FM

On January 1, an investment fund was opened with an initial balance of 5000. Just after the balance grew to 5200 on July 1, an additional 2600 was deposited. The annual effective yield rate for this fund was 9.00% over the calendar year. Calculate the time-weighted rate of return for the year

"The annual effective yield rate for this fund was 9.00% over the calendar year." means 5000(1.09) + 2600(1.09)^1/2, because that 2600 accumulated for half of the year Time-weighted: separated year into regions separated by deposits or withdrawls (Fund balance right before next activity/fund balance rights after previous activity)

Relationsip between a nominal semiannual discoutn rate convertible quarterly and an annual effective interest rate

(1 + d/4)^-4 = 1+i

s-double dot-angle-n

(1 + i) s-angle-n OR [(1 + i)^n - 1] / d

Convert a nominal quarterly annual discount rate to a nomical semiannual interest rate

(1 - d(4)/4)^-4t = (1 - i(2)/2)^2t

Convert effective discount rate to eir

(1-d)^-t = (1+i)^t

If the effective semiannual interest rate is .03, what is the accumulation function for 1 dollar in n years? If the effective monthly interest rate is .05, what is the accumulation function for 1 dollar in n years?

(1.03)^2n (1.05)^12n

Accumulated value of unit decreasing annuity

(Ds)angle-n = [(n(1+i)^n - s-angle-n) / i]

(Ia)-angle-inf

(a-double dot-angle-inf) / i

MacD of a level perpetuity

(can use the same formulas, or:) 1/d

Decreasing unit annutiy formulas

(n - a-angle-n) / i

How to find the outstanding loan balance at any time?

1. Present value at t of any remaining payments after t 2. Accumulated value of the loan amount to t minus the accumulated value of any payments made so far

v

1/(1+i) (1+i)^-1

Present value of perpetuity-due

1/d OR 1 + a-angle-inf

MacD of an inflating perpetuity (payments growing by g%)

1/d (still), but with a modified j, such that j = (i - g) / (1 + g)

A common stock pays dividends at the end of each year into perpetuity. Assume that the dividend increases by 2% each year. Using an annual effective interest rate of 5%, calculate the Macaulay duration of the stock in years.

1/d, with a modified j: j = (i - g) / (1 + g) j = (.05 - .02) / (1.02) 1/d --> 1 / [j / 1 + j]

Rewrite: 10(1+k)v^6 + 10(1+k)^2 v^7 + ... (inf) where i=9%

10(1+k)(1.09)^-6 + 10(1+k)^2 1.09^-7 + 10(1.09)^-5 [SUM(1 to inf) [(1+k)/1.09]^n] 10(1.09^-5 * [((1+k)/1.09) / (1 - ((1+k)/1.09))]

Accumulated value of 100 from time 3 to time 6 with a force of interest A(t)

100 * (a(6)/a(3)), where a(t) = e^integral 0 to t of A(t)

What is $1000 accumulated over 2 years with a nomical discount rate of 2% converted semiannually?

1000(1 - .02/2)^-4

John made a deposit of 1000 into a fund at the beginning of each year for 20 years. At the end of 20 years, he began making semiannual withdrawals of 3000 at the beginning of each six months, with a smaller final withdrawal to exhaust the fund. The fund earned an annual effective interest rate of 8.16%. Calculate the amount of the final withdrawal.

1000s-doubledot-angle-20 is the accumulated value of the fund after 20 years. So this value (50382) must fund the annuity due for the 2000 withdrawls: 50382 = 3000a-doubledot-angle-n [using the monthly effective rate] to solve for n=26.8 MAKE SURE YOU MAKE THE PV NEGATIVE ON CALCULATOR BECAUSE THE PAYMENTS ARE POSITIVE if n=26, he makes 26 withdrawls of 3000 and one final withdrawl as the 27th withdrawl. The 27th withdrawls is 26 periods away from the beginning of the annuity due, so: 50382 = 3000-a-dotdot-26 + Xv^26

Sinking fund loan of 20,000 is paid back over 20 years. Lender gets 8% interest and sinking fund earns j percent. Sinking fund deposits are 215. Find j

20,000 = 215 s-angle-20 j

Payments are made to an account at a continuous rate of (8k + tk), where 0 <= t <= 10 Interest is credited at a force of interest 1/(8+t) After time 10, the account is worth 20,000.

