PMB LO 2.4-2.6 and 2.16-2.18: interest, rate conversions, payments, balances, renewals, refinancing and penalties. The PMB exam requires the Qualifier Plus calculator; these notes give the formulas so that every result can be checked by hand or in a spreadsheet.
1. A private lender advances a $25,000 bridge loan at 7.25% per annum simple interest, with interest and principal due after 120 days. Using a 365-day year, how much interest is owed at maturity?
PMB Blueprint LO 2.4.2: simple interest = principal x annual rate x time in years, so $25,000 x 0.0725 x 120/365 = $595.89 (rounded to the cent). A 360-day year gives $604.17, counting both the first and the last day (121 days) gives $600.86, and compounding daily gives $602.99, which the loan terms do not provide for.
2. A borrower paid $1,350 of interest on a $45,000 interest-only private loan that ran for exactly six months. Assuming simple interest, what annual interest rate did the loan carry?
Under PMB Blueprint LO 2.4.2, rate = interest / (principal x time in years) = $1,350 / ($45,000 x 0.5) = 6.00%. Dividing $1,350 by $45,000 without annualizing gives 3.00% (a six-month rate), entering the time as 6 instead of 0.5 gives 0.50%, and treating six months as 182/365 of a year gives 6.02% although the loan ran exactly half a year.
3. A private second mortgage of $60,000 carries 7.80% per annum simple interest, with all principal and interest due in a single payment after exactly 9 months. What amount must the borrower pay at maturity?
PMB Blueprint LO 2.4.2: maturity value = principal x (1 + rate x time) = $60,000 x (1 + 0.078 x 9/12) = $63,510.00. Charging a full year of interest gives $64,680.00, compounding monthly gives $63,602.66 although the loan bears simple interest, and treating 9 months as 270/365 of a year gives $63,461.92.
4. A private lender offers a $75,000 loan for 4 years with all interest due at maturity, either at 7.00% simple interest or at 7.00% compounded semi-annually. How much more interest does the borrower pay under the compound-interest version?
PMB Blueprint LO 2.4.2-2.4.3: simple interest is $75,000 x 0.07 x 4 = $21,000.00, while semi-annual compounding over 8 half-years gives $75,000 x (1.035^8 - 1) = $23,760.68, a difference of $2,760.68. Compounding annually gives $2,309.70 more and compounding monthly $3,154.04 more, neither of which is the stated compounding; $23,760.68 is the total compound interest, not the difference.
5. A payout statement is prepared for a mortgage with a balance of $268,400 right after its last regular payment. The lender charges per diem interest as simple interest at the 4.69% contract rate on a 365-day basis. What interest accrues over the 17 days to the payout date?
PMB Blueprint LO 2.4.2 (simple interest): $268,400 x 0.0469 x 17/365 = $586.29, rounded to the cent. A 360-day basis gives $594.43, counting 18 days instead of 17 gives $620.78, and prorating 17/30 of one month's interest at the monthly rate equivalent to 4.69% compounded semi-annually gives $588.71, which is not the per diem method the lender stated.
6. A buyer's $50,000 down-payment savings are invested at 6.00% compounded monthly for 3 years, with no deposits or withdrawals. What is the balance at the end of the 3 years?
PMB Blueprint LO 2.4.3: future value = $50,000 x (1 + 0.06/12)^36 = $59,834.03. Simple interest gives $59,000.00, annual compounding gives $59,550.80 and semi-annual compounding gives $59,702.61; each understates the growth because interest is credited less often than monthly.
7. A $120,000 private loan accrues interest at 8.00% compounded semi-annually, not in advance, and no payments are made; unpaid interest is added to the debt. How much interest has accrued after 2 years?
Under PMB Blueprint LO 2.4.3, compound interest over 4 half-years is $120,000 x (1.04^4 - 1) = $20,383.03. Simple interest gives $19,200.00, annual compounding gives $19,968.00, and monthly compounding gives $20,746.55, which overstates the charge on a semi-annual loan.
8. A couple wants $40,000 available for a down payment in exactly 5 years. Their savings earn 4.50% compounded annually. What single deposit must they make today?
PMB Blueprint LO 2.4.3 (compound interest, used in reverse): present value = $40,000 / 1.045^5 = $32,098.04. Deducting 4.5% simple interest for 5 years gives $31,000.00, a simple-interest present value gives $32,653.06, and discounting at 2.25% per half-year gives $32,020.41, which assumes semi-annual compounding the account does not offer.
9. A 70-year-old homeowner takes a $200,000 lump-sum reverse mortgage at 7.50% compounded semi-annually, not in advance, with no payments required while she lives in the home. What will the balance be after 10 years?
