# Bond price, yield and duration calculator

> Calculate a bond's price from its yield to maturity, or the YTM from its price, with current yield, Macaulay and modified duration and convexity.

ಸಂವಾದಾತ್ಮಕ ಆವೃತ್ತಿ: https://www.calcopenly.com/kn/finance/bond-calculator
ವಿಷಯ: ಹಣಕಾಸಿನ ಕ್ಯಾಲ್ಕುಲೇಟರ್‌ಗಳು

A bond's price is the present value of its coupons and its face value, each discounted at the yield to maturity for the coupon period. With semi-annual coupons, a 5% coupon on 1,000 pays 25 every six months and the annual yield is halved per period, the bond-equivalent convention. Solving the same equation for the rate turns a market price into a yield. Duration is the present-value-weighted average time until the cash flows arrive; modified duration and convexity estimate how far the price moves when the yield changes.

With the defaults, a 10-year 5% semi-annual bond with a face value of 1,000, priced to yield 6%, is worth 925.61: a discount, because its coupon is below the yield. Its Macaulay duration is 7.895 years and its modified duration 7.665, so a 0.01% rise in yield lowers the price by about 0.71.

The price assumes settlement on a coupon date, so there is no accrued interest. Credit risk, call features and taxes are not modeled.

## ಒಳಾಂಶಗಳು

- **Solve for** (ಆಯ್ಕೆಗಳು: Price from yield, Yield from price)
- **Face value**
- **Coupon rate (per year)**
- **Years to maturity**
- **Coupons per year** (ಆಯ್ಕೆಗಳು: Annual, Semi-annual, Quarterly, Monthly)
- **Yield to maturity (per year)**
- **Market price**

## ಫಲಿತಾಂಶಗಳು

- Price — ಮುಖ್ಯ ಫಲಿತಾಂಶ
- Yield to maturity (annual, bond-equivalent)
- Effective annual yield
- Current yield
- Price as % of face
- Macaulay duration (years)
- Modified duration (years)
- Convexity (years²)
- Price change per 0.01% yield change

## ಸೂತ್ರ

$$
P = \sum_{t=1}^{n} \frac{c}{(1+y)^t} + \frac{F}{(1+y)^n},\quad D_{mac} = \frac{1}{P\,f}\sum_{t=1}^{n} \frac{t\,CF_t}{(1+y)^t},\quad D_{mod} = \frac{D_{mac}}{1+y}
$$

## ಪರಿಹರಿಸಿದ ಉದಾಹರಣೆಗಳು

### 10-year 5% semi-annual bond at 6%

- Solve for: Price from yield
- Face value: 1000
- Coupon rate (per year): 5%
- Years to maturity: 10 years
- Coupons per year: Semi-annual
- Yield to maturity (per year): 6%
- **Price: 925.61**
- **Macaulay duration: 7.894997 years**
- **Modified duration: 7.665046 years**
- **Convexity: 71.785398 years²**
- **Current yield: 5.401828%**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Python decimal (prec 50): cash flows discounted term by term; Fabozzi convexity Σt(t+1)PV/((1+y)²·P·f²); current yield 50/925.6126 = 5.4018278%

### Microsoft DURATION example

- Solve for: Price from yield
- Face value: 100
- Coupon rate (per year): 8%
- Years to maturity: 29.5 years
- Coupons per year: Semi-annual
- Yield to maturity (per year): 9%
- **Macaulay duration: 10.919145 years**
- **Modified duration: 10.448943 years**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Microsoft DURATION documentation (settlement 2018-07-01 is a coupon date, maturity 2048-01-01): 10.9191453; Python decimal agrees

### Microsoft MDURATION example

- Solve for: Price from yield
- Face value: 100
- Coupon rate (per year): 8%
- Years to maturity: 8 years
- Coupons per year: Semi-annual
- Yield to maturity (per year): 9%
- **Modified duration: 5.73567 years**
- **Price: 94.38**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Microsoft MDURATION documentation (2008-01-01 to 2016-01-01, a coupon date): 5.73567; Python decimal price 94.382992

### Par bond: coupon equals yield

- Solve for: Price from yield
- Face value: 1000
- Coupon rate (per year): 6%
- Years to maturity: 10 years
- Coupons per year: Semi-annual
- Yield to maturity (per year): 6%
- **Price: 1,000.00**
- **Price as % of face: 100%**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: A bond whose coupon rate equals its yield prices at par (definition)

### Zero-coupon yield from price

- Solve for: Yield from price
- Face value: 1000
- Coupon rate (per year): 0%
- Years to maturity: 10 years
- Coupons per year: Annual
- Market price: 500
- **Yield to maturity (annual, bond-equivalent): 7.177346%**
- **Macaulay duration: 10 years**
- **Convexity: 95.760562 years²**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Closed form 2^(1/10) − 1 = 7.1773462536%; a zero's Macaulay duration equals its maturity; convexity n(n+1)/(1+y)² from Python decimal

### One annual period

- Solve for: Price from yield
- Face value: 1000
- Coupon rate (per year): 5%
- Years to maturity: 1 year
- Coupons per year: Annual
- Yield to maturity (per year): 5%
- **Price: 1,000.00**
- **Macaulay duration: 1 year**
- **Modified duration: 0.952381 years**
- **Convexity: 1.814059 years²**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: 1050/1.05 = 1000; D = 1; D/(1.05) = 0.952381; 2/1.05² = 1.814059 (hand calculation)

## ಪ್ರಶ್ನೆಗಳು

### How do you calculate the price of a bond?

Discount every coupon and the face value at the yield per period and add them. A 10-year 5% bond paying 25 twice a year, at a 6% yield (3% per half-year), is worth 371.94 for its 20 coupons plus 1,000 ÷ 1.03^20 = 553.68 for the face value: 925.61 in total. Excel's =PV(3%, 20, -25, -1000) gives the same price.