20,000 = integral from 0 to 10 of (8k + tk) * a(10)/a(t) dt a(10)/a(t) = e to the integral of t to 10 of the force of itnerest

If loan payments are 150% of interest due, and interest rate is 10%, what percent of that payment is used to pay down pricinpal?

5% Why? 150% of interest due is 1.5(.1)(Principal) Interest due is .1(Principal) Amount of payment applied to principal is thus 1.5(.1)(Principal) - .1(Principal) = .05(Principal)

You are given the following information with respect to a bond: (i) par value: 1000 (ii) term to maturity: 3 years (iii) annual coupon rate: 6% payable annually You are also given that the one, two, and three year annual spot interest rates are 7%, 8%, and 9% respectively. Calculate the value of the bond.

60/1.07 + 60/1.08^2 + 1060/1.09^3

College tuition is 6000 for the current school year, payable in full at the beginning of the school year. College tuition will grow at an annual rate of 5%. A parent sets up a college savings fund earning interest at an annual effective rate of 7%. The parent deposits 750 at the beginning of each school year for 18 years, with the first deposit made at the beginning of the current school year. Immediately following the 18th deposit, the parent pays tuition for the 18th school year from the fund. How to find the value in the fund immediately after the 18th payment?

750s-angle-18 Why is it s and not s-dotdot even thogh she makes the first deposit immediately (i.e., not a year from now)? Because the comparison date for the future value is THE DATE OF THE LAST PAYMENT WHICH IS NOT A DOTDOT

Write the equation of value for: To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each of the first n years and 196 at the end of each of the next 2n years.

8000 = 98 s-angle-3n + 98 s-angle-2n Why? Because 98 is paid up until n, and then afterwards, the 196 payments can be split up into 2 payments of 98. So, its like their is one level annuity of 98 from 0 to 3n, and one level annuity of 98 from n to 3n.

If a coupon bond pays 5% annual coupons for a total par value of 822,703, what is each coupon payment?

822,703(.05)

How to find a forward rate?

Accumulate a dollar according to the spot rates and has to equal accumulating a dollar for the forward rates E.g.: (1 + s2)^2 = (1 + s1) (1 +1f1) (1 + s5)^5 = (1 + s2)^2 (1 + 2f3)^3

One way to calculate the outstanding loan balance at a certain time

Accumulate the loan amount to t years, then subtract the accumulated value of all the payments made so far E.g., OB10 = L(1+i)^10 - Xs-angle-10

Accumulated value at time T of Continuous rates of payment with force of interest

Accumulated value = itegral from 0 to 10 of the rate of payment times a(T)/a(t)

Dollar-weighted rate of return

Approximates annual effective return rate i using simple interest approximations for each withdrawl/deposit E.g.: If fund starts at 75, gets deposits of 10 at end of each month, has withdrawls of 5 at end of Feb, 25 at end of June, 80 halfway through Oct, and 25 at end of Oct, and ends with 60, the formula for i is: 60 = 75(1+i) + 10[(1 + 11/12i) + (1 + 10/12i) + ... (1+0/12i)] -5(1 + 10/12i) - 25(1+6/12i) - 80(1 + 5/24i) - 35(1 + 2/12i)

A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. The annual effective rate of interest is 8%. Calculate X.

At year 5, the present value of the perpetuity is X/i = 100/,08 = 1250 This must equal the present value of the annuity because it is an excahnge The PV of the annuity at time 5 is: Xv + X(1.08)v^2 + X(1.08^2)(v^3) + ... + X(1.08^24)v^25 = X(v + (1.08)v^2 + (1.08^2)(v^3) + ... + (1.08^24)v^25) = X(v + (1.08)(1.08^-2) + (1.08^2)(1.08^-3) + ... + (1.08^24)(1.08^-25)) = X (v + 1.08^-1 + 1.08^-1 + ... + 1.08v^-1) = X (v + v + ... + v) [25 times] = X(25v) = X(25/1.08) Set equal to 1250 to solve for X

What is a sinking fund?