PMB Blueprint LO 2.4.3: the balance grows by compound interest, $200,000 x (1 + 0.075/2)^20 = $417,630.40. Annual compounding gives $412,206.31, simple interest gives $350,000.00, and monthly compounding gives $422,412.93; only the semi-annual calculation matches the stated rate.
10. A private loan accrues interest at 6.00% compounded semi-annually and no payments are made. To two decimals, how many years does it take for the debt to double?
Under PMB Blueprint LO 2.4.3, solve 1.03^n = 2: n = ln 2 / ln 1.03 = 23.45 half-years, or 11.72 years. The rule of 72 gives an approximate 12.00 years, annual compounding at 6% gives 11.90 years, and 23.45 is the number of half-years, not years.
11. A lender capitalizes all interest on a $150,000 loan with no payments, and the payout after exactly 6 years is $210,000. What nominal annual rate, compounded semi-annually, was charged?
PMB Blueprint LO 2.4.3-2.4.4: solve (1 + j/2)^12 = 210,000/150,000 = 1.4, so j/2 = 1.4^(1/12) - 1 = 2.8436% and j = 5.6872%. Dividing the $60,000 growth by 6 years gives 6.6667% (simple interest), 1.4^(1/6) - 1 = 5.7681% is an annually compounded rate, and 2.8436% is only the half-year periodic rate.
12. A 5-year fixed-rate mortgage is quoted at 5.40% per annum, compounded semi-annually, not in advance. What is its effective annual rate?
PMB Blueprint LO 2.4.4 and 2.16.9: EAR = (1 + 0.054/2)^2 - 1 = 5.4729%. 5.4000% is the nominal rate itself, 5.5357% results from compounding monthly, and 5.3290% comes from converting in the wrong direction (treating 5.40% as the effective rate).
13. A variable-rate mortgage is set at 5.70% per annum, compounded monthly. What effective annual rate does the borrower pay while that rate applies?
PMB Blueprint LO 2.4.4: EAR = (1 + 0.057/12)^12 - 1 = 5.8513%. Quoting 5.7000% ignores compounding, 5.7812% applies the fixed-rate semi-annual convention that this variable rate does not use, and 5.8651% assumes daily compounding.
14. A $300,000 private loan accrues interest at 6.00% compounded semi-annually, not in advance, and no payments are made for one year. How much of that year's interest is interest earned on interest, compared with 6.00% simple interest?
PMB Blueprint LO 2.5.3 (effects of compounding): the $9,000 of interest added after the first half-year itself earns 3%, so the year's interest is $300,000 x (1.03^2 - 1) = $18,270.00 against $18,000.00 of simple interest, and the $270.00 difference is interest on interest. $18,270.00 is the total interest, $503.34 is the gap under monthly compounding, and $0.00 ignores that added interest itself earns interest.
15. A lender's rate sheet shows an effective annual rate of 5.5756% on a fixed-rate mortgage. What nominal rate, compounded semi-annually, not in advance, corresponds to it?
PMB Blueprint LO 2.17.1 (equivalent nominal rate): j = 2 x ((1.055756)^(1/2) - 1) = 5.5000%. Keeping 5.5756% confuses effective and nominal, 5.4380% is the equivalent rate compounded monthly, and 2.7500% is the half-year periodic rate, not an annual nominal rate.
16. To calculate monthly payments, a mortgage at 5.25% compounded semi-annually, not in advance, must be restated as an equivalent nominal rate compounded monthly. What is that rate?
PMB Blueprint LO 2.17.1: j12 = 12 x ((1 + 0.0525/2)^(1/6) - 1) = 5.1935%. Using 5.2500% unchanged overcharges the borrower, 5.3189% is the effective annual rate, and 5.1278% results from treating 5.25% as an effective rate before converting.
17. Four loans all carry a nominal rate of 6.00% per annum but compound at different frequencies. Which one produces the highest effective annual rate?
PMB Blueprint LO 2.4.8: at the same nominal rate, the more often interest compounds, the higher the effective annual rate, because added interest starts earning interest sooner. At 6.00%, daily compounding gives (1 + 0.06/365)^365 - 1 = 6.1831%, against 6.1678% monthly, 6.0900% semi-annually and 6.0000% annually. Neither the Canadian semi-annual convention nor the payment frequency changes this ranking.
18. A client comparing offers asks you to restate a variable rate of 5.45% compounded monthly as an equivalent rate compounded semi-annually, so it can be set beside fixed-rate quotes. What is the equivalent rate?
PMB Blueprint LO 2.17.1: j2 = 2 x ((1 + 0.0545/12)^6 - 1) = 5.5123%. 5.4500% ignores the difference in compounding, 5.5882% is the effective annual rate, and 5.3891% converts in the wrong direction (from semi-annual to monthly).
