 581.36$ $= 1{,}102.07$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 20 N 3 I/Y 35 PMT 1050 FV CPT PV → −1,102.07
8. Question 8Bonds

   A 5-year bond with face amount 1,000 pays annual coupons of 40 and is redeemed at par. It sells for 955.

   Calculate its annual effective yield rate.

   A4.00%B4.19%C5.04%D5.13%E8.90%

   Show the solution

   1. The yield $i$ solves $955$ $= 40\,a_{\overline{5}|i}$${}+ 1{,}000\,v^5$.
   2. The price is below par, so $i$ is above the 4% coupon rate. Solving gives $i$ $= 5.04\%$.
   3. Check: at 5.04%, $40\,a_{\overline{5}|}$${}+ 1{,}000\,v^5$ $= 172.99 + 782.01$ $= 955.00$.

   On a BA II Plus

   1. 2nd CLR TVM
   2. 5 N 955 +/− PV 40 PMT 1000 FV CPT I/Y → 5.04
9. Question 9General Cash Flows, Portfolios and Asset Liability Management

   A liability requires payments of 1,000 at time 1, 1,000 at time 2 and 6,000 at time 4, with time in years.

   At an annual effective rate of 5%, calculate the Macaulay duration of the payments.

   A2.33B3.15C3.31D3.38E4.00

   Show the solution

   1. Present values at 5%: $1{,}000v$ $= 952.38$, $1{,}000v^2$ $= 907.03$, $6{,}000v^4$ $= 4{,}936.21$; total $6{,}795.63$.
   2. Time-weighted: $1(952.38)$${}+ 2(907.03)$${}+ 4(4{,}936.21)$ $= 22{,}511.28$.
   3. Macaulay duration $= 22{,}511.28 \div 6{,}795.63$ $= 3.31$ years.

   On a BA II Plus

   1. Total present value with the cash-flow worksheet: CF 0 ENTER ↓ 1000 ENTER ↓ 2 ENTER ↓ 0 ENTER ↓ ↓ 6000 ENTER NPV 5 ENTER ↓ CPT → 6,795.63
   2. Each present value with the TVM keys: 1 N 5 I/Y 0 PMT 1000 FV CPT PV → −952.38; 2 N CPT PV → −907.03; 4 N 6000 FV CPT PV → −4,936.21
   3. 952.38 + 1814.06 + 19744.84 = ÷ 6795.63 = → 3.31
10. Question 10General Cash Flows, Portfolios and Asset Liability Management

    The annual effective spot rates are 3% for 1 year, 3.5% for 2 years and 4% for 3 years.

    Calculate the one-year forward rate from time 2 to time 3.

    A3.50%B4.00%C4.50%D5.01%E9.21%

    Show the solution

    1. Investing for 3 years at the spot rate must match 2 years at the spot rate followed by the forward rate: $(1.04)^3$ $= (1.035)^2(1 + f)$.
    2. $1 + f$ $= \dfrac{1.124864}{1.071225}$ $= 1.050073$, so $f$ $= 5.01\%$.

    On a BA II Plus

    1. 1.04 yˣ 3 = → 1.124864, then STO 1
    2. RCL 1 ÷ 1.035 x² = → 1.050073
    3. − 1 = → 0.050073, which is 5.01%

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