Borrower only pays interest on the loan each period (the payment each period is thus the same) Then, at the end, the borrower pays total principal amount in one payment Each year, the borrower deposits money into a sinking fund, and uses the amount in this fund at the end of the term to pay the balance off

A loan is amortized over five years with monthly payments at an annual nominal interest rate of 9% compounded monthly. The first payment is 1000 and is to be paid one month from the date of the loan. Each succeeding monthly payment will be 2% lower than the prior payment. Calculate the outstanding loan balance immediately after the 40th payment is made.

Calculate the balance at 40 by discounting the payments from 41 to 60 to time 40. B = 1000[(.98^40) (v) + (.98^41) (v^2) + ... + (.98^59) (v^20)] B = 1000 * (.98^40)(v) [1 + (.98v)^1 + (.98v)^2 + ... + .(.98v)^19)] B = 1000 * (98^40)(v) [(1 - (.98v)^20)/(1 - (.98v))]

As of 12/31/2013, an insurance company has a known obligation to pay 1,000,000 on 12/31/2017. To fund this liability, the company immediately purchases 4-year 5% annual coupon bonds totaling 822,703 of par value. The company anticipates reinvestment interest rates to remain constant at 5% through 12/31/2017. The maturity value of the bond equals the par value. Consider two reinvestment interest rate movement scenarios effective 1/1/2014. Scenario A has interest rates drop by 0.5%. Scenario B has interest rates increase by 0.5%.

Company will get 822,703(.05)=41135 at the end of each of the 4 years, reinvest each payment, and also get 822703 at the end of year 4. So, company's total revenue is: Scenario A: 41135s-angle-4 i=4.5 + 822703 Scenario B: 41135s-angle-4 i=5.5 + 822703

An insurance company has an obligation to pay the medical costs for a claimant. Annual claim costs today are 5000, and medical inflation is expected to be 7% per year. The claimant will receive 20 payments. Claim payments are made at yearly intervals, with the first claim payment to be made one year from today. Calculate the present value of the obligation using an annual effective interest rate of 5%.

First payment is 5,000(1.07), so its PV is 5,000(1.07)v Second payment is 5,000(1.07)^2, so its PV is 5,000(1.07)^2v^2 ... 20th payment is 5,000(1.07)^20, so its PV is 5,000(1.07)^20v^20 PV = 5,000(1.07v + 1.07^2v^2 + ... + 1.07^20v^30) PV = 5,000 ((1.07v - 1.07^21v^21)/(1 - 1.07v))

The present value of a perpetuity paying 1 every two years with first payment due immediately is 7.21 at an annual effective rate of i. Another perpetuity paying R every three years with the first payment due at the beginning of year two has the same present value at an annual effective rate of i + 0.01. Calculate R.

First perp is a perp due, with effective 2 year interest rate being (1+i)^2 - 1, so: 7.21 = 1/[(1+i)^2 - 1] + 1 because the pv of a perp due is the X/i + X (and X here is the payment of 1) use that to solve for i In the next perp, find the effective 3 year rate as (1 + i + .01)^3 -1. Call this A for short. This is a perp due, so its PV is R/A + R, but then you have to discount this one period, so 7.21 = [R/A + R](1 + i + .01)^-1 solve for R

At an annual effective interest rate of i, i > 0%, the present value of a perpetuity paying 10 at the end of each 3-year period, with the first payment at the end of year 3, is 32. At the same annual effective rate of i, the present value of a perpetuity paying 1 at the end of each 4-month period, with first payment at the end of 4 months, is X. Calculate X.