19. A private lender charges 8.00% per annum calculated yearly, not in advance, but your brokerage's comparison sheet shows every rate compounded semi-annually. What figure belongs on the sheet?
PMB Blueprint LO 2.17.1: j2 = 2 x (1.08^(1/2) - 1) = 7.8461% gives the same yearly cost as 8% compounded annually. Copying 8.0000% overstates the cost, 8.1600% converts in the wrong direction (it is the effective rate of 8% compounded semi-annually), and 7.7208% is the monthly-compounded equivalent.
20. A private lender quotes 7.60% per annum compounded quarterly on a loan repaid monthly. What monthly periodic rate must be used to compute the interest in each payment?
PMB Blueprint LO 2.4.8 and 2.17.1: the quarterly rate is 0.076/4 = 1.9%, and the equivalent monthly rate is 1.019^(1/3) - 1 = 0.6294%. Dividing 7.60% by 12 gives 0.6333% (that would be monthly compounding), treating the rate as semi-annual gives 0.6235%, and the effective annual rate divided by 12 gives 0.6516%.
21. A borrower receives four fixed-rate quotes with identical terms and fees. Which quote has the lowest effective annual cost?
PMB Blueprint LO 2.4.4 and 2.4.8: convert each quote to an effective annual rate. 5.18% annual = 5.1800%; 5.07% monthly = 5.1895%; 5.12% semi-annual = 5.1855%; 5.10% quarterly = 5.1984%. The highest nominal rate is the cheapest here because it compounds only once a year, while the lowest nominal rate compounds monthly.
22. A private lender's draft mortgage quotes interest as 0.55% per month on blended monthly payments. Your brokerage restates every rate compounded semi-annually, not in advance. What is the equivalent rate?
PMB Blueprint LO 2.17.1: six monthly periods make a half-year, so j2 = 2 x (1.0055^6 - 1) = 6.6914%. Multiplying 0.55% by 12 gives 6.6000%, a monthly-compounded nominal rate, 6.8034% is the effective annual rate, and 3.3457% is only the six-month periodic rate.
23. A fixed-rate mortgage is at 4.79% per annum, compounded semi-annually, not in advance, with monthly payments. What is the monthly periodic interest rate?
PMB Blueprint LO 2.17.1: i = (1 + 0.0479/2)^(1/6) - 1 = 0.395241% per month. Dividing 4.79% by 12 gives 0.399167% and ignores the semi-annual compounding, dividing the effective rate by 12 gives 0.403947%, and treating 4.79% as an effective annual rate gives 0.390662%.
24. A mortgage at 5.60% compounded semi-annually, not in advance, is repaid every two weeks (26 payments a year). What is the periodic rate for each bi-weekly period?
PMB Blueprint LO 2.17.1: each half-year holds 13 bi-weekly periods, so i = 1.028^(1/13) - 1 = 0.212650%. Dividing 5.60% by 26 gives 0.215385%, using 24 periods (semi-monthly) gives 0.230391%, and 0.106269% is the weekly rate.
25. A mortgage at 4.50% compounded semi-annually, not in advance, is repaid weekly (52 payments a year). What is the weekly periodic interest rate?
PMB Blueprint LO 2.17.1: 26 weeks make a half-year, so i = 1.0225^(1/26) - 1 = 0.085616%. 4.50% / 52 gives 0.086538% and ignores compounding, 1.045^(1/52) - 1 gives 0.084684% by treating the rate as an effective annual rate, and 0.171305% is the bi-weekly rate.
26. A mortgage at 5.10% compounded semi-annually, not in advance, is repaid semi-monthly (24 payments a year, on the 1st and 15th). What is the periodic rate per payment?
PMB Blueprint LO 2.17.1: a half-year holds 12 semi-monthly periods, so i = 1.0255^(1/12) - 1 = 0.210056%. 5.10% / 24 gives 0.212500%, using 26 periods confuses semi-monthly with bi-weekly (0.193882%), and 0.420553% is the monthly rate.
27. A relative lends a buyer $30,000 toward closing costs under a signed agreement that says the loan bears interest but sets no rate. It is repaid in full after exactly 18 months. Under the Interest Act, how much simple interest is payable?
Interest Act s 3: when interest is payable by agreement and no rate is fixed by the agreement or by law, the rate is five per cent per annum, so $30,000 x 5% x 1.5 years = $2,250.00. $1,500.00 counts only one year, $2,306.72 compounds 5% semi-annually, as a mortgage rate often is, and $0.00 borrows the s 6 penalty for blended-payment mortgages, which does not govern this unsecured loan.