### What is the difference between yield to maturity and current yield?

Current yield is the annual coupon divided by the price; yield to maturity also counts the gain or loss to face value at maturity and the timing of every payment. A 5% bond bought at 925.61 has a current yield of 50 ÷ 925.61 = 5.40% but a yield to maturity of 6%, because the buyer also gains 74.39 when the bond repays 1,000.

### Why do bond prices fall when interest rates rise?

The coupons are fixed, so a higher yield discounts the same payments more heavily. The default 10-year 5% bond is worth 1,081.76 at a 4% yield, exactly 1,000 at 5%, 925.61 at 6% and 857.88 at 7%. When the yield equals the coupon rate the bond trades at par; above it, at a discount; below it, at a premium.

### What does bond duration tell you?

Modified duration is the approximate percentage change in price for a one-percentage-point change in yield. The default bond's modified duration is 7.665, so a rise from 6% to 7% should cut the price by about 7.67%; the actual fall, from 925.61 to 857.88, is 7.32%, because convexity cushions large moves. Macaulay duration, 7.895 years here, is the weighted average time to the cash flows.

### How often do US Treasury notes and bonds pay interest?

Every six months. The US Treasury issues notes with terms of 2, 3, 5, 7 and 10 years and bonds with terms of 20 and 30 years, and both pay interest semi-annually, which is why this page defaults to two coupons a year. Choose annual, quarterly or monthly coupons for bonds that pay on another schedule.

### “Bond price, yield and duration calculator” ಎಷ್ಟು ನಿಖರವಾಗಿದೆ?

ನಿಖರತೆ ನಿಮ್ಮ ಇನ್‌ಪುಟ್‌ಗಳು ಮತ್ತು ವಿಧಾನದ ಊಹೆಗಳನ್ನು ಅವಲಂಬಿಸಿದೆ. ದಶಮಾಂಶ ಗಣನೆ 50 ಸಾರ್ಥಕ ಅಂಕೆಗಳನ್ನು ಬಳಸುತ್ತದೆ. ಆದರೆ ಅಂದಾಜುಗಳು, ಸಂಖ್ಯಾತ್ಮಕ ವಿಧಾನಗಳು ಮತ್ತು ಮೂಲ ದತ್ತಾಂಶ ಕಡಿಮೆ ನಿಖರವಾಗಿರಬಹುದು; ಪ್ರದರ್ಶಿತ ಮೌಲ್ಯಗಳನ್ನು ರೌಂಡ್ ಮಾಡುವುದರಿಂದ ಈ ಮಿತಿಗಳು ನಿವಾರಣೆಯಾಗುವುದಿಲ್ಲ. ಸ್ವತಂತ್ರ ಮೂಲಗಳ ಪರಿಹಾರಗಳೊಂದಿಗೆ ಪರಿಶೀಲಿಸಿದ ಉದಾಹರಣೆಗಳು: 7. ಉದಾಹರಣೆಗೆ, “10-year 5% semi-annual bond at 6%” ಅನ್ನು Python decimal (prec 50): cash flows discounted term by term; Fabozzi convexity Σt(t+1)PV/((1+y)²·P·f²); current yield 50/925.6126 = 5.4018278% ಜೊತೆಗೆ ಪರಿಶೀಲಿಸಲಾಗುತ್ತದೆ.

### ಈ ವಿಧಾನದ ಮೂಲ ಯಾವುದು?

Microsoft Excel DURATION function; Microsoft Excel MDURATION function; Fabozzi — Bond Markets, Analysis, and Strategies, ch. 2 and 4 (pricing, duration, convexity).

## ಮೂಲಗಳು

- [Microsoft Excel DURATION function](https://support.microsoft.com/office/duration-function-b254ea57-eadc-4602-a86a-c8e369334038)
- [Microsoft Excel MDURATION function](https://support.microsoft.com/office/mduration-function-b3786a69-4f20-469a-94ad-33e5b90a763c)
- Fabozzi — Bond Markets, Analysis, and Strategies, ch. 2 and 4 (pricing, duration, convexity)

_ಯೋಜನೆಗಾಗಿ ಮಾತ್ರ. ಸಾಲದಾತರು, ತೆರಿಗೆ ಅಧಿಕಾರಿಗಳು ಮತ್ತು ಮಾರುಕಟ್ಟೆಗಳು ತಮ್ಮದೇ ರೌಂಡಿಂಗ್, ಶುಲ್ಕಗಳು ಮತ್ತು ನಿಯಮಗಳನ್ನು ಅನ್ವಯಿಸುತ್ತವೆ; ಬದ್ಧತೆ ಮಾಡುವ ಮೊದಲು ಅವರೊಂದಿಗೆ ಮೊತ್ತಗಳನ್ನು ಖಚಿತಪಡಿಸಿಕೊಳ್ಳಿ._