For the first perp, the present value is 10(v^3 + v^6 + ...) = 10 SUM(1 to inf) of (v^3)^n = 10 [v^3 / (1-v^3)] ... v^3 = 32/42 For second perp, present value is 1(v^1/3 + v^2/3 + ...) = 1 SUM(1 to inf) of (v^1/3)^n = v^1/3 / (1 - v^1/3) = (v^3)^1/9 / (1 - (v^3)^1/9)

A perpetuity costs 77.1 and makes end-of-year payments. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, ...., n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%.

From 1 to n, it is a unit increasing annuity immedaite, which must then be discounted one period to get to time 0. From n+1 on, it is a perpetuity with payments of n, which must be discounted n+1 periods to get to time 0. 77.1 = v(Ia)angle-n + (v^(n+1) * n/i) (Ia)angle-n = (a double dot angle n - n*v^n) / i 77.1 = v[(a double dot angle n - n*v^n) / i] + (v^(n+1) * n/i) v(a double dot angle n) is the same as a-angle-n

1000 is deposited into Fund X, which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund. At the end of the tenth year, the fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y, which earns an annual effective rate of 9%. Calculate the accumulated value of Fund Y at the end of year 10.

THe interest deposits are 6 times a decreasing unit annuity of 10 years The 100 deposits are 100 for 10 years So, accumulated value is: 6[(Ds)-angle-10] + 100s-angle-10, with i = .09 6[(n(1+i)^n - s-angle-n)/i] + 100s-angle-10

Payments made each period for a sinking fund loan

Interest payment + sinking fund deposit = Principal * interest rate for loan + sinking fund deposit

Lucas opens a bank account with 1000 and lets it accumulate at an annual nominal interest rate of 6% convertible semiannually. Danielle also opens a bank account with 1000 at the same time as Lucas, but it grows at an annual nominal interest rate of 3% convertible monthly. For each account, interest is credited only at the end of each interest conversion period. Calculate the number of months required for the amount in Lucas's account to be at least double the amount in Danielle's account.

Lucas is credited every 6 months, and is effective semi interest rate is .06/2, so his accumulation is (1.03)^t, where t is the number of semi-annuals Danielle is credited every month, and her effective monthly interest rate is .03/12=.0025, so her accumulation is (1.0025)^6t, where t is the still the number of semi's (there are 6 months in each semi) 1.03^t = 2(1.0025)^6t tln1.03 = ln2 + 6tln(1.0025) t = 47.55 half years Which is 285.3 months Since Lucase is credited every 6 months, the next actual 6 month time when he gets credit is at 288 months (that's the first month after 285.3 that is divisible by 6)

time-weighted rate of return

Multiply all accumulation ratios for whole period and then subract 1, where each period is separated by a deposit or withdrawal, with each ratio being: Beginning fund value on 2nd date / Ending fund value on 1st date

A 30-year annuity is arranged to pay off a loan taken out today at a 5% annual effective interest rate. The first payment of the annuity is due in ten years in the amount of 1,000. The subsequent payments increase by 500 each year.

Note that the deferred annuity is an anuity-due

P&Q formula

P=starting price, Q=increase each time (or decrease) PV = Pa-angle-n +Q[(a-angle-n - nv^n)/i] OR (calculator mode) PV = (P + Q/i)a-angle-n - (Qn/i)v^n payment = P + Q/i FV = -Qn/i

A perpetuity-immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and a charity receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and the charity's share is K. Calculate K.

PV of the perp is X/i Brian's share of that is 40%, and Brian gets the first n payments of X, so: .4(X/i) = X a-angle-n .4(1/i) = (1 - v^n)/i .4 = 1 - v^n v^n = .6 Charity's share is K, so: K = (X/i)v^2n (the PV of charity's chare, because chairty gets the payments after 2n) k = (X/i) (.6^2) k = (X/i) (.36) So, charity's share is 36% of the present value of the perp

Relationship between present value of perpetuity-due and perpetuity-immediate

PV(perp-dotdot) = X/i + X where X is the payment amount

If the coupon rate exceeds the yield rate, then the bond is bought at a ____

Premium

How to find amount of premium or deposit

Premium: (Fr - Ci)a-angle-n Discount: (Ci - Fr)a-angle-n

Denominator for duration is the same as the

Price

Amount of premium

Price - Redemption value (P-C) >0 P-C = (Fr - Ci)a-angle-n Where Fr is the coupon amount and Ci is the interest earned interest payment on the redemtion amount

If an n-year bond only gives you the effective semiannual yield rate and semiannual couon rate, what is the equation for price of the bond?