28. A mortgage loan is at the lender's variable rate, stated only as 6.15% per annum compounded monthly, with interest-only monthly payments and no charges other than interest. Under Alberta's Cost of Credit Disclosure Regulation, what is the APR of this agreement?
Cost of Credit Disclosure Regulation s 24(2): the rate is not calculated yearly or half-yearly, so s 23 does not apply; with no non-interest finance charges, one rate for the whole term (s 27(2) assumes the current variable rate stays fixed) and a stated annual rate that is 12 times the monthly rate on monthly payments, the APR is the stated 6.15%. 6.33% is the effective annual rate, 6.23% the half-yearly equivalent that s 23 would use for a half-yearly rate, and 8.15% adds a 2-point qualifying buffer that has no place in the APR. The 1/8-of-1% tolerance in s 27(5) only judges a disclosed figure; it does not change the APR.
29. On a $500,000 balance at a nominal 6.30% per annum, how does the first month's interest differ if the rate is compounded monthly rather than semi-annually, not in advance?
PMB Blueprint LO 2.5.3 (effects of compounding): monthly compounding charges 0.063/12 x $500,000 = $2,625.00, while semi-annual compounding charges $500,000 x (1.0315^(1/6) - 1) = $2,591.20, so monthly compounding costs $33.80 more. Dividing the gap between the two effective rates by 12 gives $35.80, and $429.54 is that gap over a whole year, not one month.
30. An interest-only $200,000 mortgage is at 6.50% compounded semi-annually, not in advance, and the borrower pays the interest monthly as it falls due, each payment rounded to the cent. What is the total interest paid over one year?
PMB Blueprint LO 2.4.6 and 2.17.1: each month the borrower pays $200,000 x (1.0325^(1/6) - 1) = $1,068.95, and 12 such payments total $12,827.40. 6.50% x $200,000 gives $13,000.00, the effective rate x $200,000 gives $13,211.25 (as if interest were left to compound, which it is not), and $13,394.37 lets 6.50% compound monthly for a year although the interest is paid each month.
31. A variable-rate mortgage compounded monthly has a balance of $390,000. Its rate drops from 5.70% to 5.45%. Ignoring principal repaid, how much does one month's interest fall?
PMB Blueprint LO 2.4.7 (impact of loan variables): with monthly compounding, the monthly rate falls by 0.25%/12, so interest falls by $390,000 x 0.0025/12 = $81.25. $975.00 is the yearly figure, $79.41 applies a semi-annual conversion this variable rate does not use, and $812.50 results from reading 0.25% as 2.5%.
32. A lender offers $100,000 for one year at '8% interest in advance': it deducts $8,000 at funding and the borrower repays $100,000 after one year. What annual rate, not in advance, is the borrower paying?
Interest Act s 6 requires a blended-payment mortgage to state its rate 'not in advance', and this loan shows why it matters: interest deducted up front costs more than its face rate. The borrower has the use of only $92,000, so the rate is 8,000 / 92,000 = 8.6957%. Quoting 8.0000% hides that cost, dividing by $108,000 gives 7.4074%, and 8% x 1.08 gives 8.6400%.
33. A private lender's draft blended-payment mortgage charges 6.00% per annum calculated half-yearly in advance, deducting each half-year's interest at the start of that half-year. What equivalent rate, compounded semi-annually, not in advance, should the mortgage state?
Interest Act s 6 requires the rate to be stated yearly or half-yearly, not in advance. A 3% half-yearly discount is worth 0.03/0.97 = 3.0928% per half-year paid in arrears, so the nominal rate is 6.1856%. Keeping 6.0000% understates the cost, 2 x 3% x 1.03 gives 6.1800%, and 6.2812% is the effective annual rate, not a semi-annual nominal rate.
34. A $425,000 mortgage is funded on March 18, 2026, and the interest adjustment date (IAD) is April 1, 2026. The lender charges interim interest as simple interest at 5.19% on a 365-day basis for each day from funding to the IAD. What is the interest adjustment?
PMB Blueprint LO 2.5.2 (interim financing interest): March 18 to April 1 is 14 days, so $425,000 x 0.0519 x 14/365 = $846.04. A 360-day year gives $857.79, and miscounting the days as 13 or 15 gives $785.61 or $906.47.
35. A fixed-rate mortgage is funded on September 12, 2026. The lender's interest adjustment date (IAD) is the first day of the following month; the first monthly payment period starts on the IAD, and each regular payment is made in arrears at the end of its period. When is the first regular payment due?
Under PMB Blueprint LO 2.5.2 and 2.5.10, interest from September 12 to the October 1 IAD is collected separately as an interest adjustment. Monthly payments in arrears then cover interest for the month that follows the IAD, so the first regular payment is due November 1, 2026. October 1 is the IAD itself, when no regular instalment has yet been earned.
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