Price = Xr a-angle-2n i + C v^2n Basically, all n's change to 2n, because the coupon rate is semiannual and the yield rate is semiannual, so there are not n payments, there are 2n payments

Bill buys a 10-year 1000 par value bond with semi-annual coupons paid at an annual rate of 6%. The price assumes an annual nominal yield of 6%, compounded semi-annually. As Bill receives each coupon payment, he immediately puts the money into an account earning interest at an annual effective rate of i. At the end of 10 years, immediately after Bill receives the final coupon payment and the redemption value of the bond, Bill has earned an annual effective yield of 7% on his investment in the bond.

Price he paid for bond is 1000 (because yield rate equals coupon rate) After everything is over, he makes 1000(1.07)^10 He gets 30 for 20 months, which are invested in the other fund at a semi-annual effective rate j. These accumulate to 30s-angle-20 j He then gets 1000 redemption at end of 20th month. So, the equation of value is: 30s-angle-20 j + 1000 = 1000(1.07)^10 calculator solves j = 4.75% That is the semiannual effective for the reonvestment fund. To find out the annual effetive i: 1 + j = (1 + i)^1/2

Formula for principal portion of a level payment at time t

Principal portion of level payment at time t = LevelPaymentAmount * v^(n-t+1)

Interest rate without cost of inflation protection

R = r + i-sub-e + i-sub-u Where i-sub-e is the expected inflation rate and i-sub-c is the unexpected inflation rate

Cost of inflation protection

R = r - c + i-sub-a Where r is the interest rate, c is the cost of inflation protection, and i-sub-a is the actual interest rate, which might not be known, in which case the R is just r-c.

Porter makes three-year loans that include inflation protection. The annual interest rate compounded continuously that must be paid is 3.2% plus the rate of inflation. The U.S. government borrows 100,000 for three years from Porter. The actual annual inflation rate during the first year was 2.4% compounded continuously. The actual annual inflation rates for the second and third years respectively was 2.8% and 4.2% compounded continuously. The U.S. government is considered a risk free borrower, which means there is no chance of default. Calculate the amount that the U.S. government will owe Porter at the end of three years.

R = r - c + i-sub-a r-c = 3.2%, because it says that R = 3.2% plus the rate of inflation R1 = 3.2% + 2.4% = 5.6% R2 = 3.2% + 2.8% = 6.0% R3 = 3.2% + 4.2% = 7.6% Amout owed over three years will be the amount accumulated over the three years compounded continously, and each year has a different R: 1000 e^R1 e^R2 e^R3

Amount of discount

Redemption value - Price (C - P) > 0 C-P = (Fr-Ci)a-angle-n

Calculate each of these person's total interest paid: Seth, Janice, and Lori each borrow 5000 for five years at an annual nominal interest rate of 12%, compounded semi-annually. Seth has interest accumulated over the five years and pays all the interest and principal in a lump sum at the end of five years. Janice pays interest at the end of every six-month period as it accrues and the principal at the end of five years. Lori repays her loan with 10 level payments at the end of every six-month period.

Seth: Interest is accumulated value of 5000 over 10 periods with i = .06 minus 5000 Janice: Interest is 5000(.6)(10) [10 equal interest payments] Lori: 5000 = Xa-angle-10 i=.06 Solve for X Multiply by 10 Subtract 5000

An investor purchases a 10-year callable bond with face amount of 1000 for price P. The bond has an annual nominal coupon rate of 10% paid semi-annually. The bond may be called at par by the issuer on every other coupon payment date, beginning with the second coupon payment date. The investor earns at least an annual nominal yield of 12% compounded semi-annually regardless of when the bond is redeemed

THe yield rate exceeds the coupon rate, so this bond is bought at a discount, so the minimum price will be at the latest possible redemtpion, which is the maturity date because there are no end times to any call periods

Process for Exact Matching

Table: each asset gets a row and list the assets in descending order of maturity (furthest maturity is first row), each time gets a column, from left to right the times are in ascendign order. At the bottom of each time column, write the liability amount for that time. Fill in table starting with top right entry: it will be the exact value of the liability for that time (say it is X). To find out how much of the asset to buy, do X/Amount asset pays that time. Fill in what that amount of the first asset you bought will pay at other times. Fill in the next row by what remaining you have to get the libaility amount at the next earlier time, then figure out how much of that asset you have to buy with the same method

You are given the following information about an investment account: (i) The value on January 1 is 10. (ii) The value on July 1, prior to a deposit being made, is 12. (iii) On July 1, a deposit of X is made. (iv) The value on December 31 is X. Over the year, the time-weighted return is 0%, and the dollar-weighted (money-weighted) return is Y.

Time-weighted: (12/10)(X/12+X) - 1 = 0 Dollar-weighted: 10(1+Y) + X(1 + /5Y) = X

You are given the following information about a loan of L that is to be repaid with a series of 16 annual payments: (i) The first payment of 2000 is due one year from now. (ii) The next seven payments are each 3% larger than the preceding payment. (iii) From the 9th to the 16th payment, each payment will be 3% less than the preceding payment. (iv) The loan has an annual effective interest rate of 7%.

The PV of first 8 payments are: 2000v + 2000(1.03)v^2 + 2000(1.03^2)v^3) + ... + 2000(1.03^7)v^8 = 2000[ (v - 1.03^8 v^9) / (1 - 1.03v)] The PV of the first 7 payments are: 2000(1.03^7)(.97)v^9 + 2000(1.03^7)(.97^2)v^10 + ... + 2000(1.03^7)(.97^8)v^16 = 2000(1.03^7) [ (.97v^9 - .97^9v^17) / (1 - .97v)]

What is a spot rate?

The annual effective yield rate for a zero-coupon bond

INterest Rate Swap

To find swap rate R, set PV(variable) = PV(fixed) X(1f0)/(1+s1) + X(1f1)/(1+s2)^2 + ... + X(1fn-1)/(1+sn)^n = XR/(1+s1) + XR/(1+s2)^2 + ... + XR/(1+sn)^n If X is a level notional amount, you can use this shortcut: 1 = R(1/(1+s1) + 1/(1+s2)^2 + ... + 1/(1+sn)^n) + 1/(1+sn)^n

Turner buys a new car and finances it with a loan of 22,000. He will make n monthly payments of 450.30 starting in one month. He will make one larger payment in n+1 months to pay off the loan. Payments are calculated using an annual nominal interest rate of 8.4%, convertible monthly. Immediately after the 18th payment he refinances the loan to pay off the remaining balance with 24 monthly payments starting one month later. This refinanced loan uses an annual nominal interest rate of 4.8%, convertible monthly.

The balance after the 18th payment, and thus the principal for the next loan, is 22000(1.007)^18 - 450.40s-angle-18 Use that as the present value of the next 24 month loan at the new interest rate to calculate the payment

What is the interest portion of a single coupon payment?

The coupon payment times the annual eir

If two funds with same deposit accumulate to the same value after the same time, what can be said about the two funds' eir?

The eir is the same!

A man turns 40 today and wishes to provide supplemental retirement income of 3000 at the beginning of each month starting on his 65th birthday. Starting today, he makes monthly contributions of X to a fund for 25 years. The fund earns an annual nominal interest rate of 8% compounded monthly. On his 65th birthday, each 1000 of the fund will provide 9.65 of income at the beginning of each month starting immediately and continuing as long as he survives. Calculate X.

The fund must accrue to enough that gets him 3000 of income at his 65th birthday For each 1000 in the fund, he gets 9.65 income, so to get 3000 income at 65, the fund must accumulate to 3000(1000/9.65) = 310880 The fund equation of value is X s-double dot-angle-300 i=.08/12 Why? the first payment is at time 0, and last payment is a month before the 300th month, so it is a double dot accumulation, and the interest rate is the effective monthly rate So, set that equation of value equal to 310880 to solve for X

An investor pays 100,000 today for a 4-year investment that returns cash flows of 60,000 at the end of each of years 3 and 4. The cash flows can be reinvested at 4.0% per annum effective. Using an annual effective interest rate of 5.0%, calculate the net present value of this investment today.

The reinvestment means that the cash flow at year 3 is not credited as a ccash inflow, rather, it is re-invested elsewhere for 4%, and will become part of the cash inflow at year 4. Year 3 cash flow of 60,000 is reinvested at 4%, so in Year 4 it will be come 62,400. So, the year 4 cash inflow is 122400. PV = -100,000 + 122400v^4

Maximum price to guarantee a minimum yield rate is

The smallest possible price obtainable with that yield rate

a 10-year loan of 2000 can be paid back in yearly level payments of 299 at 8.07%. The sum of these payments is equal to if you paid 200 each year plus interest, i, on unpaid balance each year. Calculate i

The sum of all payments from the level scenario is 2990. Since the sum of payments are equal, the total interest paid in the second scenario must be 990. So, 990 = 2000i + 1800i + 1600i + ... + 200i 990 = i (2000 + 1800 + ... + 200) because the principal is paid down 200 each year

Minimum yield rate for premium bond

Will occur at the beginning of a call period or the maturity date

Minimum yield rate for discount bond

Will occur at the ending of a call period or the maturity date

Amount of SIMPLE interest in ANY period

X(0) * i, where i is the eir for the period

Present value of perpetuity

X/i (where X is the payment amount each period)

Value or a perpetuity at any time

X/i (where X is the payment)

s-angle-n formula

[(1+i)^n - 1] / i

MacD of a level annuity-immediate

[(Ia)-angle-n] / a-angle-n

Unit increasing annuity formula

[(a double dot angle n) - nv^n] / i

Sum of geometric series

[(first term) - (first omitted term)] / [1 - (first term)]

a-angle-n formula

[1 - v^n] / i

Sum of infinite geometric series starting at 1 (SUM1 to inf of [a]^n

[common ratio] / (1 - [common ratio]) a/ (1-a)

SUM 1 to inf of a^n

a / (1-a)

v * a double dot angle n, is the same as

a angle n

Relation between a -anhle-n and a double dot

a double dot angle-n = (1 + i) a-angle-n

Accumulation function for a force of interest

a(t) = e^integral from 0 to t of the force of interest

Accumulation function between two values of t

a(t2) / a(t1)

Relationship between a-angle-n and a-doubledot-n

a-dotdot-n = (1+i)a-angle-n a-dot-dot-n = 1 + a-angle-n-1

effective discount rate formula

d = i / 1+i

Comparison date for s-angle-n

date of last payment

A bond purchased at a price less than the redemption value or an early call value is a ____ bond

discount

If redemption value exceeds price, it is a ____ bond

discount

If the yield rate exceeds the coupon rate, then the bond is bought at a ____

discount

internal rate of return

effective interest rate at which net present value of cash inflows and outflows is 0

If annual effective simple interest rate is i, what is the effective interest rate for 6 month?

i/2

WHat happenes when you take ln of a(b)^x

lna + xlnb

MacD

negative derivative of price times 1 plus the interest rate, all over the price

Comparison date for s-double dot-angle-n

one period after date of last payment

If price exceeds redemption value, it is a ___ bond

premium

Relationship between s-angle-n and s-dotdot-n

s-dotdot-n = (1+i)s-angle-n s-dotdot-n = s-angle-n+1 - 1